ProblemSolving Packet

Name: _______________________________ Period: ___________

One hundred twenty 8th graders are on the honor roll.. This represents one third of the 8th grade class. How many 8th graders are there?

Brian ate three of eight pizza slices. He paid $2.70 as his share of the cost. How much did the whole pizza cost?

An employer said that each worker received $24 in fringe benefits for every $100 in wages. At this rate, what wages were earned by a worker whose fringe benefits were valued at $5,100.

Find a number that is 96 greater than its opposite.

Three times a number, decreased by 8 is the same as twice the number, increased by 15. Find the number.

The sum of two numbers is 15. Three times one of the numbers is 11 less than five times the other. Find the numbers. (use only one variable)

A bus went 318 km farther than a car. The car went one third as far as the bus. How far did each travel?

A rectangle is 12m longer than it is wide. Its perimeter is 68 m. Find the length and the width.

The length of a rectangle is 3.5 times its width. The perimeter is 108 cm. find the length and the width.

The U.S. Senate has only Republicans and Democrats. Recently there were 12 more Democrats than Republicans. How many

Senators from each party were there at the time? (you need to know how many Senators there are in the Senate!)

Mixture Problems How many liters of water must be added to 20 L of a 24%

acid solution to make a solution that is 8% acid?

A grocer mixes 5 lb. of egg noodles costing 80 cents per pound with 2 lb of spinach noodles costing $1.50 per pound. What will be the cost per pound of the new mixture?

How many kilograms of water must be evaporated from12 kg of a 5% salt solution to produce a solution that is 30% salt?

A wholesaler has 100 kg of mixed nuts that sell for $4.00/kg. In order to make the price more attractive, she plans to mix in some cheaper nuts worth $3.20/kg. If the wholesaler wants to sell the mixture for $3.40/kg, how many kilograms of the cheaper nuts should be used?

How many liters of water must be added to 50L of a 30% acid solution in order to produce a 20% acid solution?

A spice mixture is 25% thyme. How many grams of thyme must be added to 12 g of mixture to increase the thyme content to 40%?

A farm stand owner mixes apple juice and cranberry juice. How much should he charge if he mixes 8 L of apple juice selling for $0.45/L with 10 L of cranberry juice selling for $1.08/L?

How many liters of water must be evaporated from a 10 L of a 40% salt solution to produce a 50% salt solution?

Fraction and investmentOne third of a number is four more than one fifth of a number. Find the number.

The sum of a number and it’s reciprocal is 2512. Find the number.

The sum of a number and it’s reciprocal is 136 . Find the number.

The sum of the reciprocals of two consecutive odd integers is 815. Find the integers.

The sum of the reciprocals of two consecutive even integers is 1160. Find the integers.

The numerator of a fraction is 1 more than the denominator. If the numerator and the denominator were both increased by 2, the new fraction will be ¼ less than the original fraction.

The numerator of a fraction is 1 less than the denominator. If the numerator and the denominator are both increased by 4, the new fraction witl be 1/8 more than the original fraction. Find the original fraction.

Emily is a real estate broker who earns 12% commission on each house she sells. If she earned $21,600 on a sale of a house, what was the sale price?

The number of students at Westwood High School with a driver’s license is 558. This is 24% more than last year. How many students had driver’s license last year?Jessica invests $6000. Some of the money is invested in stocks paying 6% a year and some in bonds paying 11% year. She receives a total of $550 each year from these investments. How much was invested in stocks and bonds?

Half of Sammy’s money was invested at 12% interest, one third at 11% and the rest at 9%. Her total annual income from investments is $1340. How much money has Sammy invested?

Work Problems.

12

13

1

x

Work Rate formula: Maren can paint a room in 4 h. Her work rate is the part of

the job she can get done in 1 h. So, what is her work rate? One printing press can finish a job in 8 h. The same job would take a

second press 12 h. How long would it take both presses working together?

An installer can carpet a room in 3 h. An assistant takes 4 ½ h to do the same job. If the assistant helps for 1 h and then is called away, how long will it take the installer

to finish?

Work rate Time Work done

Instllr

Assist

Work rate

Time Work done

1st

2nd

A mail handler needs 3 hours to sort an average day’s mail, but with an assistant, it takes 2 h. How long would it take the assistant to sort the mail working alone?

One printing machine works twice as fast as the other. When both machines are used, they can print a magazine in 3 h. How many hours would each machine require to do the job alone

Brad and Angelina are addressing invitations for the 8th grade dance. Brad can address one every 30 sec and Angelina can do one every 40 sec. How long will it take them to address 140 invitations?

Wyatt can load his truck in 24 min. If his brother helps him, it takes them 15 min. to load the truck. How long does it take Wyatt’s brother alone?

Jake can wallpaper a room in 10h and Maura can do it in 8 h. What part of the job can they do working together for 2.5 h and for h hours.

It takes Sally 15 min to pick apples from the tree in her backyard. Lisa can do it in 25 min. How long will it take them working together?

A roofing contractor estimates that he can shingle a house in 20 h and that his assistant can do it in 30 h. How long will it take them to shingle the house working together?

Arthur can do a job in 30 min, Bonnie can do it in 40 min, and Claire can do it in 60 min. How long will it take them if they can work together?

Phyllis can rake our lawn in 50 min, and I can do it in 40 min. If she rakes for 5 min before I join her, how long will it take us to finish?

Pipe A can fill a swimming pool in 12 h. After it has been used for 4.5 h, Pipe B is also used, and the pool is filled in another 4.5 h. How long would it take Pipe B to fill pool by itself?

A mail handler needs 3 h to sort an average day’s mail, but with an assistant it take 2 h. How long would it take the assistant to sort the mail working alone?

Digit Problems

The sum of the digits in a two-digit number is 15. If the digits are reversed, the new number obtained is 9 more than the original. Find the original number.

Let t = the tens digit of the original number Let u = the ones digit of the original number

Tens Units Value

Original # t u 10t+u

Number w/ digits u t 10u+t

reversed

Examples Notes:

A two digit number is four times the sum of the digits. The tens digit is 3 less than the units digit. What is the number?

When the digits of a two digit number are reversed, the new number is 54 less than the original number. If the sum of its digits is 8, what is the original number?

The sum of the digits of a three digit number is 12. The tens digit exceeds the hundreds digit by the same amount that the units digit exceeds the tens digit. If the digits are reversed, the new number exceeds the original number by 198. Find the original number.

The sum of the digits of a two-digit number is one third of the number. The units digit is 5 more than the tens digit. What is the number?

When the digits of a two-digit number are reversed, the new number is 36 more than the original number. The units digit is twice the tens digit. What is the original number?

When the digits of a two digit number are reversed, the new number is 54 less than the original number. If the sum of its digits is 8, what is the original number?

Age Problems Nine years ago Frank was half as old as Alexis. Now his age is two

thirds her age. Find their ages now?Chart:

Nicole is 5 years older than Jamie. Last year she was twice as old as he was. How old is each now?

Chart:

Next year, Lyle will be twice as old as Sean will be. Four years ago, he was three times as old as Sean was. How old is each now?

Chart:

Fraction Problems The denominator of a fraction is 2 more than the numerator. If 3 is

added to both the numerator and denominator, the new fraction is equal to 5/6. Find the original fraction

The denominator of a fraction is 8 more than the numerator. If 3 is added to both the numerator and denominator, the value of the resulting fraction is ½. What is the original Fraction

The two digits in the numerator of a fraction whose value is 4/7 are reversed in its denominator. The reciprocal of the fraction is the value obtained when 16 is added to the original numerator and 5 is subtracted from the original denominator. Find the original fraction.

The two digits in the numerator of a fraction are reversed in its denominator. If 1 is subtracted from both the numerator and denominator, the value of the resulting fraction is ½ . The fraction whose numerator is the difference and whose denominator is the sum of the units and the tens digit equals 2/5. Find the original fraction.

Cecilia is 24 years younger than Joe. Six years ago, she was half as old as he was. How old is each now?

Steve is three times as old as Theresa. If four years he will be twice as old as she will be. How old is each now?

Seven years ago, my father’s age was five times mine. Right now, his age is the sum of our two ages 7 years ago. How old am I?

The denominator of a fraction is 9 more than the numerator. If the numerator is decreased by 3 and the denominator is increased by 3, the value of the resulting fraction is ¼. What is the original fraction?

If the numerator of a fraction is increased by 4, the value of the resulting fraction is ¾. If the denominator of the original fraction is increased by 2, the values of the resulting fraction is ½. Find the original fraction.

Next year, Lyle will be twice as old as Sean will be. Four years ago, he was three times as old as Sean was. How old is each now?

Mary is three years older than her twin brothers. Next year, the sum of the ages of the three will be exactly 102. How old are they now?

The sum of three digits is 12. The tens digit exceeds the hundreds digit by the same amount that the units digit exceeds the tens digit. If the digits are reversed, the new number exceeds the original number by 198. Find the original number.

The two digits in the numerator of a fraction whose value is 4/7 are reversed when 16 is added to the original numerator and 5 is subtracted from the original denominator. Find the original fraction.