Ch 5

Proportional Reasoning and Probability

5.1

Solving Proportions

Proportions

• Ratio – a comparison of two numbers by division

• Proportion – an equation stating two ratios are equal– Cross products will be equal

Property of Proportions

Example

35639

m

• Solve the proportion.

Example

549 zz

• Solve the proportion.

Example

8135

cc

• Solve the proportion.

Example

24183

a

• Solve the proportion.

Equivalent Measurements

• Proportions can be used to find equivalent measurements Converting quarts to pints

• Conversions on PA16 and A18 if you’ve forgotten them

Example

• Convert 15 pints to quarts

Example

• Convert 4750 millimeters to meters

Example

• Convert 96 centimeters to meters

Example

• Convert 2 gallons to quarts

Rates

• Rate – a ratio of two measurements having different units 200 miles / 4 hours

• Unit Rate – a rate whose denominator is 1 50 miles / hour

Dimensional Analysis

• A process of carrying units and canceling units to get the correct units

Example

• Density is a measure of the amount of matter that occupies a given volume. The density of wood is 0.71 gram per cubic centimeter. Suppose you have a piece of wood whose volume is 50 cubic centimeters. How many grams of wood does it contain?

Example

• The density of copper is 8.96 grams per cubic centimeter. Suppose you have a piece of copper whose volume is 25 cubic centimeters. How many grams of copper do you have?

Example

• A trucker drove 210 miles in 5 hours. At this rate, how far will she travel in 8 hours?

Example

• Juanita’s family drove 400 miles in 8 hours. The next day, they need to drive another 700 miles. At the same rate, how long will it take them to drive the 700 miles?

Assignments

• #1 – due today P191: 3 – 15

• #2 – due next time P192: 16 – 46 even, 47 – 50, 54 – 60 even

5.2

Scale Drawing and Models

Scale Drawing or Scale Model

• A drawing or model used to represent an object that is too large or too small to be drawn or built to actual size

• Scale – The ratio of a length on the drawing or model to

the corresponding length of the real object

Scale Model

Example

• The scale on a map of the upper Midwest is 1 inch = 15 miles. Find the distance between Chicago and Milwaukee on the map if the distance between the two cities is 90 miles.

Example

• A railroad car is 36 feet long and a scale model of the railroad car is 1.5 feet long. What is the scale for the model?

Assignments

• #1 – due today P196: 2 – 5

• #2 – due next time P196: 6 – 15, 19 – 22

5.3

The Percent Proportion

Percent

• A ratio that compares a number to 100 Per hundred

40% complete would be 40 of the hundred problems are complete

Example

• Express each fraction or ratio as a percent.

Example

• Express each fraction or ratio as a percent. In a classroom, 18 out of 24 students used a

computer at home last week.

Example

• Express each fraction or ratio as a percent.

7350

Percent Proportion

• Percentage: the part of the whole

• Base: the whole or total

Example

• What percent of 175 is 35?

Example

• 20 is 40% of what number?

Example

• 7 is what percent of 20?

Example

• 60 is 15% of what number?

Example

• Find 25% of 66.

Example

• What number is 10% of 88?

Example

• A family recently moved to a new home. The table below shows the number of days they spent packing boxes, cleaning the two homes, and then unpacking boxes. What percent of the total did they spend on each activity?

Assignments

• #1 – due today P201: 1 – 15a

• #2 – due next time P202: 16 – 42, 45 – 52

5.4

The Percent Equation

Percent Equation

• Rate: in decimal form!

Examples

• Find 17% of $250.

• 35% of what number is 105?

Examples

• Find 62% of 120.

• 75% of what number is 12?

Example

• A store collects 8.5% sales tax on all purchases. How much tax must be paid on a purchase of $188?

Simple Interest

• I=prt I = interest

p = principal (amount you start with)

r = annual interest rate (in decimal form)

t = time in years

Example

• Jessica deposited $3000 in a savings account that pays an interest rate of 6%. How long should she leave the money in the account if she wants to earn $90 in interest?

Example

• Riona serves food at a restaurant where she is paid 18% of the diners’ bills. She earned $126 last weekend. What was the total of the diners’ bills?

Mixture Problems

• Problems that involve combining two or moreparts into a whole Actually mixing items like meat, nuts, chemical

solutions

Example

• The Mars candy company was getting ready to introduce the blue M&M. In their marketing meeting, they began to explore pricing for its new promotion. If brown M&M's cost $.25 per pound and blue M&M's cost $.85 per pound, how many pounds of brown M&M's must be added to 300 lbs. of blue M&M's to obtain a mixture that would sell for $.45 per pound?

Example

• A chemistry experiment calls for a 20% solution of copper sulfate. You have 150 milliliters of a 30% solution. How many milliliters of an 18% solution should you add to et a 20% solution?

Example

• All 208 freshmen at a school went on a field trip. For transportation, busses that each hold 64 students and vans that each hold 8 students were used. Every bus and van was completely filled, and there were 5 vehicles used. How many buses and vans were used?

Example

• Crystal sold tickets to the Drama Club’s spring play. Adult tickets cost $8.00, and student tickets cost $5.00. Crystal sold 35 more student tickets than adult tickets. She collected a total of $1475. How many of each type of ticket did she sell?

Assignment

• P207: 1 – 18, 25, 26, 30

Correct Dose

• The dosing amount for Augmentin, a common antibiotic, is 51.4 milligrams for each kilogram of body weight. If 1 kilogram is equal to 2.2 pounds, how much would you prescribe for a child who weighs 66 pounds? How much would you prescribe for you?

5.5

Percent change

Percent Increase or Decrease

• When a number increases or decreases, the percent it changed is called the percent of increase or the percent of decrease

• The beginning or original number is used as the base

• The amount it changed (not it’s new value) is the percentage

Example

• Find the percent of increase or decrease. Round to the nearest percent. Original: 100

New: 140

Original: 150

New: 125

Example

• Find the percent of increase or decrease. Round to the nearest percent. Original 12

New: 20

Original: 50

New: 49

Applications

• Sales Tax: tax is added to the cost of the item Percent of increase

• Discount: amount by which the regular price is reduced Percent of decrease

Example

• A family bough a home computer for $890. a sales tax of 5.5% on the purchase was then added. What was the total price?

Example

• What is the total cost of a basketball that sells for $45 if the sales tax rate is 7%?

Example

• All long-sleeved T-shirts are on sale for 40% off. If the original was $19.95, what is the discount price?

Assignment

• P215: 4 – 34

5.6

Probability and Odds

Outcomes with Two Die

• 36 equal to occur outcomes

• Find chances of an event – probability

Probability

Ranges of Probability

• Between 0 and 1 inclusive (can’t be any other numbers) 0 means event is impossible

1 means the event is certain to happen

The closer to 1, the more likely to occur

Random

• If every event has an equally likely chance, the events are random

Example

• If a person is chosen at random, what is the probability that the person is age 5 – 17? 55 –64?

Theoretical vs. Experimental Probability

• Theoretical Probability – What should occur

Rolling two die• How many outcomes were there?

• Each happening once.

• So probability of rolling a 3 then a 4

• Experimental Probability – What actually occurred

136

Experimental v. Theoretical Probability

• Theoretical probability of tossing two die 36 times and landing on two 1s:

• Experimental probability of tossing two die 36 times and landing on two 1s:

Dice 1Dice 2 1 2 3 4 5 6

1

2

3

4

5

6

Example

• A systems technician at a large corporation monitored their e-mail system for one hour and found that employees sent 550 e-mail messages, of which 375 were sent to e-mail accounts outside the company.

Example

• A systems technician at a large corporation monitored their e-mail system for one hour and found that employees sent 550 e-mail messages, of which 375 were sent to e-mail accounts outside the company.

• Find the experimental probability that a randomly-chosen e-mail message is being sent outside the company.

Odds

Example

• A coin is randomly removed from a change purse that contains 7 pennies, 8 nickels, and 5 quarters. What are the odds that the coin is a nickel? A quarter?

Example:

• A bag contains 6 red marbles, 3 blue marbles, and 1 yellow marble. Find the odds of choosing a blue marble.

Assignments

• #1 – due today P222: 4 - 8

• #2 – due next time P222: 9 – 22, 27 – 31

5.7

Compound Events

Compound Events

•• Compound Event Compound Event –– Two or more simple events that are connected by Two or more simple events that are connected by

the words the words andand or or oror..

•• Example: Example: Rolling a dice and spinning a spinnerRolling a dice and spinning a spinner

•• Does the outcome of one affect the outcome of the Does the outcome of one affect the outcome of the other?other?

•• Independent Event Independent Event –– Events in which the outcomes do not affect each Events in which the outcomes do not affect each

otherother

Probability of Independent EventsProbability of Independent Events

ExampleExample

•• The two spinners shown below are spun at the The two spinners shown below are spun at the same time. Find the probability that the left same time. Find the probability that the left spinner lands on green and the right spinner spinner lands on green and the right spinner lands on a number greater than 2.lands on a number greater than 2.

ExampleExample

•• Two dice are rolled. Find the probability that Two dice are rolled. Find the probability that an even number is rolled on the first dice and an even number is rolled on the first dice and a number greater than 4 on the second die.a number greater than 4 on the second die.

Mutually ExclusiveMutually Exclusive

•• Mutually exclusive Mutually exclusive –– Two events that cannot occur at the same timeTwo events that cannot occur at the same time

•• If drawing a card from a deck of cards, it can’t be both a If drawing a card from a deck of cards, it can’t be both a jack and a queenjack and a queen

Probability of Mutually Exclusive EventsProbability of Mutually Exclusive Events

ExampleExample

•• A marble is selected at random from a bag A marble is selected at random from a bag that contains 5 red marbles, 3 blue marbles, that contains 5 red marbles, 3 blue marbles, and 2 yellow marbles. What is the probability and 2 yellow marbles. What is the probability that the marble is either red or yellow?that the marble is either red or yellow?

ExampleExample

•• Jamal has 4 quarters, 2 dimes, and 4 nickels in Jamal has 4 quarters, 2 dimes, and 4 nickels in his pocket. He takes one coin out at random. his pocket. He takes one coin out at random. Find the probability of Jamal’s choosing a Find the probability of Jamal’s choosing a quarter or a nickel.quarter or a nickel.

Inclusive EventsInclusive Events

•• Inclusive event Inclusive event –– Events connected by the word Events connected by the word oror that are not that are not

mutually exclusive.mutually exclusive.

These events have an overlapping partThese events have an overlapping part

Probability of Inclusive EventsProbability of Inclusive Events

ExampleExample

•• If there is a 90% chance of snow in January If there is a 90% chance of snow in January and a 95% chance of snow in February, find and a 95% chance of snow in February, find the probability that it will snow sometime in the probability that it will snow sometime in January or February.January or February.

ExampleExample

•• If there is a 40% chance of rain on Saturday If there is a 40% chance of rain on Saturday and a 60% chance of rain on Sunday, find the and a 60% chance of rain on Sunday, find the probability that it will rain on either Saturday probability that it will rain on either Saturday or Sunday.or Sunday.

Concept SummaryConcept Summary

AssignmentsAssignments

•• #1 #1 –– due todaydue today P228: 3 P228: 3 –– 9 9

•• #2 #2 –– due next timedue next time P228: 10 P228: 10 –– 19, 23 19, 23 –– 27 27