ASVAB Study Guide

Electronics Information

General

Note you only will have about a half of a min to answer each question so be mindful not to take to much time on a single questions as to maximize your score.

Definitions

General

Electricity – form of energy that can travel invisibly through conductors. Electricity is carried by moving charged particles, especially by electrons. Electrons are tiny negative charges that orbit the nucleus of an atom.

Conductor – a material that allows an easy flow of electrons. Silver, Copper, Aluminum are all good conductors.

Insulator – a material that resists the flow of electrons. Rubber, Plastic, and Ceramic are good insulators

Circuit – loop of conductors that takes electricity from its source to the load and back to the source.

Load – anything in a circuit, such as a heater, a light, or a motor, that uses power.

Direct Current (DC) – steady flowing type of electricity, produced by batteries and used in flashlights, boom boxes, and computers.

Alternating Current (AC) – type of current that changes direction many times per second. AC is used in home wiring, mainly because it can be transported long distances over the transmission of wires.

Electric Current - the amount of electrons flowing through a conducting material.

Electric Power – amount of power consumed by an electrical device.

Voltage – Annotated as a capital V, force that affects the rate at which electricity flows through a conductor (aka: electrical pressure). The higher the voltage the more likely electricity will “leak” across an insulator or an air gap.

Voltage Drop – how much electrical pressure is used in a part of the circuit.

Frequency – number of complete alternations from one direction to the other and then back a gain, that alternating current makes per second. Each complete alteration is called a cycle.

Resistance – is the opposition of a material to the flow of electricity through it. ALL circuits must have a resistance. If they do not they are then considered short circuits which can lead to wires overheating.

Amperes – Annotated as a capital A, represents current strength

Ohms - Annotated as Ώ, represents the measurement of the resistance in any circuit.

Watts - Annotated as a capital W, represents the measurement of power consumed by an electrical device.

Electricity and Magnetism

General Knowledge – There exists a close relationship between electricity and magnetism. This close relationship explains electromagnets, transformers, motors, and generators.

Electromagnet – a current passing through a conductor creates a magnetic field around it. The conductor (Wire) is wrapped around an iron core.

Transformer – essentially is two electromagnets placed next to each other. Transformers change the voltage and amperage of a current.

Step up Transformer – has more turns of wire on the output side which creates a stronger output voltage than input voltage.

Step Down Transformer – the exact opposite of a step up transformer.

Motor – changes electric energy into kinetic energy. In a motor each magnet has two poles (North and South – Note North repels south but attracts North), a rotor that spins inside the stationary magnet is an electromagnet. The rotor turns due to the repelling of the magnet which creates electricity.

Generator – changes kinetic energy into electric energy

Series, Parallel, and Series Parallel Circuits

Series Circuit – All moving electrons pass through every part of the circuit, to include all loads and switches. Given this the total voltage of the loads must equal the total voltage of the circuit.

Parallel Circuit – loads are placed between two supply wires, so that they all get the same voltage. Additionally current can flow through any of the loads even if one is switched off.

Circuit Rule – Current is the same at all points in a series circuit. Voltage is the same at all points in a parallel circuit.

Practical Electricity

Circuit Breaker Box – (1) Breaks up the load in a building into a number of circuits. (2) Prevents excess current from flowing into the circuits.

Typical Wiring – 3 Wires (black, white, bare) Black- hot conductor, White – grounded conductor aka neutral, Bare – Alternative route utilized in case of an emergency.

Miscellaneous

Semiconductor – Can act as an insulator or a conductor. Silicon is the main semiconductor and is the basis for computer memory and logic boards. Chemicals called dopants are applied to the silicon to determine whether it will act as an insulator or as a conductor.

Transistors – Device that can switch a current, regulate its flow, or amplify a current, all based on the presence of a smaller unit.

Diodes – Devices that allow a current to flow in one direction only.

Formulas

Ohm’s Law – Describes the relationship between voltage, amperage, and ohms: A = V/ Ώ

The Law of Electrical Power – used to calculate the amount of power consumed by an electrical or electronic device:

W = V x A

Finding Resistance in a Circuit – Series – add the resistance of each loadParallel – add the inverses of the resistance

Auto Information

General

Note you only will have about a min to answer each question so be mindful not to take to much time on a single questions as to maximize your score.

Definitions

Automobile Engines

General Information – Cars use internal combustion engines, meaning the fuel is burned inside the engine. Steam engines are the opposite they are external combustion engines meaning the fuel is burned outside the engine. All car engines to include diesel engines utilize the Otto Cycle.

Otto Cycle Engine – A mix of fuel and air is brought inside a closed space, called a cylinder. The mix is compressed and then explodes. The explosion moves a piston, which rotates the crankshaft. The crankshaft is connected through the drive train to the driving wheels, which move the car. Waste heat from the explosions is removed by the cooling system. Memorize this order of operations in an Otto cycle engine:

INTAKE – The piston moves down, creating a partial vacuum in the cylinder. The fuel air mixture enters the cylinder through the open intake valve. The exhaust valve is closed.

COMPRESSION - Both valves are closed. The piston moves up, compressing the fuel-air mixture to about ten times atmospheric pressure.

POWER - The spark plugs fire, starting an explosion inside the cylinder. The resulting high pressure pushes the piston down.

EXHAUST - The piston moves up again, with the exhaust valve open and the intake valve closed. The piston pushes burned exhaust gases into the exhaust manifold and out of the engine.

Cylinder and Piston – The cylinder is the heart of the internal combustion engine, since it is where the combustion takes place. The cylinder is a finely machined chamber that holds a piston as it slides up and down. Thin rings called piston rings seal the gap between the cylinder and the piston, containing the explosions and increasing efficiency.

Cylinder Head – complex metal casting that closes the top of the cylinders. The head is bolted to the engine block.

Head Gasket – separates the head and the block. Like all gaskets, the head gasket creates a seal between two rigid objects that would otherwise leak.

Spark Plug – These electrical devices create a spark when they get a high voltage jolt of electricity from the ignition system. Spark plugs are screwed into the cylinder head and should be replaced periodically.

Connecting Rods and Crankshaft – change linear motion into rotary motion. Connecting rods are attached to the crankshaft by the main bearings. The crankshaft itself rotates on journal bearings attached to the engine block.

Valves and Valve Train – Valves play a critical part in the Otto engine because they admit fresh fuel and air and discharge burned fuel and air. Please reference the suggested ASVAB Prep Book for further information and diagrams.

Firing Order – to make the engine run more smoothly, nearby cylinders do not ignite in sequence, rather the engine’s firing order is spread amongst the engine.

Lubrication System – Engines have a complicated set of tubes and internal passages that bring oil to the contact points. A gear driven oil pump pushes oil through these passages. Oil also splashes onto the cylinder walls, lubricating the piston rings and making compression more effective.

Oil – **Important Notes** Oil gets dirty and wears out, so it must be replaced periodically. Oil filters clean

the oil, but they must also be replaced. Oil and filter manufacturers estimate their product lifetime in miles and/or in

months. Replace the oil and filter when you first reach one of these milestones. To change oil, warm up the engine, place a pan under the oil plug in the crank

case, remove the plug, drain the oil. Replace the plug and refill the oil through the filler cap on top.

Recycle used oil; do not dump it down the drain or in a field. When a car starts “burning oil”, blue smoke indicates wear of the rings and/or

cylinders. If the oil pressure gauge comes on while you are driving, pull over as soon as

possible and check for trouble. If you are lucky the oil level may simply be low, and adding oil should alleviate the issue. Otherwise to prevent severe engine damage, get the car towed to a shop. Sudden loss of oil pressure can also result from oil pump failure, or other serious engine problems.

Viscosity – measures the thickness of the oil. Measured by S.A.E. numbers which range from very light SAE 5 to SAE 90, typically for regular driving oil is rated at SAE 30 to 40.

Cooling System - Only about 30% of the energy in gasoline is converted into energy used to make the car drive, the remaining energy all becomes wasted heat. If the engine cannot get rid of this excess heat the car will overheat. The cooling systems runs off of coolant,

not just water due to the fact that water would freeze in the colder temperatures. In order to combat this, a chemical known as antifreeze is added to the water. Additionally water would eventually cause rusting to occur which is why a rust inhibitor is added to the coolant. Unfortunately over time this rust inhibitor breaks down which is why coolant needs to be replaced.

Water Jackets – tiny holes in the engine and cylinder block that allow the coolant to reach all parts of the engine.

Radiator – Coolant flows through the water jackets and to the radiator, the radiator has many small tubes covered by fins, additionally the radiator is located near the front of the engine to maximize its exposure to fresh air.

Radiator Fan – pulls air through the radiator, removing heat from the tubes.

** Special Note** - If you always add coolant (50-50 antifreeze and water), your coolant should stay liquid down to about -30 Degrees Fahrenheit.

Water Pump – Often located on the front of the crankshaft, circulates coolant through the engine. If the water pump fails for even a few min the engine can overheat causing immense damage to the engine.

Automobile Engine Troubleshooting

1. A “ping” or “knocking” sound on acceleration usually means that you need a higher octane gasoline. The noise indicates that the fuel-air mixture is igniting too soon inside the cylinder.

2. A squealing noise that increases with engine speed indicates a loose or worn fan belt.

3. An engine that runs extremely rough may have a failed spark plug or some other problem within the ignition system.

4. General sluggishness, roughness, or poor fuel mileage all indicate the need for a tune-up.

5. Loud clanking sounds may indicate major engine problems that will only get worse if you ignore them.

Drive Train

Drive Train – gets power from the engine to the wheels. The drive train includes the transmission, driveshaft, differential, and axels on the driving wheels, which may be in front, in back, or both. The drive train must:

1. Provide different gears so that the engine can always work at an efficient RPM, no matter what the driving speed.

2. Allow the car to move in reverse.

3. Allow the engine to run when the car is not moving.4. Drive two or four wheels.5. Allow the car to turn without tire slippage.

Manual Transmission – in a manual transmission you change gears with a clutch and a gear shift.

Clutch – disconnects the engine from the transmission, so that you can shift gears. The clutch is engaged when the clutch pedal is up and disengaged when the pedal is down.

Automatic Transmission – changes gears automatically to suit driving conditions.

Differential – When a car turns a corner, the outside wheels must drive further than the inside wheels. If both driving wheels were locked to the axel, the tires would be forced to slip against the pavement. The differential solves this problem. The differential has three jobs:

1. Allow different (differential) movement of the two axels, so that the car can turn a corner.

2. Change drive directions. In rear-wheel-drive cars, the differential connects to the drive shaft that runs from front to back and powers the axels, which run from side to side.

3. Increase Power. The differential usually reduces the drive speed, so that three to four input revolutions create one turn of the axels.

Electrical System

Ignition System – The key components of the ignition system are the breaker points, the coil, the distributor cap, and the distributor rotor.

Breaker Points – When the breaker points close they complete a circuit, and a pulse of 12-volt current goes through the primary winding of the ignition coil, creating a brief magnetic field.

Ignition Coil – direct current (DC) transformer. The induced current, which creates the spark at the spark plugs, is at 10,000 volts or higher. This high voltage output goes via heavy, high voltage cable to the center of the distributor cap.

Distributor Cap and Rotor – direct the high voltage current to the spark plugs.

Spark Plugs – receives the high voltage from the distributor cap and then creates an electric spark that sets off the explosion in the cylinder. Important notes regarding spark plugs:

1. If a spark plug is coated with a greasy black substance, the cylinder is leaking oil, probably because of worn piston rings. Oil burns incompletely in the cylinders, leaving this black deposit.

2. Spark plugs must have the correct gap, measured in thousandths of an inch. Feeler gauges are used to set the correct gap.

3. When spark plug electrodes get thin, the plug should then be replaced.

Electronic Ignition – electronic ignitions have been introduced over the last 20 years to eliminate the many weak points of the distribution style ignition. Instead of one centrally located coil, there is a coil at each spark plug, and instead of a mechanical distributor, an electronic unit directs a low voltage current to those coils. This system also allows precise spark timing, which increases power and MPG, all the while reducing pollution.

Battery – cars use a 12-volt battery to store electric energy for starting. Inside a lead-acid battery, a chemical reaction between sulfuric acid (the electrolyte) and lead plates (the electrodes) creates extra electrons at the negative pole. When you connect the starter motor to the negative and positive electrodes a current flow is initiated, discharging the battery and turning the starter motor. **Important Notes**:

1. Lead-acid batteries eventually wear out, however this is not the biggest problem, the biggest problem is that corrosion on the battery terminals or battery cables can break the circuit, causing a fully charged battery to appear dead.

2. Corrosion-preventing chemicals can avoid problems blamed on “dead batteries”, these chemical often show up as red spray on the terminal.

Starter Motor – Large direct current (DC) motor. When the starter spins, its gear is inserted against the teeth on the flywheel, cranking the engine until it starts. When you release the ignition key, the starter motor disengages from the flywheel and the motor stops. If you have ever tried to start a car that was already running, you’ve heard a loud grinding noise. This racket comes from the starter-motor gear clashing against the spinning wheel.

Alternator - Batteries don’t create electricity, they only store it, and they must be recharged while the engine runs. An alternator is basically an electric motor working in reverse. Instead of converting electricity into rotary motion, it makes electricity from rotary motion. The rotation comes from the engine, via a fan belt. Alternators make alternating current (AC), which must be rectified to direct current (DC) for use by a car.

Engine Control Unit (ECU) – auto engines are currently run by the ECU which is a small computer system within the car often referred to as the brains of the car. ECU’s get information from sensors that direct:

1. The mass of the air going into the engine.2. Engine speed3. The position of the throttle4. The amount of oxygen in the exhaust5. Coolant Temp.6. Pressure in the intake manifold.7. Alternator voltage

Electrical System Troubleshooting

1. Loose and corroded connections are a major source of intermittent problems. Corrosion is an insulator; remove it with fine sandpaper to ensure a good contact.

2. Many times, the best diagnosis for electrical problems comes from hooking a car up to a diagnostic computer.

Fuel System

Fuel System – contains the fuel tank, the fuel filter, and the carburetor or fuel injector.

Fuel Tank - generally located in the rear of the car, stores the fuel. A charcoal filter in the tank absorbs gasoline fumes, reducing pollution.

Carburetor – it is important to note that this is an old component which has largely been replaced. This component combines gasoline with air in a venture, where a rapid stream of air flows past a small fuel port. The partial vacuum in the fast moving air draws fuel into the air stream, where it vaporizes and mixes with the air. The exact ratio of air to fuel is critical to performance, a mixture with too much fuel, aka “rich mixture”, will waste gas and increase pollution, a mixture with too little fuel, aka “lean mixture”, will burn to hot and be short on performance.

Fuel Injector – electronically controlled valves that squirt fuel into the cylinder. The valve is closed until the injector receives an electric current and an electromagnet opens the valve. Fuel sprays into the cylinder until the valve closes. Although problems are rare some causes could be explained by:

1. Water or dirt in the fuel can clog lines or filters.2. Water can freeze in the fuel line.3. Vapor lock, a condition caused by overheating fuel lines, is no longer a problem

with modern vehicles.

Exhaust System

Exhaust system – After combustion, burned gas enters the exhaust system. The exhaust manifold connects the cylinders to the muffler, tailpipe, and parts of the pollution control equipment.

Pollution Control

Pollution Types – 1. Carbon Monoxide – a product of partial combustion is colorless, odorless, but

poisonous gas.2. Partially Burned Hydrocarbons – In large quantities, small pieces of hydrocarbon

make black soot that you can see.3. Volatile Organic Compounds - a product of partial combustion, some cause

cancer, others irritate lung tissue, and many are the source of smog.

Positive Crank Case Ventilation - The bottom of an internal-combustion engine is full of smoky, polluted gas made from the burning of oil and gas. To prevent crankcase gases from polluting the atmosphere, the positive crankcase ventilation system pipes this gas to the intake manifold, where the gas is burned in the cylinders.

Catalytic Converter – Catalysts-rare metals like platinum, rhodium, and palladium-cause other chemicals to react without being consumed themselves. Pollutants in the exhaust gases are converted to simpler, less toxic compounds in the catalytic converter.

Suspension

Steering Gear - Connects the steering wheel to the front wheels.

Rack and Pinion – a small pinion gear at the end of the steering shaft turns against a flattened gear called a rack which is attached to the axel and thusly turns the wheels.

Power Assisted Steering – Generally larger cars have power steering; power steering can provide varying levels of assistance through the use of a hydraulic system that runs off of power steering fluid.

Suspension – Autos use a wide variety of suspension types, however the two key categories are leaf spring and coil spring. Springs are made of a kind of steel that is tough and resilient.

Shock Absorbers – prevents the axels from bouncing back when the wheels encounter a severe decline or incline, which makes for a safer, more controllable, and comfortable ride. NOTE: When shocks wear out, a car tends to wander on the road because the springs aren’t limited by the shocks.

Bearings – Two types, ball bearings and roller bearings. Bearings are mechanical gadgets that allow a part to rotate with almost no friction. Cars have bearings wherever

rotating parts are found: steering wheel, engine, transmission, drive shaft, differential, and axels.

Wheels – usually made of steel, are bolted to the axel (for a driving wheel) or the breaking apparatus (for a non driving wheel).

Tires – most tires are called radial-ply, meaning that their plies (the invisible reinforcing belts that hold the tire together) run radially-in a line starting at the center of the wheel. Tires are made of artificial rubber, inflated to a pressure of 30 or 40 lbs. per sq. inch. Important Notes:

1. Proper inflation is crucial for top mileage per gallon, tire life, and safe handling.2. Given the principles of air and temperature (hot air expands, cold air contracts) it

is vital to always measure your tire pressure when the tire is cold, due to the recommended tire pressure being applicable only to a cold tire. If a tire seems to be over pressurized yet it is hot it is not recommended to release air rather to measure the tire pressure again when the tire is cold and then assess whether or not the tire pressure is still over or not.

Brake System General Overview – Brake systems have two key components: the hydraulics and the breaking mechanism at the wheel that does the actual stopping. The hydraulics transfer motion from the driver’s foot to the braking mechanism. Hydraulic fluid is used because it cannot be compressed therefore however pump one reacts pump two must also act in the exact manner and so forth.

Drum Brakes – operate by expanding two brake shoes inside a large cylinder called a drum. The drum rotates with the wheel, but the shoes do not rotate. A wheel cylinder operates the brake by moving the shoes towards the drum. Friction between the shoes and the drum creates the stopping power. Notes:

1. When the shoes wear, a self adjusting mechanism moves them closer to the drum. Wheel cylinders have rubber pistons that must be replaced when they wear out.

2. Brake shoes and drums must also be replaced occasionally. If you let the shoes wear too far, they will damage the drums, forcing a more expensive repair.

Disk Brakes – has a rotor, or a disk, that revolves with the wheel and a pair of stationary pads that create friction when they are pressed against the rotor. Disk brakes are an immense improvement over drum brakes because they offer shorter stopping distances and better control. Notes:

1. Over time, pads wear out. Many disk brakes make a warning noise when the pads are wearing out. You usually will hear the noise when your foot is not on the brake pedal.

2. Eventually if not replaced, the pads wear concentric grooves in the disk. Before putting on new pads, the disks must be machined flat so that the new pads press against a flat surface. Eventually the disk can become to thin which due to it being a safety hazard must be immediately replaced.

Anti-lock Brakes (ABS) - uses sensors to detect wheel rotation. If a wheel starts to skid during braking, the ABS reduces pressure to that brake, and then starts pumping that brake.

Shop Information

General

Note as this is part of the auto shop section of the exam you only will have about a min to answer each question so be mindful not to take to much time on a single questions as to maximize your score.

Definitions and Descriptions

Measuring Tools

Inside Caliper Outside Caliper Vernier Caliper

Micrometer Carpenter’s Square Sliding Bevel

Carpenters Level Carpenters Tape Measure

Definitions – Measuring Tools

Inside Caliper - used to measure the internal size of an object

Outside Caliper - used to measure the external size of an object

Vernier Caliper - Vernier calipers can measure internal dimensions (using the uppermost jaws in the picture above), external dimensions using the pictured lower jaws, and depending on the manufacturer, depth measurements by the use of a probe that is attached to the movable head and slides along the centre of the body. This probe is slender and can get into deep grooves that may prove difficult for other measuring tools.

Micrometer – Very similar in purpose to the Vernier Caliper however more precise in measurement.

Carpenters Square – In carpentry, a square or set square is a guide for establishing right angles (ninety-degree angles), usually made of metal and in the shape of a right triangle.

Sliding Bevel – Used to measure and/or mark any angle on another object. Carpenters Level – The easiest way to tell if something is level (horizontal) or vertical (plumb) is with a level, sometimes called a spirit level. Carpenters Tape Measure - flexible form of a ruler. It consists of a ribbon of cloth, plastic, fiber glass, or metal with linear-measure markings, often in both imperial and metric units.

Cutting and Shaping

Wood Working Tools

Hand Saw Keyhole Saw Circular Saw

Jack Plane Wood Chisel

Definitions – Wood Working Tools

Important Notes – Sharp hand saws are the most basic way to cut wood. Saws cut a kerf that is wider than the blade itself; the kerf allows the saw to move freely through the cut. Crosscut saws are designed to cut at 90 degree to the grain, while rip saws cut parallel to the grain. Rip saws have larger teeth. Back saws have ridged steel backing that improves accuracy; they are used in miter boxes that guide them for 45 degree or 90 degree cuts.

Hand Saw – Reference Notes for information regarding different types of hand saws.

Keyhole Saw – made to cut complicated profiles. An electric version is called the Jigsaw. A coping saw, which is very similar to the above referenced picture, has a thin blade held in a P-Shaped Handle. This saw is generally used to cut molding.

Circular Saw – Usually has a 7-1/4 inch-diameter blade, is much faster for cutting wood, especially for rip sawing, and for sawing plywood or other panels. These saws are dangerous so make sure you reference the manual for user guidance.

Jack Plane – A hand plane removes thin strips of wood and is used to shape, smooth, or reduce the size of boards. It’s especially useful for removing saw marks from the edge of a board. The “Jack” plane is a general purpose type of hand plane.

Wood Chisel – Sold in widths from ¼ inch to 1-1/2 inches, cut wood when they are struck with a hammer or mallet.

Metal Working Tools

Hacksaw Tin Snips Pipe Cutter

Thread Cutting Die Thread-Cutting Tap

Definitions – Metal Working Tools

Hacksaw – has a replacement blade with small teeth and is used for cutting iron, steel, and other, softer metals. Choose a blade with finer teeth for thinner metal, and one with larger teeth for thicker metal. The hacksaw should cut on the forward stroke.

Tin Snips – cut steel, copper, or aluminum sheet metal, using a shearing action. Some snips have replaced blades; others can be sharpened. Special snips are designed to cut curves.

Pipe Cutter – used for copper, not steel, pipe – has a sharp cutting wheel. Gradually tighten the handle as you rotate the tool around the pipe.

Thread cutting Die & Taps – cut or restore threads in metal. A die cuts threads on a rod; a tap cuts threads in a hole frilled in a plate. Either tool can be used to restore mangled threads. Both taps and dies cut only one diameter and pitch (number of threads per inch). To select a die, you must know the outside diameter (O.D.) of the pipe.

Drills and Drilling Tools

Drill Chuck & Chuck Key 3/8 Inch Drill Drill Bit

Center Punch Auger Bit Brace and Bit

Counter Sink Hole Saw

Definitions – Drills and Drilling Tools

General Information – Electric Drills have become the centerpiece of all workshops, for making holes, driving screws, and other purposes. Drills are sized by the largest diameter of bit that will fit in the chuck (the rotating clamp that holds the bit). You may sometimes see ¼ inch drills, but 3/8 inch drills are the all around tool for the home workshop. Larger drills handle bits of ½ or ¾ inch diameter. However, the shank (the part that gets grabbed in the chuck) can be smaller than the tip of the drill bit, so it’s possible to drill 1 inch holes in wood with a 3/8 inch drill. Iron and steel are much harder to drill than wood, and thus call for a larger more powerful drill. Often it’s best to drill a small “pilot” hole in metal. This is because metal-cutting drill bits have a blind spot near the center where they do not drill very well. While drilling metal, it often helps to oil the bit for cooling; excess heat can destroy the heat treatment that makes a bit hard enough to cut metal.

Drill Chuck with Chuck Key – the chuck may be tightened with a chuck key. Newer, self-tightening chucks do not use a key. Instead, the parts tighten when they are turned against each other.

Center Punch – makes a dimple in the metal to locate the bit as it starts to drill. Use a hammer to hit the center punch.

Auger Bit – made only for drilling wood, which they cut much faster than twist drills. Auger bits were originally driven by a brace and bit, but they can also be used in electric drills.

Countersink – a conical depression in a surface that allows a flat head screw to sit flush (flat) to the surface. A special drill bit, also called a counter sink, makes the countersink.

Hole Saw – makes large diameter holes in wood and some metals. The type of hole saw shown screws into a mandrel, allowing one mandrel to handle several size saws. Hole saws are more economical than big drills for drilling wood, but they do not work with hard metal.

Pounding Tools

Claw Hammer Ball-Peen Hammer Hand Sledge Rubber Mallet

Definitions – Pounding Tools

Claw Hammer – pounds nails with the face and pulls them with the claw. A straight claw is better for longer nails, and is also handier for doing demolition. Some hammer faces have a checkered pattern, called a waffle head, to increase the grip on the nails. The standard size of a claw hammer is 16oz. Hammers 24oz. in size are used for larger nails.

Ball-Peen Hammer – Metal workers use this style of hammer. One face is flat, like a claw hammer, the other face has a ball peen, used for shaping metal and riveting. Ball peen hammers may weigh up to 3 lbs.

Sledge Hammers – used for heavy purposes. True sledgehammers have a 32-inch handle and require two hands

Rubber Mallet – may be used to adjust parts without damage, or to drive wooden handled chisels. Many chisels however have a steel handle thus making it acceptable to utilize a normal hammer.

Turning and Grabbing Tools

Adjustable Wrench Combination Wrench Deep Socket Ratchet Wrench

Torque Wrench Allen Wrench Arc-Joint Pliers Locking Pliers

Needle Nose Pliers Bar Clamp Pipe Wrench Screwdriver

Definitions – Turning and Grabbing Tools

Adjustable Wrench – Also known as a “Crescent Wrench” can be adjusted to hold various sizes of hexagonal or square bolts.

Combination Wrench – combines two basic types of wrench: the box end and the open end wrench. Each end of the combination wrench fits the same size bolt.

Deep Socket – A socket wrench holds a bolt or nut from above. The deep socket, shown, allows you to grab a nut even if some threads are sticking out above it. Socket wrenches connect to the socket with a square drive; possible drive dimensions include ¼, 3/8, ½, or ¾ inch.

Ratchet Wrench – is one way to drive a socket. It will grab when it swings in one direction and slip in the other, for convenient tightening or loosening.

Torque Wrench – Fits a socket and drives a bolt to a specific tightness. You can buy them marked in American or metric units.

Allen Wrench – fits screws with hexagonal recess in the head. Allen wrenches are sold in inches and millimeters.

Arc Joint Pliers – Also known as “Channel Locks” are used to grab various sizes of material. To adjust the jaws, open them wide and engage a different set of arcs. Arc-Joint pliers are not good for grabbing bolts or nuts, as they will scar the metal.

Locking Pliers – Also known as “Vise Grips” have a lever system that gives a very strong, locking grab. They arte one of the handiest tools in the box, but they can damage bolts and nuts.

Needle-Nose (Long Nose) Pliers – gets at small parts and is especially handy for electrical work.

Bar Clamp – among other types of clamp, can hold parts in position while you work or hold joints while the glue sets.

Pipe Wrench – has steel teeth that hold steel pipe. They are sold in lengths ranging from 12 to 48 inches.

Flat Bladed (Standard) Screwdriver – Come in many sizes. A long shank will protect your hands when you are pressing hard.

Phillips Screwdriver – drives Phillips screws (cross shaped head), also sold in in various sizes.

Fasteners

Threaded Fasteners

Round-Head Wood Screw Sheet-Metal Screw Machine Screw Hex Bolt

Carriage Bolt

Definitions – Threaded Fasteners

Wood and Sheet Metal Screws - Some threaded fasteners (screws/bolts) are designed to cut threads in the material. Wood/Sheet Metal screws are both designed to accomplish this task. These screws are measured in length and diameter. Length is measured by inches; diameter by a numbering system (#6, #8, #10, ect.) Larger numbers indicate a larger diameter.

Machine Screws (Hex & Carriage Bolt) – unlike wood/sheet metal screws, machine screws must be screwed into a nut with the same diameter and number of threads per inch (also called pitch). Hex bolts range in diameter from ¼ inch on up, and in length from ½ inch on up. Carriage Bolt’s have a round head and a square shank. The shank fits a square slot, so the bolt does not turn while being tightened.

Definitions – Nails

General Information – Nails are sized by length and by pennies (“d”). A 4d nail is 1 ½ inches long, while a 16d nail is 3-1/2 inches long. Nails are sometimes coated with zinc (galvanized) to resist rust.

Brad – A small, thin nail with a small head, used for picture frames and other light fastening

Tack – A small nail with a large head, used to tack carpet and upholstery.

Finishing Nail – A sturdy, small-diameter nail with a small head, used to attach trim and molding.

Common Nail – A big, large-headed nail used for rough construction.

Sinker – A small-diameter version of the common nail that causes less splitting.

Spike – Heavy large nail for fastening timber

Ring-Shank – a nail with rings that improves grip

Spiral – A nail with a spiral on the shank, used to increase grip, for example; flooring.

Welding, Soldering, and Brazing

General Information - Welding, soldering, and brazing are fastening methods that use molten metal. Welding melts the metal that is being fastened. Soldering and brazing metal a separate metal, called solder or brazing rod, to make the attachment. Soldering is used for plumbing, for electrical or electronic equipment. To solder, clean the parts, apply flux, heat, and then touch the solder to the heated parts. Brazing is similar to soldering with the exception that the joining material is bronze.

Mechanical Comprehension

General Information

The ASVAB Mechanical Comprehension test is all about the basic materials and mechanical devices that you see around you everyday. The questions deal with things like levers, pipes, water wheels, gears, pulleys, and the like. You will have 20 minutes to complete 25 questions, so like every part of the test be mindful of your time.

Definitions and Descriptions

Basic Concepts: Work, Energy, and Force

Work – refers to a specific force applied over a specific distance. For example, your arm does work when it uses force to lift a heavy object. Thusly a lever does also work when it uses force to lift an object.

Energy – is defined as the ability to do work. Energy comes in several different forms:1. Kinetic Energy: Energy in a moving object.2. Potential Energy: Energy that can be released under certain conditions. For

example, potential energy is stored in objects when they are lifted off the ground. It is released when the object falls.

3. Chemical Energy: Energy stored in chemicals, such as in a flashlight battery. Chemical energy is potential until it is released in a chemical reaction.

4. Electric Energy: Energy in moving electrons in an electric current.5. Nuclear Energy: Energy released by reactions in the nucleus of an atom.

Forces – are powers that push or pull objects. A force has a magnitude (Strength) that you can measure, and it has a direction. Some forces are obvious-when a bat hits a baseball, you can even hear the force being applied. Other forces, like gravity or air pressure, are much less obvious, but they are still real.

Gravity – gravity is an attractive force between objects. All objects create a gravitational attraction to each other. On Earth, gravity causes all objects to fall toward the center of the Earth. Falling objects accelerate (fall faster) as they fall.

Acceleration – is defined as the change in velocity, or speed in a particular direction. If you discount air resistance, all objects fall at the same rate. In real life however, air resistance often disguises the fact that objects fall at the same rate.

Friction – a force that results from the interaction between two surfaces that are touching each other. Friction acts as a resistance to the movement of an object. For example, friction makes it harder to push a heavy crate up a ramp.

Compression – Force that pushes two or more materials together (i.e. air pressure, water pressure).

Tension – Force that pulls to materials apart.

Principles of Mechanical Devices

General Information – Machines are devices that multiply force or motion. Some machines are simple devices that involve only a single force. A lever is an example. Other machines involve combinations of devices working together, a bicycle is an example.

Mechanical Advantage

Defined – the amount your force is multiplied by a machine is called the mechanical advantage, or MA. There are 2 ways in which to calculate MA.

1. Divide the output force (called the load or sometimes the resistance) by the input force (called the effort): Load/Effort = MA

2. Divide the length of the effort (called the effort distance) by how far the load moves (called the load distance): Effort Distance/Load Distance = MA

Simple & Compound Machines

General – group of very common, basic devices that have all been in use for a very long time. They are called simple because each one is used to multiply just one single force. The simple machines include the lever, the pulley, the incline plane, the gear, the wedge, the wheel and axel, and the screw.

Levers – device that helps the user apply force to lift a heavy object. To understand levers you will need to know the following terms:

1. Fulcrum: The stationary element that holds the lever but still allows it rotate.2. Load: The object to be lifted or squeezed.3. Load Arm (load distance): The part of the lever from load to fulcrum.4. Effort: The force applied to lift or squeeze5. Effort Arm (effort distance): The part of the lever from the force to the fulcrum.

Class 1 Lever -

First-class levers have the fulcrum placed between the load and the effort, as in the seesaw, crowbar, and balance scale. If the two arms of the lever are of equal length, as with the balance scale, the effort must be equal to the load. If the effort arm is longer than the load arm, as in the crowbar, the effort travels farther than the load and is less than the load.

MA = Load/Effort

Class 2 Lever –

Second-class levers have the load between the effort and the fulcrum. A wheelbarrow is a second-class lever. The wheel’s axle is the fulcrum, the handles take the effort, and the load is placed between them. The effort always travels a greater distance and is less than the load.

MA = effort distance/load distance

Class 3 Lever –

Third-class levers have the effort placed between the load and the fulcrum. The effort always travels a shorter distance and must be greater than the load. A hammer acts as a third-class lever when it is used to drive in a nail: the fulcrum is the wrist, the effort is applied through the hand, and the load is the resistance of the wood. Another example of a third-class lever is the human forearm: the fulcrum is the elbow, the effort is applied by the biceps muscle, and the load is in the hand.

MA = Load/Effort

Pulleys – Another kind of simple machine that helps you lift a heavy object is the pulley, also called block and tackle. In pulleys, the mechanical advantage is in either of the following two ways:

1. MA = effort distance/load distance2. MA = number of supporting strands. Supporting strands of rope or cable get

shorter when you hoist the load, but you can’t always just count strands as some do not shorten as you hoist.

Gears – are simple machines used to multiply rotating forces. Finding the MA of a gear is simplicity itself. Identify the driving gear (the one that supplies the force) and count the teeth. Once you count the teeth on the driving gear then use this formula:

Number of teeth on Driven gear/ Number of teeth on the Driving gear = MA

Sheaves – sheaves and belts are a simple machine closely related to gears. To calculate the MA of a sheave system, divide the diameter of the driven sheave by the diameter of the drive sheave. Whenever the driven sheave is larger than the drive sheave, you get a MA.

MA = Driven diameter/Drive diameter

Inclined Plane – is a fancy term for a “ramp”. An incline plane is another simple machine that is used to lift heavy objects. The formula for finding the mechanical advantage of an inclined plane is a follows:

MA = horizontal length/vertical rise

NOTE** to find the mechanical advantage, measure horizontally and vertically (Don’t measure diagonally across the ramp)

Wedge – Type of inclined plane. It is one of the rarer simple machines. As always, MA = Effort distance/Load distance.

Wheel & Axel – Wheels are a common and essential part of daily life, but most of these wheels are not simple machines. Instead, they are a way to reduce friction by the use of a bearing. A wheel and axel is a simple machine only when the wheel and the axel are fixed and rotate together. A typical wheel and axel simple machine is a screwdriver. The screwdriver’s handle is the wheel, and the screwdrivers blade is the axel. For wheel and axel machines, MA is calculated as follows:

MA = effort distance (Radius of the wheel) / load distance (Radius of the axel)

NOTE** a wheel and axel can also give a mechanical disadvantage. In a car or a bicycle, where the axel drives the wheel instead of the wheel driving the axel, a small rotation at the axel creates a large motion at the circumference of the rim. In these cases, you need a larger force, but you get more motion in return.

Compound Machine – is one in which two or more simple machines work together. For example, a screwdriver (wheel and axel) driving a screw is a compound machine. To find the MA of a compound machine, multiply the MA of the simple machines together.

Structural Support – Some mechanical Comprehension questions ask about the load carried by support structures such as beams or bridges-or sometimes people. You will be given a diagram showing support structures and asked which one is the strongest or weakest, or which support in the diagram is bearing the lesser or the greater amount of the load. To answer this question, keep in mind: When a load of any kind is supported by

two support beams, posts, or people, the load is perfectly balanced if it is exactly centered. In that case each beam, post, or person is bearing the exact same amount of the load. However, if the load is not centered, then the beam, post, or person nearer to the load is bearing the greater part of the weight.

Fluid Dynamics

Fluids – substances that take the shape of their container. Gases and liquids are both fluids.

Air Pressure – measured in pounds per square inch. Atmospheric pressure at sea level is 14.7 lb/in^2, which is actually quite a bit of pressure.

Pneumatics and Gas Laws – Systems that use compressed air to do work are called pneumatic systems. Air is easily compressed, and the calculations are more complicated then they are with liquids, which usually can’t be compressed. The larger the driven cylinder, the more air pressure it is exposed to, and the greater force it can exert. The gas laws apply to air as it is compressed and expanded.

1. When a gas is compressed, it gains thermal energy-it warms up. The gas also gains potential energy, which is why compressed air can be used to drive nail guns and pneumatic hammers.

2. When a given amount of gas expands, its pressure drops and the gas cools.3. When a gas cools without a change in outside pressure, it looses volume.

Water Pressure – On the ASVAB, water pressure questions often involve flow through pipes. Keep these principles in mind:

1. Total flow through a pipe system must be the same everywhere because water cannot be compressed.

2. When liquid speeds up, pressure drops.3. When liquid slows down, pressure rises

Note: Water in a container also exerts pressure on the bottom of the container, the deeper the water, the greater the pressure. To find the amount of water pressure in a tank, calculate the total weight of the water and divide by the area of the tank.

Example:A tank with a base that measures 2 feet x 4 feet holds 1,600 pounds of water. What is the water pressure at the bottom of the tank?

2 ft x 4 ft = 8 ft^21,600/8 = 200 lb/ft^2

**Remember to that 1 ft^2 = 144 in^2. To convert pressure between pounds per square inch and pounds per square foot, divide or multiply by 144. **

Filling and Emptying Tanks – “GUARANTEED TYPE QUESTION” Example Question: Water is being piped into the tank at a rate of 2 gallons per second. At the same time, it is being piped out of the tank at a rate of 60 gallons per minute. How many gallons will be added in 5 min?

1. Convert the inflow rate so that you are only working with gallons per min. gal/sec x 60 = gal/min 2 gal/sec x 60 = 120 gal/min

2. Subtract: 120 gal/min inflow – 60 gal/min outflow = 60 gal/min net gain Thusly the net gain in 5 min is 5 x 60 = 300 gallons

Arithmetic Reasoning

General Information

The ASVAB Arithmetic Reasoning test measures your ability to solve the kinds of arithmetic problems that you encounter every day at home or on the job. The topics covered will be: place value, arithmetic operations, positive and negative numbers, multiplying and dividing by zero, factors, multiples, fractions, mixed numbers, decimals, percent, finding the percent of a number, finding the percent increase or decrease, exponents and square roots, scientific notation, mean, median, mode, graphs, units, of measure, as well as word problems. The test will consist of 30 questions and you will be given 36 min to answer these questions.

Definitions and Descriptions

Place Value - In our decimal number system, the value of a digit depends on its place, or position, in the number. Each place has a value of 10 times the place to its right.A number in standard form is separated into groups of three digits using commas. Each of these groups is called a period. For the purpose of the ASVAB you will need to understand that in the number 23,456,789.247 (going left to right):

2 – (first 2) ten millions place 3 – millions place 4 – (first 4) hundred thousands place 5 - ten thousands place 6 – thousands place 7 – (first 7) one hundreds place 8 – tens place 9 – ones place 2 – tenths place 4 – hundredths place 7 – thousandths place

Addition & Subtraction - Here's how to add two positive integers:

4 + 7 = ?

If you start at positive four on the number line and move seven units to the right, you end up at positive eleven. Also, these integers have the same sign, so you can just keep the sign and add their absolute values, to get the same answer, positive eleven.

Here's how to add two negative integers:

-4 + (-8) = ?

If you start at negative four on the number line and move eight units to the left, you end up at negative twelve. Also, these integers have the same sign, so you can just keep the negative sign and add their absolute values, to get the same answer, negative twelve.

Here's how to add a positive integer to a negative integer:

-3 + 6 = ?

If you start at negative three on the real number line and move six units to the right, you end up at positive three. Also, these integers have different signs, so keep the sign from the integer having the greatest absolute value and subtract the smallest absolute value from the largest.

Subtract three from six and keep the positive sign, again giving positive three.

Here's how to add a negative integer to a positive integer:

5 + (-8) = ?

If you start at positive five on the real number line and move eight units to the left, you end up at negative three. Also, these integers have different signs, so keep the sign from the integer having the greatest absolute value and subtract the smallest absolute value from the largest, or subtract five from eight and keep the negative sign, again giving negative three.

To subtract a number, add its opposite:

5 - 8 = ?

Because they give the same result, you can see that subtracting eight from five is equivalent to adding negative eight to positive five. The answer is - 3.

To subtract a number, add its opposite:

-3 - (-6) = ?

Because they give the same result, you can see that subtracting negative six from negative three is equivalent to adding positive six to negative three. The answer is 3.

Multiplication & Division - You multiply or divide integers just as you do whole numbers, except you must keep track of the signs. To multiply or divide signed integers, always multiply or divide the absolute values and use these rules to determine the sign of the answer.

When you multiply two integers with the same signs, the result is always positive. Just multiply the absolute values and make the answer positive.

Positive x positive = positive Negative x negative = positive

When you multiply two integers with different signs, the result is always negative. Just multiply the absolute values and make the answer negative.

Positive x negative = negativeNegative x positive = negative

When you divide two integers with the same sign, the result is always positive. Just divide the absolute values and make the answer positive.

Positive ÷ positive = positiveNegative ÷ negative = positive

When you divide two integers with different signs, the result is always negative. Just divide the absolute values and make the answer negative.

Positive ÷ negative = negativeNegative ÷ positive = negative

Examples

1.

2.

3.

4.

Order of Operations - When an expression contains more than one operation, you can get different answers depending on the order in which you solve the expression. Mathematicians have agreed on a certain order for evaluating expressions, so we all arrive at the same answers. We often use grouping symbols, like parentheses, to help us organize complicated expressions into simpler ones. Here's the order we use:

1. First, do all operations that lie inside parentheses. 2. Next, do any work with exponents or roots. 3. Working from left to right, do all multiplication and division. 4. Finally, working from left to right, do all addition and subtraction.

In Example 1, without any parentheses, the problem is solved by working from left to right and performing all the addition and subtraction. When parentheses are used, you first perform the operations inside the parentheses, and you'll get a different answer!

Example 1 - Parenthesis

Without Parenthesis With Parenthesis8 - 7 + 3 =

1 + 3 =4

8 - (7 + 3) = 8 - 10 =

-2

Example 2

Order of Operations Explanation22 x 20/4 - 7 x 3 + 55 = Calculate the exponent4 x 20/4 - 7 x 3 + 55 =

4 x 5 - 21 + 55 =Working from left to right, do all multiplications and divisions. When

OR80/4 - 21 + 55 =

(4 x 5 and 80/4 both = 20)

there are several of these operations in the same term, the order within the term doesn't matter

20 - 21 + 55 = Add and subtract from left to right54 The correct answer!

Rounding - To round a number you must first find the rounding digit, or the digit occupying the place value you're rounding to. Then look at the digit to the right of the rounding digit. If it is less than 5, then leave the rounding digit unchanged. If it is more than five, add one to the rounding digit. If it is five, the rule is to always round up (add one to the rounding digit). This rule was created to "break the tie" when you are rounding a number that is exactly between two other numbers. These kinds of rules are called "conventions", and are important so we all get the same answer when doing the same problems.

If you're dealing with a decimal number, drop all of the digits following the rounding digit.

If you're dealing with a whole number, all the digits to the right of the rounding digit become zero.

This sounds a lot more complicated than it really is!

It's easiest to learn rounding by studying examples.

To round the number 16,745.2583 to the nearest thousandthFirst find the rounding digit. This is the "8". You are trying to get rid of the all the digits to the right of the 8, but you want the result to be as accurate as possible.

Now look one digit to the right, at the digit in the ten-thousandths place which is "3". See that 3 is less than 5, so leave the number "8" as is, and drop the digits to the right of 8. This gives 16,745.258.

To round 14,769.3352 to the nearest hundredFind the rounding digit, "7". Look at the digit one place to right, "6". Six is more than 5, so this number needs to be rounded up. Add one the rounding digit and change all the rest of the digits to the right of it to zero. You can remove the decimal part of the number too. The result is 14,800.

To round 365 to the nearest tenFind the rounding digit, "6". Look at the digit to the right of the six, "5". Since 365 is exactly halfway between 360 and 370, the two nearest multiples of ten, we need the rule to decide which way to round. The rule says you round up, so the answer is 370.

Factors & Multiples - A factor is simply a number that is multiplied to get a product. Factoring a number means taking the number apart to find its factors--it's like multiplying in reverse. Here are lists of all the factors of 16, 20, and 45.

16 --> 1, 2, 4, 8, 1620 --> 1, 2, 4, 5, 10, 20 45 --> 1, 3, 5, 9, 15, 45

12 --> 12, 24, 36, 48, 60, . . . 5 --> 5, 10, 15, 20, 25, . . . 7 --> 7, 14, 21, 28, 35, . . .

Factors are either composite numbers or prime numbers. A prime number has only two factors, one and itself, so it cannot be divided evenly by any other numbers. Here's a list of prime numbers up to 100. You can see that none of these numbers can be factored any further.

PRIME NUMBERS to 100

2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61,67,71,73,79,83,89,97

A composite number is any number that has more than two factors. Here's a list of composite numbers up to 20. You can see that they can all be factored further. For example, 4 equals 2 times 2, 6 equals 3 times 2, 8 equals 4 times 2, and so forth.

By the way, zero and one are considered neither prime nor composite numbers-they're in a class by themselves!

COMPOSITE NUMBERS up to 20

4,6,8,9,10,12,14,15,16,18,20

You can write any composite number as a product of prime factors. This is called prime factorization. To find the prime factors of a number, you divide the number by the smallest possible prime number and work up the list of prime numbers until the result is itself a prime number. Let's use this method to find the prime factors of 168. Since 168 is even, we start by dividing it by the smallest prime number, 2. 168 divided by 2 is 84.

84 divided by 2 is 42. 42 divided by 2 is 21. Since 21 is not divisible by 2, we try dividing by 3, the next biggest prime number. We find that 21 divided by 3 equals 7, and 7 is a prime number. We know 168 is now fully factored. We simply list the divisors to write the factors of 168.

168 ÷ 2 = 84 84 ÷ 2 = 42 42 ÷ 2 = 2121 ÷ 3 = 7 Prime numberprime factors = 2 × 2 × 2 × 3 × 7

To check the answer, multiply these factors and make sure they equal 168.

Here are the prime factors of the composite numbers between 1 and 20.

4 = 2 × 26 = 3 × 28 = 2 × 2 × 29 = 3 × 310 = 5 × 212 = 3 × 2 × 2

14 = 7 × 215 = 5 × 316 = 2 × 2 × 2 × 218 = 3 × 3 × 220 = 5 × 2 × 2

Fractions - One way to think of a fraction is as a division that hasn't been done yet. Why do we even use fractions? Why don't we just divide the two numbers and use the decimal instead? In this day of cheap calculators, that's a very good question. Fractions were invented long before decimal numbers, as a way of showing portions less than 1, and they're still hanging around. They're used in cooking, in building, in sewing, in the stock market - they're everywhere, and we need to understand them.

Just to review, the number above the bar is called the numerator, and the number below the bar is called the denominator.

We can read this fraction as three-fourths, three over four, or three divided by four.

Every fraction can be converted to a decimal by dividing. If you use the calculator to divide 3 by 4, you'll find that it is equal to 0.75.

Here are some other fractions and their decimal equivalents. Remember, you can find the decimal equivalent of any fraction by dividing.

Here are some terms that are very important when working with fractions.

Proper fraction When the numerator is less than the denominator, we call the expression a proper fraction. These are some examples of proper fractions.

Improper fraction An improper fraction occurs when the numerator is greater than or equal to the denominator. These are some examples of improper fractions:

Mixed number When an expression consists of a whole number and a proper fraction, we call it a mixed number. Here are some examples of mixed numbers:

We can convert a mixed number to an improper fraction. First, multiply the whole number by the denominator of the fraction. Then, add the numerator of the fraction to the product. Finally, write the sum over the original denominator. In this example, since three thirds is a whole, the whole number 1 is three thirds plus one more third, which equals four thirds.

Convert 1-1/3 to an improper fraction:

Equivalent fractionsThere are many ways to write a fraction of a whole. Fractions that represent the same number are called equivalent fractions. This is basically the same thing as equal ratios. For example, �, 2/4, and 4/8 are all equivalent fractions. To find out if two fractions are equivalent, use a calculator and divide. If the answer is the same, then they are equivalent.

ReciprocalWhen the product of two fractions equals 1, the fractions are reciprocals. Every nonzero fraction has a reciprocal. It's easy to determine the reciprocal of a fraction since all you have to do is switch the numerator and denominator--just turn the fraction over. Here's how to find the reciprocal of three-fourths.

To find the reciprocal of a whole number, just put 1 over the whole number. For example, the reciprocal of 2 is

Adding and Subtracting Fractions - It's easy to add and subtract like fractions, or fractions with the same denominator. You just add or subtract the numerators and keep the same denominator. The tricky part comes when you add or subtract fractions that have different denominators. To do this, you need to know how to find the least common denominator. In an earlier lesson, you learned how to simplify, or reduce, a fraction by finding an equivalent, or equal, fraction where the numerator and denominator have no common factors. To do this, you divided the numerator and denominator by their greatest common factor.

In this lesson, you'll learn that you can also multiply the numerator and denominator by the same factor to make equivalent fractions.

Example 1

In this example, since 12 divided by 12 equals one, and any number multiplied by 1 equals itself, we know 36/48 and 3/4 are equivalent fractions, or fractions that have the same value. In general, to make an equivalent fraction you can multiply or divide the numerator and denominator of the fraction by any non-zero number.

Since only like fractions can be added or subtracted, we first have to convert unlike fractions to equivalent like fractions. We want to find the smallest, or least, common denominator, because working with smaller numbers makes our calculations easier. The least common denominator, or LCD, of two fractions is the smallest number that can be divided by both denominators. There are two methods for finding the least common denominator of two fractions:

Example 2

Method 1: Write the multiples of both denominators until you find a common multiple.

The first method is to simply start writing all the multiples of both denominators, beginning with the numbers themselves. Here's an example of this method. Multiples of 4 are 4, 8, 12, 16, and so forth (because 1 × 4=4, 2 × 4=8, 3 × 4=12, 4 × 4=16, etc.). The multiples of 6 are 6, 12,…--that's the number we're looking for, 12, because it's the first one that appears in both lists of multiples. It's the least common multiple, which we'll use as our least common denominator.

Method 2: Use prime factorization.

For the second method, we use prime factorization-that is, we write each denominator as a product of its prime factors. The prime factors of 4 are 2 times 2. The prime factors of 6 are 2 times 3. For our least common denominator, we must use every factor that appears in either number. We therefore need the factors 2 and 3, but we must use 2 twice, since it's used twice in the factorization for 4. We get the same answer for our least common denominator, 12.

Example 3

prime factorization of 4 = 2 × 2 prime factorization of 6 = 2 × 3

LCD = 2 × 2 × 3 = 12

Now that we have our least common denominator, we can make equivalent like fractions by multiplying the numerator and denominator of each fraction by the factor(s) needed. We multiply 3/4 by 3/3, since 3 times 4 is 12, and we multiply 1/6 by 2/2, since 2 times 6 is 12. This gives the equivalent like fractions 9/12 and 2/12. Now we can add the numerators, 9 + 2, to find the answer, 11/12.

Example 4

Multiplying and Dividing Fractions - To multiply fractions, first we simplify the fractions if they are not in lowest terms. Then we multiply the numerators of the fractions to get the new numerator, and multiply the denominators of the fractions to get the new denominator. Simplify the resulting fraction if possible.

Note that multiplying fractions is frequently expressed using the word "of." For example, to find one-fifth of 10 pieces of candy, you would multiply 1/5 times 10, which equals 2. Study the example problems to see how to apply the rules for multiplying fractions.

Example 1

Dividing by fractions is just like multiplying fractions, except for one additional step.

To divide any number by a fraction:

First step: Find the reciprocal of the fraction.

Second step: Multiply the number by the reciprocal of the fraction.

Third step: Simplify the resulting fraction if possible.

Fourth step: Check your answer: Multiply the result you got by the divisor and be sure it equals the original dividend.

Note that you can only divide by non-zero fractions.

Example 1

Example 2

Adding, Subtracting, Multiplying, & Dividing Mixed Numbers - As you may recall, a mixed number consists of an integer and a proper fraction. Any mixed number can also be written as an improper fraction, in which the numerator is larger than the denominator, as shown in the following example:

Example 1

To add mixed numbers, we first add the whole numbers together, and then the fractions.

If the sum of the fractions is an improper fraction, then we change it to a mixed number. Here's an example. The whole numbers, 3 and 1, sum to 4. The fractions, 2/5 and 3/5, add up to 5/5, or 1. Add the 1 to 4 to get the answer, which is 5.

Example 2

If the denominators of the fractions are different, then first find equivalent fractions with a common denominator before adding. For example, let's add 4 1/3 to 3 2/5. Using the techniques we've learned, you can find the least common denominator of 15. The answer is 7 11/15.

Subtracting mixed numbers is very similar to adding them. But what happens when the fractional part of the number you are subtracting is larger than the fractional part of the number you are subtracting from?

Here's an example: let's subtract 3 3/5 from 4 1/3. First you find the LCD; here it's 15.

4 1/3 - 3 3/5 Write both fractions as equivalent fractions with a denominator of 15.

4 5/15 - 3 9/15

 3 + 1 5/15 - 3 9/15

3 + 20/15 - 3 9/15

 

Since you're trying to subtract a larger fraction from a smaller one, you need to "borrow" a one from the integer 4, change it to 15/15, and add it to the fraction.

3 20/15 - 3 9/15

11/15

 

Now the problem becomes 3 20/15 minus 3 9/15 and the answer is 11/15.

Multiplying mixed numbers is just like multiplying fractions. In fact, it is multiplying fractions, since you first change the mixed numbers into improper fractions. Let's work through a "word problem" example.

Melinda is building a birdhouse and needs three pieces of wood that are each 9-5/8 inches long. She has a piece of wood that is 29 inches long. Will this be long enough?

Let's multiply 3 times 9-5/8. First we have to change both numbers to improper fractions. Three is easy; it becomes 3/1.

The next one is a little trickier. Keep the 8 in the denominator. For the numerator, multiply 9 times 8 and add 5. This gives 77 in the numerator, so the improper fraction is 77/8. This can't be simplified any further, so we go on with our multiplication.

Now multiply the numerators and then the denominators. This gives 231 in the numerator, and 8 in the denominator.

We want our answer in lowest terms, so we divide 231 by 8, and get 28-7/8 inches. Since this is less than 29 inches, Melinda will be able to saw her three pieces from this board, as long as she measures and cuts very carefully!

Lastly, Dividing mixed numbers is very similar to multiplying mixed numbers. You just add one step—after changing the divisor into an improper fraction, you then find its reciprocal and multiply. Let's work through a "word problem" example.

The SuperQuik Market has just installed new scanners for its check-out lanes. They claim the average time to check out a customer is 2 ½ minutes. How many customers, on average, can they check out in half an hour?

To solve this problem, we have to know that half an hour is the same as 30 minutes. Then we can divide 30 by 2 ½.

First step: Write the whole number and the mixed number as improper fractions.

Second step: Write the reciprocal of the divisor, 2/5, and multiply.

Third step: Simplify, if possible. Notice that we can simplify our problem at this step, to make our calculations easier. Five goes evenly into 30, so we can divide both 5 and 30 by 5, to give 1 and 6.

Fourth step: Perform the simple multiplication of the numerators and the denominators. We find that the market can check out 12 customers in 30 minutes with its new scanners.

Fifth step: Put the answer in lowest terms, and check the answer. Our answer is already in lowest terms, so there is nothing left to do but check the answer, to be sure it makes sense. We can use estimation and rounding to do our check. If we round 2 ½ minutes to 3 minutes and divide 3 into 30, we get 10 customers in 30 minutes. So it is reasonable that 2 more customers per half hour, or 12 customers, can be checked, since 2 ½ minutes per customer is less than 3 minutes per customer.

Decimal - Our decimal system of numbers lets us write numbers as large or as small as we want, using a secret weapon called the decimal point. In our number system, digits can be placed to the left and right of a decimal point, to indicate numbers greater than one or less than one. The decimal point helps us to keep track of where the "ones" place is. It's placed just to the right of the ones place. As we move right from the decimal point, each number place is divided by 10.

We can read the decimal number 127.578 as "one hundred twenty seven and five hundred seventy-eight thousandths". But in daily life, we'd usually read it as "one hundred twenty seven point five seven eight."Here is another way we could write this number:

Notice that the part to the right of the decimal point, five hundred seventy-eight thousandths, can be written as a fraction: 578 over 1000. However, you will hardly ever see a decimal number written like this. Why do you think this is? You can see that our decimal code is a very handy and quick way to write a number of any size!

Examples

Here's how to write these numbers in decimal form:

Three hundred twenty-one and seven tenths 321.7

(6 x 10) + (3 x 1) + (1 x 1/10) + (5 x 1/100)63.15

Five hundred forty-eight thousandths0.548

Five hundred and forty-eight thousandths 500.048

Hint #1: Remember to read the decimal point as "and" -- notice in the last two problems what a difference that makes!

Hint #2: When writing a decimal number that is less than 1, a zero is normally used in the ones place:

0.526 not .526

Percent - We use the percent symbol (%) to express percent. Percents are used everywhere in real life, so you'll need to understand them well. Here are three ways to write the same thing:

15% = 15/100 = 0.15

Fifteen percent is the same as the fraction 15/100 and the decimal 0.15. They all simply mean "fifteen out of a hundred." A percent can always be written as a decimal, and a decimal can be written as a percent, like this:

0.85 = 85%

We can find any percent of a given number by changing the percent to a decimal and multiplying. One hundred percent of a number is just the number itself. Two hundred percent of a number is twice that number.

100% of 50 -> 50200% of 50 -> 2 x 50 = 100

Let's find 30 percent of 400:

First change 30% to a decimal by moving the decimal point 2 places to the left.

30% = 0.30

Then multiply.

0.30 x 400 = 120

30% of 400 is 120.

Mental MathThere's an easy way to find 10% of a number without multiplying. Just move the decimal point in the number left by one place. Let's try it with these numbers:

895 27 10,411

10% of 895 = 89.510% of 27 = 2.7

10% of 10,411 = 1,041.1

ExamplesWrite each as a percent:

$6 out of $12 -> $6/$12 = 0.5 = 50% 34 out of 100 -> 34/100 = 0.34 = 34%

0.55 = 55%1.2 = 120%

Find the value of n for each problem:

n is 50% of 60 -> n = 0.5 x 60 = 30n is 4% of 33 -> n = 0.04 x 33 = 1.32n is 150% of 24 -> n = 1.5 x 24 = 36

Exponents, Square Roots, Scientific Notation - An exponent tells you how many times the base number is used as a factor. A base of five raised to the second power is called "five squared" and means "five times five." Five raised to the third power is called "five cubed" and means "five times five times five." The base can be any sort of number--a whole number, a decimal number, or a fraction can all be raised to a power.

Here are some simple rules to use with exponents.

1. a1 = aAny number raised to the power of one equals the number itself.

2. For any number a, except 0, a0 = 1Any number raised to the power of zero, except zero, equals one.

3. For any numbers a, b, and c, ab x ac = ab+c

This multiplication rule tells us that we can simply add the exponents when multiplying two powers with the same base.

ALERT! These are mistakes that students often make when dealing with exponents.

Mistake! Do not multiply the base and the exponent. 26 is not equal to 12, it's 64!

Mistake! The multiplication rule only applies to expressions with the same base. Four squared times two cubed is not the same as 8 raised to the power two plus three.

Mistake! The multiplication rule applies just to the product, not to the sum of two numbers.

Scientific NotationWhat happens when you're using a calculator and your answer is too long to fit in the window? Use a calculator to multiply these 2 numbers:

60,000,000,000,000 x 20,000,000,000You'll discover a short way of writing very long numbers. This is called scientific notation, or E notation on a calculator ("E" stands for "Exponent"). A number written in scientific notation is written as a product of a number between 1 and 10 and a power of 10.

For example, to write 127,680,000 in scientific notation, change the number to a number between 1 and 10 by moving the decimal point 8 places to the left. Then multiply by 10 raised to the power of the number of places you had to move the decimal point--that is, 108:

127,680,000 = 1.2768 x 108

On your calculator window, the base of 10 is not shown; the E means "10 raised to the following power."

Examples7 x 7 x 7 x 7 = ?    74

2 x 2 x 2 x 2 x 2 x 2 = ?   26

110 =    1 53 =    5 x 5 x 5 = 125

Write the following numbers in scientific notation.565,000  =  5.65 x 105

7,325,000  =  7.325 x 106

91,247,000,000  =  9.1247 x 1010

Square Root - Many mathematical operations have an inverse, or opposite, operation. Subtraction is the opposite of addition, division is the inverse of multiplication, and so on. Squaring, which we learned about in a previous lesson (exponents), has an inverse too, called "finding the square root." Remember, the square of a number is that number times itself. The perfect squares are the squares of the whole numbers: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 …

The square root of a number, n, written

is the number that gives n when multiplied by itself. For example,

because 10 x 10 = 100

Examples

Here are the square roots of all the perfect squares from 1 to 100.

Finding square roots of numbers that aren't perfect squares without a calculator

1. Estimate - first, get as close as you can by finding two perfect square roots your number is between.

2. Divide - divide your number by one of those square roots.

3. Average - take the average of the result of step 2 and the root.

4. Use the result of step 3 to repeat steps 2 and 3 until you have a number that is accurate enough for you.

Example: Calculate the square root of 10 ( ) to 2 decimal places.

1. Find the two perfect square numbers it lies between.

Solution:32 = 9 and 42 = 16, so lies between 3 and 4.

2. Divide 10 by 3. 10/3 = 3.33 (you can round off your answer)

3. Average 3.33 and 3. (3.33 + 3)/2 = 3.1667

Repeat step 2: 10/3.1667 = 3.1579Repeat step 3: Average 3.1579 and 3.1667. (3.1579 + 3.1667)/2 = 3.1623

Try the answer --> Is 3.1623 squared equal to 10? 3.1623 x 3.1623 = 10.0001

If this is accurate enough for you, you can stop! Otherwise, you can repeat steps 2 and 3.

Note: There are a number of ways to calculate square roots without a calculator. This is only one of them.

Mean – also known as the average. The mean of a set of numbers is the average. Add the numbers and divide by the number of numbers in the set given to achieve the average.

Median – Order a set of numbers from least to greatest. If there is an odd number of numbers, the median is the number in the middle of that sequence of numbers. If there is an even number of numbers, the median is the mean, or average of the two middle numbers.

Mode – The mode of a set of numbers is the number that appears most frequently.

Word Problems - There are two steps to solving math word problems:

1. Translate the wording into a numeric equationthat combines smaller "expressions"

2. Solve the equation!

Math expressions (examples):

addition:  5+x subtraction: 5-x

multiplication: 5*x; 5x division:  5 ÷ x; 5/x

Key words for addition + :increased by; more than; combined together; total of; sum; added to

("mouse over" the block for answer)

What is the sum of 8 and y? 8 + y

Express the number (x) of apples increased by two x + 2

Express the total weight of Alphie the dog (x) and Cyrus the cat (y) x + y

Key words for Subtraction - :less than, fewer than, reduced by, decreased by, difference of

What is four less than y y - 4

What is nine less than a number (y) y - 9

What if the number (x) of children was reduced by 6? x - 6

What is the difference of my weight (x)and your weight (y) x - y

Key words for multiplication  * x or integers next to each other (5y, xy) :of, times, multiplied by

What is y multiplied by 13 13y or 13 * y

Three runners averaged "y" minutes.Express their total running time: 3y

I drive my car at 55 miles per hour.How far will I go in "x" hours? 55x

Key words for division  ÷ /per, a; out of; ratio of, quotient of; percent (divide by 100)

What is the quotient of y and 3 y/3 or y ÷ 3

Three students rent an apartment for $ "x" x/3 or x ÷ 3

/month.What will each have to pay?

"y" items cost a total of  $25.00.Express their average cost: 25/y or 25 ÷ y

Word problems are a series of expressions that fits into an equationAn equation is a combination of math expressions.Suggestions:

Read the problem entirelyGet a feel for the whole problem

List information and the variables you identifyAttach units of measure to the variables (gallons, miles, inches, etc.)

Define what answer you need,as well as its units of measure

Work in an organized mannerWorking clearly will help you think clearly

o Draw and label all graphs and pictures clearly o Note or explain each step of your process;

this will help you track variables and remember their meanings Look for the "key" words (above)

Certain words indicate certain mathematical operations:

More vocabulary and key words:

"Per" means "divided by"as "I drove 90 miles on three gallons of gas, so I got 30 miles per gallon" (Also 30 miles/gallon)

"a" sometimes means "divided by"as in "When I tanked up, I paid $3.90 for three gallons, so the gas was 1.30 a gallon, or $1.30/gallon

"less than"If you need to translate "1.5 less than x", the temptation is to write "1.5 - x".  DON'T!  Put a "real world" situation in, and you'll see how this is wrong:  "He makes $1.50 an hour less than me."  You do NOT figure his wage by subtracting your wage from $1.50.  Instead, you subtract $1.50 from your wage

"quotient/ratio of" constructions  If a problems says "the ratio of x and y",it means "x divided by y" or x/y or x÷y

"difference between/of" constructionsIf the problem says "the difference of x and y",it means "x - y"

Examples: 

What if the number (x) of children was reduced by six,

and then they had to share twenty dollars?How much would each get?

20/(x - 6)

What is 9 more than y? y + 9

What is the ratio of 9 more than y to y? (y + 9)/y

What is nine less than the total of a number (y) and two (y + 2) - 9  = y - 7

The length of a football field is 30 yards more than its width "y".

Express the length of the field in terms of its width y

y + 30

Word Knowledge

General Information

The ASVAB Word Knowledge test measures your ability to understand the meaning of words through synonyms. You will have 11 minutes to complete 35 questions, so like every part of the test be mindful of your time.

Study AidsCommon Word Roots –

Root Meaning Example Definitionagri field agronomy field-crop production and soil managementanthropo man anthropology the study of manastro star astronaut one who travels in interplanetary spacebio life biology the study of lifecardio heart cardiac pertaining to the heartcede go precede to go beforechromo color chromatology the science of colors

demos people democracy government by the peoplederma skin epidermis the outer layer of skindyna power dynamic characterized by power and energygeo earth geology the study of the earthhelio sun heliotrope any plant that turns toward the sunhydro water hydroponics growing of plants in water reinforced with nutrientshypno sleep hypnosis a state of sleep induced by suggestionject throw eject to throw outmagni great, big magnify to enlarge, to make biggerman(u) hand manuscript written by handmono one monoplane airplane with one wingortho straight orthodox right, true, straight opinionpod foot pseudopod false footpsycho mind psychology study of the mind in any of its aspectspyro fire pyrometer an instrument for measuring temperaturesscript write manuscript hand writtenterra earth terrace a raised platform of earththermo heat thermometer instrument for measuring heatzoo animal zoology the study of animals

Common Prefixes -

ante- before antebellum before the waranti- against antifreeze liquid used to guard against freezingauto- self automatic self-acting or self-regulatingbene- good benefit an act of kindness; a giftcircum- around circumscribe to draw a line around; to encirclecontra- against contradict to speak against

de- reverse, remove defoliate remove the leaves from a tree

dis- apart dislocate to unlodgedys- bad dysfunctional not functioningecto- outside ectoparasite parasite living on the exterior of animalsendo- within endogamy marriage within the tribeex- out excavate to dig outequi- equal equidistant equal distanceextra- beyond extraterrestrial beyond the earthhyper- over hypertension high blood pressurehypo- under hypotension low blood pressurein- in interim in between

inter- between intervene come betweenintra- within intramural within bounds of a schoolintro- in, into introspect to look within, as one's own mindmacro- large macroscopic large enough to be observed by the naked eyemal- bad maladjusted badly adjusted

micro- small microscopic so small that one needs a microscope to observe

multi- many multimillionaire one having two or more million dollarsneo- new neolithic new stone agenon- not nonconformist one who does not conformpan- all pantheon a temple dedicated to all godspoly- many polygonal having many sidespost- after postgraduate after graduatingpre- before precede to go beforepro- for proponent a supporterproto- first prototype first or original modelpseudo- false pseudonym false name; esp., an author's pen-namere-, red- back again rejuvenate to make youngre-, red- together reconnect to put together againretro- backward retrospect a looking back on thingssemi- half semicircle half a circlesub- under submerge to put under watersuper- above superfine extra finetele- far telescope seeing or viewing afartrans- across transalpine across the Alps

Number prefixes -

uni- one tetra- four oct- eightmono- one quint- five nov- ninebi- two pent- five dec- tenduo- two sex- six lat- sidedi- two hex- six ped- foottri- three sept- seven pod- footquad- four hept- seven  

Math & Science Affixes and Roots -

Root or Affix Exampleaqua (water) aquariumhydro (water) hydroplanehemi (half) hemisphere

semi (half) semicircleequi (equal) equivalenttele (far off) telescopemicro (small) microfilmonomy (science of) astronomyology (study of) geologyuni (one) universebi (two) bicycletri (three) triangleocta (eight) octagondec (ten) decadecenti (hundred) centimetermilli (thousand) millimeterbio (life) biologyastro (star) astronautthermo (heat) thermodynamicmeter (measure) diameterped (foot) pedestrianpod (foot) tripod

Prefixes that mean "no": a- de- dis-, in- non- un-, contra -

Examples: disqualify, nondescript, unscrupulous, contradict, inadvertent

Prefix Meaning Examplesa-, an- without, not asexual, atypical, amoral, anarchyde- reverse action, away defrost, demystify, desensitize, deductdis-, dif-, di- not, apart dissatisfied, disorganized, different, divertin-, il-, it-, im- not inappropriate, invisible, illegal, impossiblenon- not nonproductive, nonessential, nonsenseun- not unlikely, unnoticeable, unreliablecontra-, counter- against contrary, contradict, counterproductive

Prefixes that indicate "when," "where," or "more": pre-, post-, ante-, inter-, infra-, traps-, sub-, circum-, ultra -

Examples: premature, postscript, anteroom, intervene, transformation

Prefix Meaning Examples

pre-, pro- before pre-dinner,  preliminary,  previous,  prologue

post- after postwar,  postoperative,  postponeante- before antecedent,  antechamber

inter- between, among interstate,  intercept,  interfereintra- within intramural,  intrastate,  intravenoustrans- across transcontinental,  transparent,  transactionsub- under submarine,  submerge,  subjugatecircum- around circumnavigate,  circumference

ultra- beyond, on the far side of, excessive ultrasonic,  ultraviolet,  ultraconservative

Paragraph Comprehension

General Information

The ASVAB Word Knowledge test measures your ability to obtain and retain information from written material. You will have 13 minutes to complete 15 questions, so like every part of the test be mindful of your time.

Study Tips

A lot of people are thrown off by the Paragraph Comprehension section of the ASVAB. Here you are, cruising along, answering short, simple questions, and all of a sudden you're hit in the face with lengthy sentences — and often they're about boring subjects! Then you're told to answer numerous questions on each paragraph. It can seem overwhelming. Don't panic. Just keep in mind that the questions in this section are divided into four types.

A. Detail QuestionsA question might ask you about a detail from the paragraph you just read — in other words, a fact check. For example, in the following paragraph:

"Using bulldozers to slice bunkers and a helicopter landing pad out of a mountainside, U.S. special operations forces dug in Tuesday on a peak overlooking Pakistan, fortifying the area for the intensifying battle against al-Qaida and Taliban forces. Special operations forces — who include Green Berets, Navy SEALs, and CIA operatives — are playing a secretive but leading role in the battle against al-Qaida and Taliban suspects believed to be hiding out in the mountains of Pakistan's tribal areas."

You might be asked the following question about what you just read:

1. Which of the following is NOT being done by U.S. special operations forces?

A. Building a helicopter landing pad B. Fortifying the area for battle C. Fighting against al-Qaida and Taliban rebels D. Hiding in Pakistan's tribal areas

The answer would be "D."

B. What's the Point? The second type of question is a general question about what you just read — in other words, the summary.

If we use the paragraph above as an example once again, you might be asked:

2. The best title for this selection is:

A. Rebels Have Not Been Caught B. Special Forces Prepare to Battle Rebels C. Building a Helicopter Pad D. Who's in Special Operations?

The answer would be "B." Note that almost all of the choices are true statements, but they aren't the focus of the paragraph. The main point is that Special Forces teams are digging in, preparing to attack the rebels.

C. Context and Word MeaningEarlier we talked about synonyms and context. Here's where you get tested on them. As you know, words have different meanings depending on how and where they're used (the context). Again, using the above paragraph as an example, answer this question:

3. In this paragraph's context, the word "leading" means:

A. Principal B. In Front C. Popular D. Stellar

This one’s a bit tricky; you could argue that "leading" means "in front" in this paragraph, but the paragraph also states that the Special Operations forces are "secretive," meaning they aren’t on the front lines, but working behind the scenes. Thus, the correct answer would be "A," as "principal" also means "most important" or "main." See how the context and your knowledge of synonyms come into play?

D. Inference QuestionsTo infer is to take the information that is given to you and come to a conclusion about what it means, even though you’re not told directly. For example, if the weatherman says there’s a 80 percent chance of rain tomorrow, you can infer that he’s recommending you break out your umbrella.

Take this paragraph as an example:

"Within a few hours after it is born, a young wild horse can run fast enough to keep up with the herd. It is able to do this because its legs are long for its size — almost as long as they will be when the horse is fully grown. If young horses could not run so soon after birth, they would be quickly eaten by predators. Usually, only one foal is born at a time. In the case of Mongolian wild horses, the coat of a newborn foal is often quite light in color. After four or five weeks, this is shed and replaced by a darker coat. Foals usually stay close to their mothers. When there is danger, they are moved to the center of group and protected by all of the adults."

An inference question for this paragraph might go like this:

4. According to the author, young wild horses run fast soon after birth because: A. They have long legs B. They must escape predators C. All horses run fast D. It’s easier for adult horses to protect them

This is a tricky one; you could argue that the answer is "A," because long legs help the horses run fast. This is true, but it doesn’t explain why they run fast. The author suggests that if they don’t run fast after birth, they will be caught by predators. Thus, the answer would be "B."

General Tips for Reading Comprehension -

1. The Quick Read. Read each paragraph quickly and get the essential details first. What is the main point being made?

2. The Closer Look. Read the paragraph again, understanding all the information, and how it relates to the main point.

3. Look at ALL Questions and Answers. Make sure you know exactly what you’re being asked to answer. Small words like least, greatest, except, not, all, always, every, never, etc. in a question can change what you’re being asked. For example, "What does the author always eat at dinner?" is very different from "What does the author never eat at dinner?" Also, read ALL the possible answers. Even if you think the answer to a question is "A" right off the bat, make sure to look at all the choices.

4. No Opinions. You may not agree with what the paragraph says. It doesn’t matter. Don’t let your opinions get in the way when you answer the questions. Base your answers only on what you’ve read, and not what you think, or something you read somewhere else.

5. Fast and Efficient. You are being timed, so don’t get caught up if you’re stuck on a question. If all else fails, guess and move on. In the actual ASVAB, you can only answer questions in order, so give yourself time to answer ALL questions.

Mathematics Knowledge

General Information

The ASVAB Mathematics Knowledge test measures your ability solve problems using concepts taught in high school math courses. You will have 24 minutes to complete 25 questions, so like every part of the test be mindful of your time.

Algebra & Probabilities

The Language of Algebra – Learning algebra is a little like learning another language. In fact, algebra is a simple language, used to create mathematical models of real-world situations and to handle problems that we can't solve using just arithmetic. Rather than using words, algebra uses symbols to make statements about things. In algebra, we often use letters to represent numbers.

Since algebra uses the same symbols as arithmetic for adding, subtracting, multiplying and dividing, you're already familiar with the basic vocabulary.

In this lesson, you'll learn some important new vocabulary words, and you'll see how to translate from plain English to the "language" of algebra.

The first step in learning to "speak algebra" is learning the definitions of the most commonly used words.

Algebraic ExpressionsAn algebraic expression is one or more algebraic terms in a phrase. It can include variables, constants, and operating symbols, such as plus and minus signs. It's only a phrase, not the whole sentence, so it doesn't include an equal sign.

Algebraic expression:

3x2 + 2y + 7xy + 5

In an algebraic expression, terms are the elements separated by the plus or minus signs. This example has four terms, 3x2, 2y, 7xy, and 5. Terms may consist of variables and coefficients, or constants.

VariablesIn algebraic expressions, letters represent variables. These letters are actually numbers in disguise. In this expression, the variables are x and y. We call these letters "variables" because the numbers they represent can vary—that is, we can substitute one or more numbers for the letters in the expression.

CoefficientsCoefficients are the number part of the terms with variables. In 3x2 + 2y + 7xy + 5, the coefficient of the first term is 3. The coefficient of the second term is 2, and the coefficient of the third term is 7.

If a term consists of only variables, its coefficient is 1.

ConstantsConstants are the terms in the algebraic expression that contain only numbers. That is, they're the terms without variables. We call them constants because their value never changes, since there are no variables in the term that can change its value. In the expression 7x2 + 3xy + 8 the constant term is "8."

Real NumbersIn algebra, we work with the set of real numbers, which we can model using a number line.

Real numbers describe real-world quantities such as amounts, distances, age, temperature, and so on. A real number can be an integer, a fraction, or a decimal. They can also be either rational or irrational. Numbers that are not "real" are called imaginary. Imaginary numbers are used by mathematicians to describe numbers that cannot be found on the number line. They are a more complex subject than we will work with here.

Rational NumbersWe call the set of real integers and fractions "rational numbers." Rational comes from the word "ratio" because a rational number can always be written as the ratio, or quotient, of two integers.

Examples of rational numbersThe fraction ½ is the ratio of 1 to 2.

Since three can be expressed as three over one, or the ratio of 3 to one, it is also a rational number.

The number "0.57" is also a rational number, as it can be written as a fraction.

Irrational NumbersSome real numbers can't be expressed as a quotient of two integers. We call these numbers "irrational numbers". The decimal form of an irrational number is a non-repeating and non-terminating decimal number. For example, you are probably familiar with the number called "pi". This irrational number is so important that we give it a name and a special symbol!

Pi cannot be written as a quotient of two integers, and its decimal form goes on forever and never repeats.

Translating Words into Algebra LanguageHere are some statements in English. Just below each statement is its translation in algebra.

the sum of three times a number and eight 3x + 8

The words "the sum of" tell us we need a plus sign because we're going to add three times a number to eight. The words "three times" tell us the first term is a number multiplied by three.

In this expression, we don't need a multiplication sign or parenthesis. Phrases like "a number" or "the number" tell us our expression has an unknown quantity, called a variable. In algebra, we use letters to represent variables.

the product of a number and the same number less 3 x(x – 3)

The words "the product of" tell us we're going to multiply a number times the number less 3. In this case, we'll use parentheses to represent the multiplication. The words "less 3" tell us to subtract three from the unknown number.

a number divided by the same number less five

The words "divided by" tell us we're going to divide a number by the difference of the number and 5. In this case, we'll use a fraction to represent the division. The words "less 5" tell us we need a minus sign because we're going to subtract five.

Addition, Subtraction, Multiplication, and Division w/Unknown –

Solving an equation is like solving a puzzle! It means finding a value for the variable that makes the equation true. Using the properties of real numbers that you've learned, you can rearrange the terms of an equation and use inverse operations to help you find the value of the variable. But be careful! You can think of an equation like a balance scale—whatever you do to one side of the scale, you must also do to the other side, to keep it in balance.

First, let's look at a simple addition equation, x + 15 = 30.

To solve the equation we must try to get the variable x alone on one side. We can use the inverse of adding 15 - or subtracting 15 - to get x alone on the left side. Now we have x alone on the left side, since 15 – 15 = 0, but the scale is not in balance. To balance the scale, we must also subtract 15 from the right side of the equation.

x + 15 - 15 = 30 - 15         x = 15

30 – 15 = 15, so we find that x = 15.

We can check this solution by substituting the value 15 for x in the original equation. When we evaluate for x = 15 we get 30 = 30, which is a true statement. We know our solution is correct!

      x + 15 = 30(15) + 15 = 30

                             30 = 30 Correct!

Now, let's look at a subtraction equation, y – 9 = 3

To solve this equation we must try to get the variable y alone on one side. We can use the inverse of subtracting 9, or adding 9, to get y alone on the left side.

y - 9 + 9 = 3 + 9   y - 0 = 12        y = 12

Now we have y alone on the left side, since –9 + 9 = 0, but the scale is not in balance. To balance the scale, we must also add 9 to the right side of the equation.

Now we have y alone on the left side of the equation. Three plus nine is twelve, so we find that y = 12.

We can check this solution by substituting the value 12 for y in the original equation. When we evaluate for y = 12, we get 3 = 3, which is a true statement. Our solution is correct!

      y - 9 = 3(12) - 9 = 3

                          3 = 3 Correct!

Mt. Everest in Nepal is the world's tallest mountain, about 29,000 ft. high. It is twice as high as Mount Whitney in California. How high is Mount Whitney?

We can write a multiplication equation to find the answer to problems like this. Our unknown number is the height of Mount Whitney. Let x represent this height. We know that 2x is the height of Mount Everest. We can write our equation like this:

2x = 29,000 ft

To solve this equation, we can use the inverse of multiplying by 2, which is dividing by 2.

If we divide the left side of the equation by 2, we will get x alone on the left. Remember, any operation done to one side must also be done to the other side, so we must also divide the right side by 2.

We divide, and find that x is equal to 14,500 ft. This is very close to the actual height of Mount Whitney, which is 14,494 ft.

Mental Math

You don't always need to use the calculator or pencil and paper to solve equations. You can solve many of them mentally. Use your mental muscle on these problems! Come up with an answer on your own, before looking at the answers below.

1.  10x = 350 2.  (12)(5)n = 0 3.  6 + 6 = 3x 4.  19y = 1900

Answers:

1.  x = 35 (Divide both sides by 10, an easy mental problem)2.  n = 0 (Any number divided into 0 is 0.)3.  x = 4 (6 + 6 = 12 and 12 ÷ 3 = 4) 4.  y = 100 (Divide both sides by 19, an easy mental problem)

The continental United States - the lower 48 states - has about 16,900 miles of shoreline. This is about ½ the length of the shoreline of Alaska. About how many miles of shoreline does Alaska have?

We can write a division equation to find the answer to problems like this. Our unknown number is the length of shoreline in Alaska. Let x represent this number.

We know that x divided by 2 is the approximate length of shoreline in the lower 48 states. We can write the equation like this:

To solve this equation, we can use the inverse of dividing by 2, or multiplying by 2. If we multiply the left side of the equation by 2, we will get x alone on the left. Remember, any operation done to one side must also be done to the other side, so we must also multiply the right side by 2.

We multiply, and find that x, the length of the shoreline of Alaska, is equal to 33,800 miles.

To check our answer, we substitute the value of 33,800 into the original equation, like this.

Inequalities - Solving an inequality is very similar to solving an equation. You follow the same steps, except for one very important difference. When you multiply or divide each side of the inequality by a negative number, you have to reverse the inequality symbol! Let's try an example:

-4x > 24

Since this inequality involves multiplication, we must use the inverse, or division, to solve it. We'll divide both sides by –4 in order to leave x alone on the left side.

When we simplify, because we're dividing by a negative number, we have to remember to reverse the symbol. This gives "x is less than –6," not "x is greater than –6."

x < -6

Why do we reverse the symbol? Let's see what happens if we don't. Think about the simple inequality –3 < 9. This is obviously a true statement.

-3 < 9

To demonstrate what happens when we divide by a negative number, let's divide both sides by –3. If we leave the inequality symbol the same, our answer is obviously not correct, since 1 is not less than –3.

We must reverse the symbol in order to find the correct answer, which is "1 is greater than –3."

Let's go back to the original problem and graph the solution x < –6 on a number line. To graph the solution for an inequality, you start at the defining point in the inequality. Here, it's –6. Then you graph all the points that are in the solution.

The red arrow shows that all the values on the number line less than –6 are in the solution. The open circle at –6 shows us that –6 is not in the solution. If the solution had been "x is less than or equal to –6," the circle would be a dark, or filled in, circle.

How can we check our answer? We can't use –6 to substitute in the inequality, because it lies outside our solution. To check, we can choose any value that lies in the solution. Let's use –7.

-4x > 24-4(-7) > 24

28 > 24 Correct!

Our substitution gave a true result, so the solution is correct.

Ratios - We use ratios to make comparisons between two things. When we express ratios in words, we use the word "to"--we say "the ratio of something to something else." Ratios can be written in several different ways: as a fraction, using the word "to", or with a colon.

Let's use this illustration of shapes to learn more about ratios. How can we write the ratio of squares to circles, or 3 to 6? The most common way to write a ratio is as a fraction, 3/6. We could also write it using the word "to," as "3 to 6." Finally, we could write this ratio using a colon between the two numbers, 3:6. Be sure you understand that these are all ways to write the same number.

Which way you choose will depend on the problem or the situation.

ratio of squares to circles is 3/6 ratio of squares to circles is 3 to 6 ratio of squares to circles is 3:6

There are still other ways to make the same comparison, by using equal ratios. To find an equal ratio, you can either multiply or divide each term in the ratio by the same number (but not zero). For example, if we divide both terms in the ratio 3:6 by the number three, then we get the equal ratio, 1:2. Do you see that these ratios both represent the same comparison? Some other equal ratios are listed below. To find out if two ratios are equal, you can divide the first number by the second for each ratio. If the quotients are equal, then the ratios are equal. Is the ratio 3:12 equal to the ratio 36:72? Divide both, and you discover that the quotients are not equal. Therefore, these two ratios are not equal.

Some other equal ratios: 3:6 = 12:24 = 6:12 = 15:30

Are 3:12 and 36:72 equal ratios?

Find 3÷12 = 0.25 and 36÷72 = 0.5

The quotients are not equal —> the ratios are not equal.

You can also use decimals and percents to compare two quantities. In our example of squares to circles, we could say that the number of squares is "five-tenths" of the number of circles, or 50%.

Here is a chart showing the number of goals made by five basketball players from the free-throw line, out of 100 shots taken. Each comparison of goals made to shots taken is expressed as a ratio, a decimal, and a percent. They are all equivalent, which means they are all different ways of saying the same thing.

Proportions - A proportion is simply a statement that two ratios are equal. It can be written in two ways: as two equal fractions a/b = c/d; or using a colon, a:b = c:d. The following proportion is read as "twenty is to twenty-five as four is to five."

In problems involving proportions, we can use cross products to test whether two ratios are equal and form a proportion. To find the cross products of a proportion, we multiply the outer terms, called the extremes, and the middle terms, called the means.

Here, 20 and 5 are the extremes, and 25 and 4 are the means. Since the cross products are both equal to one hundred, we know that these ratios are equal and that this is a true proportion.

We can also use cross products to find a missing term in a proportion. Here's an example. In a horror movie featuring a giant beetle, the beetle appeared to be 50 feet long. However, a model was used for the beetle that was really only 20 inches long. A 30-inch tall model building was also used in the movie. How tall did the building seem in the movie?

First, write the proportion, using a letter to stand for the missing term. We find the cross products by multiplying 20 times x, and 50 times 30. Then divide to find x. Study this step closely, because this is a technique we will use often in algebra. We are trying to get our unknown number, x, on the left side of the equation, all by itself. Since x is multiplied by 20, we can use the "inverse" of multiplying, which is dividing, to get rid of the 20. We can divide both sides of the equation by the same number, without changing the meaning of the equation. When we divide both sides by 20, we find that the building will appear to be 75 feet tall.

Note that we're using the inverse of multiplying by 20-that is, dividing by 20, to get x alone on one side.

Solving Equations for Two Unknowns - We have already found that there will be an infinite number of solutions in most situations when we have one equation with two unknowns. However, if we have two equations with the same two unknowns (e.g., both equations have variables x and y), we know already that there is only one solution for x

and y that solves both of these equations at the same time. We call this situation a system of two simultaneous equations, since we want to make both of these equations true at the same time. For example, suppose we want to solve the following system of equations:

y = 2 + 4xy = 10 + 3x

Substitution Method.

SUBSTITUTION METHOD FOR SOLVING A TWO-EQUATION SYSTEMWITH TWO UNKNOWNS (x and y)

1. Use one of the equations to derive a formula for y in terms of x. (To dothis, use the Principle of Equality to isolate y on one side of theequation.)

2. Substitute the resulting formula into the other equation wherever yappears.

3. The result from step 2 is a single equation with a single unknown (x). Usethis equation to solve for the value of x.

4. Insert the value of x into the expression for y from step 1 to determinethe value of y.

For example, going back to our original example:y = 2 + 4xy = 10 + 3x

Since both equations are already derived for y in terms of x, we can use thesubstitution property to transform both equations into a single equation:

2 + 4x = 10 + 3x

Subtract 3x from both sides of the equation:

2 + 4x – 3x = 10 + 3x - 3x2 + x = 10

Subtract 2 from both sides of the equation:

2 + x - 2 = 10 - 2x = 8

Now that we know the value of x, we can use either equation to pin down the value of y. For example, we can use the first equation to tell us that when x = 8 then y = 2 + 4 • 8 = 34. We can also use the second equation to see that when x = 8, then y = 10 + 3 • 8, which is also 34. Therefore, the ordered pair x = 8, y = 34 (written (8, 34)) is the solution to the two equation simultaneous-equation system.

In some cases you can simplify a two-equation system merely by adding the two equations together. For example, let’s examine the following system of simultaneous equations:

4x + 3y = 386x – 3y = 12

Let’s add the second equation to the first equation to form a new equation, like this: (4x + 3y) + (6x – 3y) = 38 + 12

Combining like terms we obtain:(4x + 6x) + (3y – 3y) = 50

Simplifying we obtain:

10x = 50x = 5

We also find that y = 6, when substituting 5 for the variable x in either of the two original equations. This method of eliminating one of the variables to form a new equation is termed the Elimination Method and is summarized below.

Elimination Method

1. Multiply both sides of the first equation by a number chosen so that the coefficient of y in the first equation becomes the negative of the coefficient of y in the second equation.

2. Add the second equation to the first equation.

3. The result from step 2 will be a new equation from which y has been eliminated. Solve that equation for x.

4. Insert the value for x into either one of the two original equations and then solve for the value of y.

Monomials: Addition, Subtraction, Division, Multiplication –

Monomial is a product of two or some factors, each of them is either a number, or a letter, or a power of a letter. For example,

3 a 2 b 4 ,    b d 3 ,    – 17 a b c

are monomials. A single number or a single letter may be also considered as a monomial. Any factor of a monomial may be called a coefficient. Often only a numerical factor is called a coefficient. Monomials are called similar or like ones, if they are identical or differed only by coefficients. Therefore, if two or some monomials have identical letters or their powers, they are also similar (like) ones. Degree of monomial is a sum of exponents of the powers of all its letters.

Addition of monomials. If among a sum of monomials there are similar ones, he sum can be reduced to the more simple form:

a x 3 y 2  – 5 b 3 x 3 y 2 + c 5 x 3 y 2 = ( a – 5 b 3 + c 5 ) x 3 y 2 .

This operation is called reducing of like terms. Operation, done here, is called also taking out of brackets.

Multiplication of monomials. A product of some monomials can be simplified, only if it has powers of the same letters or numerical coefficients. In this case exponents of the powers are added and numerical coefficients are multiplied.E x a m p l e:

5  a x 3 z 8 ( – 7 a 3 x 3 y 2 ) =  – 35 a 4 x 6 y 2 z 8 .

Division of monomials. A quotient of two monomials can be simplified, if a dividend and a divisor have some powers of the same letters or numerical coefficients. In this case an exponent of the power in a divisor is subtracted from an exponent of the power in a dividend; a numerical coefficient of a dividend is divided by a numerical coefficient of a divisor. E x a m p l e :

35 a 4 x 3 z 9 : 7 a x 2 z 6 = 5 a 3 x z 3 .

Geometric Formulas –

FORMULA REFERENCE SHEET 

Shape Formulas for Area (A) and Circumference (C)

Triangle  A = ½bh = ½ x base x height

Rectangle  A = lw = length x width

Trapezoid  A = ½(b1 + b2)h = ½ x sum of bases x height

Parallelogram  A = bh = base x height

Circle 

Figure Formulas for Volume (V) and Surface Area (SA)

Rectangular Prism

  V = lwh = length x width x heightSA = 2lw + 2hw + 2lh      = 2(length x width) + 2(height x width) + 2(length x height)

General Prisms   V = Bh = area of base x heightSA = sum of the areas of the faces

Right Circular Cylinder

  V = Bh = area of base x heightSA = 2B + Ch = (2 x area of base) + (circumference x height)

Square Pyramid

Right Circular Cone

Sphere

Equations of a Line    Coordinate Geometry Formulas

Standard Form:

Ax + By = C

where A and B are not both zero

Slope-Intercept Form:

   y = mx + b or y = b + mx

where m = slope and b = y-intercept

Point-Slope Formula:

   

Let (x1,y1) and (x2,y2) be two points in the plane.

Distance Traveled

d = rtdistance = rate x time

Polygon Angle Formulas

Sum of degree measures of the interior angles of a polygon:

180 (n - 2) Degree measure of an interior angle of a regular polygon:

where n is the number of sides of the polygon

  Simple Interest

I = prt

interest = principal x interest rate x time

Formulas for Right Triangles Special Triangles