Equations and Inequalities - San Dieguito Union High ...teachers.sduhsd.net/slesan/Math A Honors/Mod...

HOMEWORK KEY –

SELECTED ANSWERS

Equations and Inequalities

Math A Honors

Module #4

2015 - 2016

Created in collaboration with Utah Middle School Math Project

A University of Utah Partnership Project

San Dieguito Union High School District

SDUHSD Math A Honors Module #4 – TEACHER EDITION 2

4.1A Homework: Model and Solve Equations* Name: Period:

1 =

x =

-1 =

-x =

Use Algebra Tiles to model and solve the following equations. Show your algebraic process in the right column. Write all fractional answers in simplest form. Be sure to check your solutions using tiles.

1.

=

x 6 = 9

=

x 6+6 = 9+6

Additive Inverse

=

x = -3

Be sure to check using tiles


2.

=

–15 = x – 14

=

=

3.

=

m + 2 = –11

=

m + 2 - 2 = -11 - 2

Additive Inverse

=

m = -13



4. 4n = –12

=

=

=

5. –15 = –3m

=

–15 = –3m

=

(–15) = – 3m

Multiplicative Inverse

=

=

5 = m or

m = 5

TEACHER NOTE: To explain this last move, you

may want to add and 5 to both sides of the equation to avoid flipping tiles.



6.

=

.

7.

=

x = -6



Solve the following equations without the use of a model. Write all fractional answers in simplest form. Be sure to check your solutions.

8.

9.

10. 11.

12.

13.

14.

15.


4.1B Homework: More Modeling and Solving One- and Two-Step Equations* Name: Period:

Students can show checks on the back of an available page. Assignment is long. Consider shortening. Use Algebra Tiles to model and solve the following equations. Show your algebraic process in the right column. Write all fractional answers in simplest form. Be sure to check your solutions using tiles.

1.

Show work x = –8


2.


3.

x = 2

4.


5.

6.



7.

8.


9.

10.

11.

12.


13.

c =

14.


4.1C Homework: Create Equations for Word Problems and Solve* Name: Period:

Determine which of the equations will work to answer the question. Explain your reasoning. 1. Three pounds of fruit snacks cost $4.25. How much does one pound of fruit snacks cost?

What is the unknown/variable? Let y = the cost of one pound of fruit snacks

Equation Will it work? Explain in words what each equation is actually representing.

A y + 3 = 4.25 No This says the cost of one pound of fruit snacks plus 3 pounds is

$4.25.

B 3y = 4.25

C y + y + y = 4.25 Yes This shows adding the cost of one pound of fruit snack three times

would result in the total cost of $4.25.

D 4.25 = 3y

Define each variable. Write and solve an algebraic equation. Write your answer in a complete sentence. 2. The sum of a number and its double is eighteen. What is the number?

3. The $5 bill was worth $3 more than the cost of the notebook. How much did the notebook cost? Let n = the cost of the notebook 5 = n + 3 n = 2 The notebook costs $2.


4. Bill has twice as much money as I do. Together we have $9. How much money do I have?

5. A large popcorn and a drink together cost the same as the movie ticket. I spent a total of $10 for my night at the movies. How much did my ticket cost? Let t = the cost of a movie ticket t + t = 10

The cost of a movie ticket (t) is $5. 6. A 109-cm long board is to be cut into two pieces so that one part is 9 cm more than 4 times as long as

the shorter part. Find the length of each part.

7. Sally worked 7 more than 5 times as many hours as Sid did. How many hours did each work if

together they worked 97 hours? Sid: 15 hours and Sally: 82 hours


8. A pile of money totaling $224 was divided among 3 people so that the second person received $1 less

than double the first person, and the third received $11 more than the second. How much did each person receive?

9. The sum of the ages of Greta and her mother Olga is 76 years. The difference in their ages is 32

years. How old are they? Greta: 22 and Olga: 54

10. A pile of 180 marshmallows was divided among 3 people so that the second person received 6 less

than double the first person, and the third received 7 more than the second. How many marshmallows did each person receive?


4.1D Homework: Practice Solving Equations* Name: Period:


1. n = 13

2.

3. x= 3

4.

5. c = 4.5

6.

7.

8.

9. n = -120


Students in Mr. Timon’s class were making frequent errors in solving equations. Help analyze their errors. Examine the problems below.

A) Circle the line in which the mistake first occurs. B) Explain the mistake. Be specific. C) Solve the equation correctly showing all of the steps. Be prepared to present your thinking in

class.

10. A.

B.

C.

11. A. 3(2x + 1) + 4 = 10

B. The student added terms that were not like

terms.

C.


4.1E Homework: Solving Equations with Variables on Both Sides Name: Period:


1.

2.

3. x = 1

4.

5.

6.

7. No solution 8.

9. x = 0

10.


4.1F Homework: Clearing Fractions and Decimals in Equations Name: Period:


1.

2.

3.

4.

5. Infinitely many solutions 6.

7.

8.

9.

10.


4.1G Homework: Applications of Equations* Name: Period:

Define each variable. Write and solve an algebraic equation. Write your answer in a complete sentence. 1. Harry spent his birthday money on a flying lesson. Margo spent $75 more than Harry. Together they

spent $143. How much did Harry get for his birthday? Let x = the amount Harry spent x + 75 = the amount Margo spent x + x + 75 = 143 Harry received $34 for his birthday.

2. The Spanish Club raised $131.25 by having a garage sale and car wash. The amount raised at the

car wash was half the amount raised at the garage sale. How much was raised at each event?

3. Dan has saved 235 trading stamps. He needs a total of 685 stamps to get a pair of Rollerblades. If he

gets 10 stamps for every dollar he spends at the supermarket, how much must he spend to earn enough stamps for the Rollerblades? Let x = the amount Dan needs to spend 10x + 235 = 685 Dan needs to spend $45 at the supermarket to get his rollerblades.

4. A piece of wire 60 inches long is cut into 2 pieces so that the larger piece is 10 inches longer than the shorter piece. How long is each piece of wire?


5. After using of a bag of fertilizer on his garden, Ken gave 8 pounds to a neighbor. If he had 42 pounds left, how much had been in the full bag? Let x = the amount of fertilizer in a full bag

6. A certain number of minutes divided by sixty gives us 3 hours. How many minutes did we start with?

7. When the gas gauge on Lucy’s car was on the mark, Lucy pumped 15 gallons of gas into the tank in order to fill it. How many gallons of gas does the tank in Lucy’s car hold? Let x = the total number of gallons the tank holds


Section 4.1 Review* Name: Period:

Read each set of directions carefully. You must show ALL of your work to earn credit.

Key for Tiles: = 1

= x

= –1

= –x

Use Algebra Tiles to model and solve the following equations. Show your algebraic process in the right column. Write all fractional answers in simplest form. Be sure to check your solutions using tiles.

1.

n = 6

2.

– X

X 1

-1



3.

4.

5. x = -3

6.

7.

8.

9.

10.


11.

12.

13.

14.

15. Which of the following word problems can be solved using the equation ? Circle all that apply. a. Julio has 3 erasers. He buys more erasers at a store that sells erasers in packs of 6. How many

packs, x, of erasers does Julio buy if he ends up with 24 erasers?

b. A bug crawls away from a wall at a constant rate of 3 inches per minute. If the bug is already 6

inches away from the wall, how many minutes, x, would it take the bug to be 24 inches away from

the wall?

c. A class reading list contains 24 books. Renee has read 6 of the books and is planning on reading

the same number of books each month for 3 months. How many books, x, does Renee need to

read each month to complete the reading list?

16. Jeanie buys 4 pairs of jeans at x dollars each and a t-shirt for $5.99. Jeanie states that the total cost, before taxes, of her purchases is $77.95. Write an equation to determine the value of x, solve, and write the answer as a complete sentence.


17. At a bookstore, Mrs. McIntyre buys three books, one for each of her children. Each book costs the

same amount. She also buys a $1.50 magazine for herself. Her purchases have a total price of

$13.47. The equation represents this situation.

a. Solve this equation for x. x = 3.99

b. Explain what this value of x means in terms of the bookstore purchases.(Define the variable.) x represents the cost of a book at the bookstore.

18. Frieda writes the expression: Frieda states that the value of her expression is -8. Determine the value of x that makes Frieda’s statement true and explain your reasoning.

19. Determine which equation(s) has No Solution: a. 3(x + 4) = x + 2(x + 5) + 2 Infinitely Many Solutions

b.

c. 2 + x + 4 = 3x + 8 - 2x No Solution

d. 2x + 6 = 18 x = 6


4.2A Homework: Review of Inequality Statements* Name: Period:

Define a variable. Write an inequality for each statement below. 1. Children younger than age 5 can get in free.

Let x = the age children can get in for free x < 5

2. To join the FBI, you must be at least 23, but younger than 37 years old.

3. To run the art class, they must have no less than 12 participants registered.

x is the number of participants to run the art class.

4. For a healthy diet a person should eat more than1200 calories and at most 2000 calories daily.

5. To run the track at the community gym, you must be at least 16 years old. x is the age to run the track.

Graph each solution set on the provided number line.

6.

7.

8.

9.


Write an inequality for the solution set shown on the number line. 10.

11.

12.

For each problem below, graph both statements on the same number line. What integer values for x make both statements true at the same time?

13. and Students should be able to list values that make

true.

14. and

15. and No value will make both statements true at the same time.

Spiral Review: 16. There are a total of 128 cars and trucks on a lot. If there are four less than twice the number of trucks

than cars, exactly how many cars and how many trucks are on the lot? Be sure to define your variable, write an equation, solve, and answer in a complete sentence.


17. Place the fractions on the number line below.

18. A mouse can travel 1.5 miles in ¾ of an hour. Write an equation showing how far it travels ( ) for any

given time ( ).

19. Solve:

1.28 20. Fill in the equivalent fraction and percent for this decimal:

Fraction Decimal Percent

0.8


4.2B Homework: One-Step Inequalities Review* Name: Period:

Find the solution set for each inequality showing ALL steps. Graph the solution set. Remember to write fractional answers in simplest form and scale the number lines appropriately.

1.

2.

3.

4.

5.


6.

7.

8.

9.

10.


4.2C Homework: Solve and Graph Multi-Step Inequalities* Name: Period:


1.

2.

3.


4.

5.

6.

7.

Solve and graph each inequality. Write fractional answers in simplest form.


8. 9.

10.

11. d = 99

12.

13. n = 101


14.

15.


4.2D Homework: Solving Inequalities with Variables on Both Sides* Name: Period:


1.

2.

3.

4.


5.

6.

Spiral Review: Solve each equation. Write fractional answers in simplest form.

7.

8.


9.

10.

11. Solve:



Define each variable. Write and solve an inequality. Write your answer in a complete sentence.

1. A popular pizza restaurant charges a flat rate of $60 for a birthday party plus $2.25 for each person. Juan can’t spend more than $120. How many friends can he invite?

He can invite 26 friends.

2. Vicki wants to play a video game that charges you $0.12 per minute. If she has $15 to spend, how many minutes can she play at most?

3. Yellow Cab Taxi charges a flat rate of $3.50 for every cab ride, plus $0.95 per mile. Tofi needs a ride from the airport. He only has $30 cash. How many miles can he go?

, He can go 27 miles.

Determine if each statement below is always true, sometimes true, or never true. Explain your reasoning.

4. If and , then .

5. If and , then .

Sometimes true because and , which means when and , . (This is just one example, other examples work too.)



6.

7.

8.

9.

10.


11.

12.

13.

14.


15.


16.

17.

18.

Determine if each statement below is always true, sometimes true, or never true. Explain your reasoning.

19. If , then x is a positive number. Sometimes true because the solution to the

inequality is .

20. If x is a positive number, then .

21. Determine whether the statement is true or false. If false, give a counterexample.

, when and ,

False, 6 < 10, but .

22. If and , what can we conclude about and ?


Spiral Review: 23. The spinner to the right contains three letters of the alphabet.

a. How many outcomes are possible if the spinner is spun three times?

b. List the sample set for spinning three times. You may make a tree diagram or array on the opposite page. AAA, AAB, AAC, ABA, ABB, ABC, ACC, ACB, ACA, BBB, BBA, BBC, BAA, BAB, BAC, BCB, BCA, BCC, CCC, CCA, CCB, CBA, CBB, CBC, CAA, CAB, CAC

c. What is the probability of getting exactly one A in three spins? Write as a fraction, decimal, and percent.

24. Roll a pair of 6-sided dice. Draw a sums chart below (or on the back of the previous page) and answer

the questions that follow. a. The probability the sum is a multiple of 8.

b. The probability the sum is a multiple of 3.

c. Probability the sum is a multiple of 2.

A BV C

H


4.3A Homework: Write and Solve Equations and Inequalities for Word Problems* Name: Period:

Define a variable. Write an equation or inequality. Solve. State your answer in a complete sentence. 1. Kimberly is taking her 6 nieces and nephews to

a hockey game. She wants to buy them snacks. What is the most she can spend on snacks for each child if Kimberly wants to spend no more than $33 in total? Let m = the amount of money Kimberly can spend

Kimberly can spend at most $5.50 on each child.

2. The school is having a carnival fundraiser. Tickets sell for $0.50 each. They are planning on buying supplies for the carnival that cost $50. How many tickets must they sell to raise at least $200?

3. Billy needs to read 300 minutes this week for his English class. He is going to read 6 days. If he already reads 15 minutes every day, how many additional minutes does he need each day to read at least 300 minutes? Let m = the number of additional minutes Billy needs to read each day

Billy needs to read at least 35 minutes more each day.

4. Erin is buying cupcakes for her birthday party. Each cupcake costs $1.50. What is the maximum number of guests can she invite if her budget is $80 and she has already spent $16 on paper cups and plates? By the way, Erin thinks that each guest will want two cupcakes to eat but as the birthday girl she gets her own cupcakes for free.


5. Lauren got money from various relatives on her birthday. She put 20% of the money into her savings account, and plans to spend at least $256.80 on new clothes. How much money did she get from her relatives? Let m = the amount of money Lauren got from her relatives

Lauren got at least $321 from her relatives.

6. Mrs. Brown is ordering pictures of her new baby. There is a $20 sitting fee and each 5x7 portrait she orders is $4. She also has a coupon for $10 off. If she wants to spend less than $50, how many 5x7 portraits can she order?

Spiral Review: 7. Solve the following inequality:

8. Solve the following equation.

9. There are 36 red and 44 blue marbles in a bag. What is the probability of randomly drawing a red

marble?


10. State whether the inequality is always, sometimes, or never true. Justify your reasoning.

when and

11. Give an example of a whole number that is not natural number.

0 12. Give an example of an integer that is also a whole number.


4.3B Homework: Review of Integer, Separation, Age Word Problems * Name: Period:

Define each unknown using one variable. Write an algebraic equation or inequality. Solve and state your answer in a complete sentence. 1. The sum of three consecutive integers is 53 more than the least of the integers. Find the integers.

x + x + 1 + x + 2 = x + 53 25, 26, 27

2. The sum of three consecutive even integers is at most 139. Find the greatest set of integers.

3. The sum of three consecutive odd integers is 17 less than four times the smallest integer. Find the integers. x + x+ 2 + x + 4 = 4x - 17 23, 25, 27

4. The sum of three consecutive odd integers is -219. Find the integers.

5. Michal’s age is 10 years more than twice Talia’s age. The sum of their ages is 64. How old is each person? Talia: 18 Michal: 46


6. The sum of two numbers is 16. The greater of the two numbers is one more than four times the lesser number. What are the numbers?

7. The number of boys in the 7th grade is the number of girls. If there are 360 boys and girls altogether in the 7th grade, how many girls are there? 200 girls

8. Todd’s age is ten years greater than half Ali’s age. If the sum of their ages is 55, how old is Todd?

9. Last week, Amy sold three times as many sports magazines as cooking magazines and five fewer news magazines than sports magazines. If Amy sold 58 magazines altogether, how many of each type did she sell? 9 cooking 27 sports 22 news

Spiral Review: 10. One added to the quotient of a number and two is thirty-four. Find the number.

11. Write an expression to model the following situation in two different ways: The price of the car was

reduced by 15%.

Possible answers include: 12. Juliana bought 3 bags of chips and 3 sodas for herself and two friends. The chips were $0.85 a bag.

Define a variable and write an equation to find the price of each can of soda if she spent a total of $6, then solve.

13. Simplify the following expression:

14. Solve:


4.3C Warm Up: Ages & Integers

1. Doris is 3 years older than Jennifer. The sum of their ages is 29. What are their ages?

2. Terry is 12 years old. Her mother is 40. How many years ago was her mother exactly 5 times as old as Terry?

3. The sum of four consecutive even integers equals seven times the greatest of the integers. Find the integers.


4.3C Warm Up: Ages & Integers – Answer Key

1. Doris is 3 years older than Jennifer. The sum of their ages is 29. What are their ages? Jennifer: 13 Doris: 16

2. Terry is 12 years old. Her mother is 40. How many years ago was her mother exactly 5 times as old as Terry? 5 years ago

TEACHER NOTE: Students will most likely problem solve their way through this and not set-up an equation. This could lead to a good discussion.

3. The sum of four consecutive even integers equals seven times the greatest of the integers. Find the integers. -10, -8, -6, and -4


4.3C Lesson: Word Problems with Multiple Unknowns Part II* Name: Period:

Define each unknown using one variable. Write an algebraic equation or inequality. Solve and state your answer in a complete sentence. 1. At the store, you find a pair of jeans and a t-shirt. Together, they’ll cost $80.20. The jeans cost three

times the cost of the t-shirt. How much does each cost? Let t = the cost of the t-shirt 3t = cost of jeans

jeans: $60.15 t-shirt: $20.05

2. Jan bought 2 shirts (same style and cost but different colors) and 2 pairs of pants (same style and cost but different colors). Each shirt was $3 less than a pair of pants. She spent $49.80 (before tax). What is the price of a shirt before tax? What is the price of a pair of pants? Let p = price of a pair of pants p – 3 = price of a shirt

Shirt: $10.95 Pants: $13.95

3. Dolores has at most $78.50 to spend on clothes. She wants to buy a pair of jeans for $26 and spend the rest on shirts. If each shirt costs $7, how many shirts can she buy? Let x = number of shirts she can buy

7 shirts

4. Saul had $30 in his wallet. He spent $18.01 including tax to buy a DVD. He needs to save $9 and he wants to buy a snack for his friends. If almonds cost 25 cents per package including tax, what is the maximum number of packages he can buy? Let x = number of packages of almonds

11 packages

5. Bob, Chuck, and Wayne combined their funds to purchase an arcade. They invested a total of $677,000. Bob invested five times as much as Chuck but $17,000 less than Wayne. How much did each person invest? Let c = the amount Chuck invested 5c = the amount Bob invested 5c + 17000 = the amount Wayne invested

Chuck invested $60,000, Bob invested $300,000 and Wayne invested $317,000.


6. Two consecutive integers have the property that five times the smaller integer is less than four times the greater integer. Find the greatest pair of integers with this property. Let x = the smaller number x + 1 = the larger number

The greatest pair of integers are 3 and 4.

7. The number of employees who use public transportation to commute to work is 200 more than twice the number of employees who drive their own cars. If there are 1400 employees, how many drive their own cars? Let x = the number of employees who drive their own cars 2x + 200 = the number of employees who take public transportation

400 employees

8. A certain molecule contains twice as many atoms of hydrogen as oxygen and one more atom of carbon than hydrogen. If there are 21 atoms altogether in the molecule, how many atoms of carbon are there? 9 atoms

9. Constellations are composed of a different number of stars. Twenty more stars are visible in Cancer

than in Aries. Sagittarius has twice as many visible stars as Cancer, and Taurus has twelve more visible stars than Sagittarius. If the number of visible stars in Aries is ten greater than one third the number of visible stars in Taurus, how many stars are visible in Cancer? 102 stars


4.3C Homework: Word Problems With Multiple Unknowns Part II* Name: Period:

Define each unknown using one variable. Write an algebraic equation or inequality. Solve and state your answer in a complete sentence. 1. At the local clothing store all shirts were on sale for the same price and sweaters for a different price.

Lonnie purchased three sweaters and two shirts for a total of $130. If the sale price of a shirt was five dollars less than the sale price of a sweater, how much did each item cost Lonnie? Let x = cost of a sweater x – 5 = cost of a shirt

x = 28; sweater costs $28; shirt costs $23

2. Cassie and Tom wanted hamburgers for lunch. Cassie ordered a hamburger for $4 and an order of fries. Tom ordered twice Cassie’s order. The total price was $16.65 (before tax). What is the cost of one order of fries before tax?

3. Lupe and Carlos work in an office. Carlos makes $16,000 less than twice Lupe’s salary. The sum of their two salaries is $104,000. How much are their salaries? Let x = Lupe’s salary 2x – 16000 = Carlos’ salary

Lupe’s salary is $40,000, Carlos’s salary is $64,000


4. A pair of consecutive integers has the property that 8 times the lesser is more than 4 times the greater. Find the least pair of integers with this property.

5. The Panama Canal is 2 km shorter than twice the length of the Suez Canal. The sum of the lengths of

the two canals is 121 km greater than the length of the Panama Canal. Find the length of each canal. Let z = length of the Suez Canal 2z – 2 = the length of the Panama Canal

Panama Canal: 160 km Suez Canal: 81 km

6. Taylor is using money from her coin jar to go to the fair. She wants to spend no more than $17 there. She has $8 in bills and some quarters to spend. At most, how many quarters does Taylor have to spend?


Spiral Review:

7. Solve:

8. Phoebe has $10. Define a variable and write an expression showing how much money she will have

left after buying 3 candy bars and a pack of pencils, if the pack of pencils is two dollars less than the cost of a candy bar. Simplify the expression.

9. Simplify:

10. Find each sum without a model.

a.

b.

11. I go to a department store with a coupon for 20% off any one item. The shoes that I want are already on sale for 40% off. Find the original price if I paid $48. Show all of your work.

Let the original price of the shoes

$100


4.3D Homework: Perimeter & Word Problems with Equations and Inequalities* Name: Period:

Define each unknown using one variable. Write an algebraic equation or inequality. Solve and state your answer in a complete sentence. 1. Container A and Container B have leaks. Container A has 800 ml of water and is leaking 6 ml per

minute. Container B has 1000 ml, and is leaking 10 ml per minute. How many minutes will it take for Container B to have less water than Container A?

After 50 minutes Container B will have less water than Container A.

2. A new giraffe weighs 150 pounds and is gaining four pounds each month. A baby rhinoceros weighs 120 pounds and is gaining ten pounds each month. a. How many months will it take for the rhinoceros to weigh more than the giraffe?

b. How many months will it take for the giraffe and the rhinoceros to weight the same amount?

3. The width of a certain rectangle is 2 m greater than half its length. Four times its length is 26 m greater than its perimeter. What are the dimensions of the rectangle? x = the length of the rectangle 0.5x + 2 = the width of the rectangle 4x – 26 = the perimeter of the rectangle

The length is 30 m and the width is 17 m.

4. The length of a rectangle is 20 m greater than its width. If the width were reduced by 20 m and the length increased by 100 m, the perimeter of the new rectangle would be twice the perimeter of the original rectangle. What are the dimensions of the original rectangle?


5. Which of the following word problems can be solved using the equation ? Select all that apply.

a. Four friends share a box of cookies. After each friend receives cookies, there are still 7 cookies left in the box. If there were initially 90 cookies in the box, how many cookies did each friend receive?

b. A student has four bags of marbles. Each bag originally has marbles. After adding 7 marbles to each bag, there is a total of 90 marbles in all. How many marbles were originally in each bag?

c. A student has four bags of marbles. Each bag originally has marbles. After taking out 7 marbles from each bag, there is a total of 90 marbles in the bags. How many marbles were originally in each bag?

d. A square has a side length of inches. Each side of the square will be increased by 7 inches to create a larger square. If the larger square has a perimeter of 90 inches, what is the side length, in inches, of the original square?

Spiral Review: 6. Solve:

a.

b.

7. Write an expression to represent the following situation: Danielle is having a birthday party and is

inviting 8 friends. She wants to give each friend a gift bag with a large candy bar and a notebook. Show the total price of the complete gift bag if each large candy bar costs $1.40.

8. Use the distributive property to simplify the following expression:

9. Express each percent as a fraction in simplest form.

a. 44%

b. 17.5%


4.3E Homework: Percents with Models and Equations* Name: Period:

Draw a model to help you solve the problems below. Then choose the algebraic equation(s) that would represent the problem. Solve the equation and answer in a complete sentence. 1. To get an A in math class, I need to get a 90% on the test. If the test has 40 questions, how many do I

need to get right in order to get an A?

a. Model:

b. Choose the appropriate equation(s). Justify your choice.

r = 36 questions correct 2. Only 25% of the Chess Club came to the meeting. 3 people were at the meeting. How many people

are in the club?

a. Model:


3. 32 of the 48 people at the gym are wearing blue. What percent of these people, given as a decimal,

are wearing blue?

a. Model:


b = 0.66… so 66 % are wearing blue


Draw a model (if needed) for each, define a variable for the unknown, and then write an algebraic equation to solve each percent problem. State your answer in a complete sentence. 4. 65% of the population needs to vote for the new law in order for it to pass. There are 800 voters. How

many need to vote for the new law in order for it to pass?

5. 5% of the apples have worms in them. 10 apples had worms in them. How many apples are there

total?

w(0.05) = 10 w = 200 total apples 6. 250 students dressed up for Spirit Day. There are 800 students. What percent dressed up for Spirit

Day?

7. After a 16% increase, the price of a pair of running shoes was $40.60. What was the old price of the

shoes?

$35

8. This year, Rainy Realty sold 289 homes. Last year they sold 340 homes. If sales decrease by the

same percent next year, many homes can Rainy Realty expect to sell?

9. In July, the price of a gallon of gasoline rose 12%. In August it fell 15% of its final July price, ending

the month at $1.19. What was its price at the beginning of July?

$1.25


4.3F Homework: Percent Problems* Name: Period:

1. Dean took his friend to lunch last week. His total bill was b dollars. He wants to tip the waitress 20%.

How much will Dean pay, including the 20% tip? (Don’t worry about tax in this problem.)

a. Draw a model to represent this situation.

b. Which of the following expressions represent the total amount that Dean will pay?

0.8b b + 0.2b 1.2b 1.8b 1 + 0. 2b

c. If Dean’s bill was $22.50, how much will Dean pay including the 20% tip? Show at least two

different ways to get your answer. Solve and answer in a complete sentence

$27

22.50(1.2) = p 22.50 + 22.50(0.2) = p 22.50(1 + 0.2) = p

2. Philip took a vocabulary test and missed 38% of the problems. There were q problems on the test.

a. Draw a model to represent this situation.

b. Which of the following expressions represent the number of problems that Philip got correct?

0.38q q – 0.38 q – 0.38q 0.62q (1 – 0.38)q

c. Which of the following expressions represent the number of problems that Philip missed?

0.38q 1q – 0.62q q – 0.38q 0.62q (1 – 0.38)q

d. If there were 150 problems on the test, how many did Philip get correct? Show at least two

different ways to get your answer. Solve and answer in a complete sentence.


3. Addie buys a book with an original price of x dollars. The book is on sale for 30% off the regular price.

a. Draw a model to represent this situation:

b. Write an expression to represent the amount she will pay. Remember to define your variable.

Let x = the original price of the book

0.7x

c. If the original price of the book was $15, what is the sale price?

Let y = the sale price of the book

0.7(15) = y

The book was on sale for $10.50.

Define a variable for the unknown, and write an algebraic equation to solve each percent problem.

State your answer in a complete sentence.

4. A group of friends go out to dinner. There is a 6% sales tax. If the bill is $89.04 after tax, how much was the original bill?

5. There were 950 students at Colt Middle School last year. The student population is expected to

increase by 40% next year. What will the new population be?

Let x = the student population next year

1.40(950) = x

The population next year will be 1330 students.


6. Chris would like to buy a picture frame for her brother’s birthday. She has a lot of coupons but is not

sure which one to use. Her first coupon is for 50% off of the original price of one item. Normally, she

would use this coupon. However, there is a promotion this week and the frame is selling for 30% off,

and she has a coupon for an additional 20% off any frame at regular or sale price. Which coupon will

get her the lower price? She is not allowed to combine the 50% off coupon with the 20% off coupon.

a. Draw a model to show the two different options.

Option 1: 50% off coupon:

Option 2: 30% off sale with additional 20% off coupon:

b. Let x represent the original price of the picture frame. Write two different expressions for each

option.

50% off coupon

30% off sale with additional 20% off coupon:

c. Which coupon will get her the lowest price? Explain how you know your answer is correct.

Define a variable for the unknown, and write an algebraic equation to solve each percent problem.

State your answer in a complete sentence. Round answers to the hundredth place value, when

necessary.

7. Carter gave the waitress a tip of $8.75. If the original price of his meal was $24.95, what percent of the

price was the tip? Let t = % of tip 24.95t = 8.75 t = 0.3507 Carter gave a 35% tip.

8. Louie paid a total of $12.55, with tax, for his new frying pan. If the original price of the pan was $11.95, what was the tax rate?


9. Robert paid $14.41, with tax, for his model airplane kit. If tax was 6%, what was the original price of

the kit? Let p = original price (1.06)p = 14.41 The original price of the kit was $13.59.

10. Two jewelry stores buy silver chains from a manufacturer for c dollars each, and then sell the chains at

a 57% markup. Store A has a sale and marks down all chains by 20% off retail. In addition, customers can use a coupon worth 15% off the price of any item, including sale items. Store B offers a coupon worth 35% off any one item.

a. At Store A, Aura used a 15%-off coupon to buy a chain already marked down by 20%. Write an

expression for the price of this chain.

b. At Store B, Tucker used a 35%-off coupon to buy a chain. Write an expression for the price of this chain.

c. Which store offers a better price on chains?



Read each set of directions carefully. You must show all of your work to earn credit. Be sure to write all fraction answers in simplest form. Complete on separate paper. Solve each equation. Show all algebraic steps.

1.

2.

3.

4.

5.

6.

7. No Solution

8.


9.

10.

11. 12.

Choose the equation that could represent each problem. Be ready to justify your answer in class. 13. Ashley went to the store, bought 3 dozen eggs and spent a total of $3.39. How much did each dozen

eggs cost?

a. b. c. d.

14. Sofas were on sale for 25% off their original price of $450. What is the sale price of each sofa?

a. b. c. d.

15. Students were asked to solve the equation . Each student described what his/her first step would be to solve the equation. Circle which student(s) described a possible correct first step, then solve the equation using one of the correct first steps you chose.

a. Toby: Subtract 4 from both sides of the equation.

b. Mario: Use multiplication to distribute to .

c. Andrea: Subtract from both sides of the equation.

d. Matt: Multiply both the and the 10 by 2 to clear the fraction.

e. Olivia: Divide both the and the 10 by 5 to clear the fraction.

Solve


Define each variable. Write and solve an algebraic equation. Write your answer in a complete sentence. 16. In a survey, four-fifths of the people questioned preferred orange juice to apple juice. If 45 people

preferred apple juice, how many total people were questioned?

17. Eve bought two books yesterday, a comedy and a drama. The price of the comedy was five more than

3 times the price of the drama. If the sum of the prices was $55, what was the price of each book? Let x = the price of the drama 3x + 5 = the price of the comedy x + 3x + 5 = 55 The drama cost $12.50 and the comedy cost $42.50.

18. For your birthday, you plan to buy 2 cupcakes for each guest and 6 additional cupcakes to have as

extras. The cupcakes cost $2.50 each. How many guests can you invite if you have $90 to spend? Since it is your birthday, you get your cupcakes for free and are not included in the guest count. (Don’t worry about tax for this problem.)

19. In a state with a 5% sales tax, a tennis racket cost $60.85, including tax. What was the selling price of the racket before tax? (Round to the hundredths place.) Let x = the selling price of the racket 1.05x = 60.85 The selling price of the tennis racket is about $57.95 before tax.

20. You paid a total of $48.75 for the jeans you’ve been wanting. They were on sale for 25% off of their

original price. What was the original price?

21. Stacy bought a photo of herself and her friends that measured 6 inches wide. She wanted it to fit into a

bigger frame, so she had it enlarged by 20%. The new print still wasn’t big enough, so she enlarged the new photo by another 20%. What is the width of the twice enlarged photo? Let x = the new width of the photo 1.20[1.20(6)] = x The new photo is 8.674 inches wide.

22. On the LCC football team, one-third of the players walk to practice and 40% are driven by their

parents. The remaining 16 players take the bus. How many members are on the football team?


23. A buyer for a computer store paid $194 each for printers. He wants to sell them at a 25% mark up.

What will be the selling price of the printer? Let x = the new price of the printer 1.25(194) = x The printer now sells for $242.50.

24. At a toy shop, 45 toys were sold in September. A bonus will be given to workers if they can increase

sales by 40% in October. How many toys must the sales team sell in October to receive the bonus?

25. Mrs. Noah’s bill, before tax, is $50. The sales tax rate is 3%. Mrs. Noah decides to leave a 2% tip for

the gate keeper based on the pre-tax amount. What will Mrs. Noah’s total bill be, including tip and tax, if she included the tip when figuring out the tax? Let x = the amount after tip Let y = the amount after tax This could also be completed as one 1.02(50) = x 1.03(51) = y equation if x = price after tip and tax: Mrs. Noah’s final bill with tip and tax will be $52.53. 1.03[1.02(50)] = x

26. Sabrina buys a jacket wholesale for $50. She sells the jacket for $56. What is Sabrina’s percent

markup?

27. An Xbox went on sale on Monday for 25% off the original price. On Tuesday, the store decided to

increase the price 10% to $330. What was the original price of the Xbox before the sale on Monday? Let x = the cost on Tuesday Let y = the cost on Monday 1.10x = 330 .75y = 300 The original price of the Xbox was $400.

28. Matt, Rosa, and Kathy are cousins. If you combine their ages the sum would be 40. Matt is one-third

of Rosa's age. Kathy is five years older than Rosa. How old are they?


29. Kim is 3 years older than Bridget. The sum of their ages is less than 16. What is the oldest Bridget

could be? Let x = Bridget’s age x + 3 = Kim’s age

Bridget could be at most 6 years old.

30. The sum of two numbers is 98. The larger number is one less than twice the smaller number. Find

both numbers.

31. The sum of two consecutive integers is no more than 90. What is the greatest possible pair? Let x = the smaller integer x + 1 = the larger integer

The greatest possible pair would be 44 and 45.

32. Loralie is having a birthday. She wants to bring treats for her friends at school. Her mom gives her $20.

How much can she spend per friend if she wants to bring treats for herself and eight friends?

33. Choose which expression(s) below is (are) equivalent to

a.

b.

c.

d.

e.

f.

34. Each choice below shows an equation and a solution for . Which of the following is/are true? Select all that apply.

a. If , then .

b. If , then .

c. If , then .

d. If , then .


e. If , then .

f. If , then .


35. Which of the following word problems can be solved using the equation ? Select all that apply.

a. Kit buys 10 packages of batteries. After using a $15 off coupon, the batteries cost $60. What was

the original price , in dollars, of each package of batteries?

b. Ursula bought 10 calculators at a store and paid $60. Each calculator was on sale for $15 off.

What was the original price , in dollars, of each calculator?

c. Siobhan originally has $15. After working as a waiter in a restaurant for $10 an hour, he now has

$60. How many ours did Siobhan work at the restaurant?

d. Gary worked for 10 hours tutoring students at the library. He uses $15 to pay for gas on his way

home. If he has $60 left after paying for gas, how much money in dollars was Gary paid per hour?

36. A teacher asks five different students to write an equation with a solution of . The students and their equations are shown below.

Student Equation

Adrian

Bert

Camille

Daniel

Etsuko

Which of the following students correctly wrote an equation with as the solution? Select all that apply. a. Adrian b. Bert c. Camille d. Daniel e. Etsuko

CHALLENGE (Yes, you need to try it!): Tom bought several appliances and a new car. He paid a sales tax of 7.5% on the appliances and a tax of 6.5% on the car. Before these taxes, the appliances and car together cost $15,200. If he paid a total of $1015 in taxes, how much did the car cost?

Equations and Inequalities - San Dieguito Union High ...teachers.sduhsd.net/slesan/Math A Honors/Mod...

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