Elementary Algebra 2e · free preview

Chapter 3 Review and Practice Test

Learning objectives: Review the key ideas of this chapter.; Apply the chapter's methods to review exercises.; Check your understanding with a practice test.

1. Learning objectives

Learning objectives

Review the key ideas of this chapter.

Apply the chapter's methods to review exercises.

Check your understanding with a practice test.

2. Worked example

Worked example

amount of discount

The amount of discount is the amount resulting when a discount rate is multiplied by the original price of an item.

discount rate

The discount rate is the percent used to determine the amount of a discount, common in retail settings.

interest

Interest is the money that a bank pays its customers for keeping their money in the bank.

list price

The list price is the price a retailer sells an item for.

mark-up

A mark-up is a percentage of the original cost used to increase the price of an item.

mixture problems

Mixture problems combine two or more items with different values together.

original cost

The original cost in a retail setting, is the price that a retailer pays for an item.

principal

The principal is the original amount of money invested or borrowed for a period of time at a specific interest rate.

rate of interest

The rate of interest is a percent of the principal, usually expressed as a percent per year.

simple interest

Key Concepts

Problem-Solving Strategy Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

Consecutive Integers Consecutive integers are integers that immediately follow each other.

\begin{array}{cccl} n & & & {1}^{\mathrm{st}}\,\mathrm{integer} \\ n+1 & & & {2}^{\mathrm{nd}}\,\mathrm{integer consecutive integer} \\ n+2 & & & {3}^{\mathrm{rd}}\,\mathrm{consecutive integer . . . etc.} \\ \end{array}

Consecutive even integers are even integers that immediately follow one another.

\begin{array}{cccl} n & & & {1}^{\mathrm{st}}\,\mathrm{integer} \\ n+2 & & & {2}^{\mathrm{nd}}\,\mathrm{integer consecutive integer} \\ n+4 & & & {3}^{\mathrm{rd}}\,\mathrm{consecutive integer . . . etc.} \\ \end{array}

Consecutive odd integers are odd integers that immediately follow one another.

\begin{array}{cccl} n & & & {1}^{\mathrm{st}}\,\mathrm{integer} \\ n+2 & & & {2}^{\mathrm{nd}}\,\mathrm{integer consecutive integer} \\ n+4 & & & {3}^{\mathrm{rd}}\,\mathrm{consecutive integer . . . etc.} \\ \end{array}

Percent Increase To find the percent increase: Step 1. Find the amount of increase. $\mathrm{increase}=\mathrm{new amount}-\mathrm{original}\,\mathrm{amount}$ Step 2. Find the percent increase. Increase is what percent of the original amount?

Percent Decrease To find the percent decrease: Step 1. Find the amount of decrease. $\mathrm{decrease}=\mathrm{original amount}-\mathrm{new}\,\mathrm{amount}$ Step 2. Find the percent decrease. Decrease is what percent of the original amount?

Simple Interest If an amount of money, P, called the principal, is invested for a period of t years at an annual interest rate r, the amount of interest, I, earned is

\begin{array}{rll} I & = & Prt \\ \mathrm{where}\,I & = & \mathrm{interest} \\ P & = & \mathrm{principal} \\ r & = & \mathrm{rate} \\ t & = & \mathrm{time} \\ \end{array}

Discount amount of discount is discount rate $\cdot$ original price sale price is original price – discount

Mark-up amount of mark-up is mark-up rate $\cdot$ original cost list price is original cost + mark up

Total Value of Coins For the same type of coin, the total value of a number of coins is found by using the model. $number\cdot{}value=total\,value$ where number is the number of coins and value is the value of each coin; total value is the total value of all the coins

Problem-Solving Strategy—Coin Word Problems Step 1. Read the problem. Make all the words and ideas are understood. Determine the types of coins involved. Create a table to organize the information. Label the columns type, number, value, total value. List the types of coins. Write in the value of each type of coin. Write in the total value of all the coins. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Use variable expressions to represent the number of each type of coin and write them in the table. Multiply the number times the value to get the total value of each type of coin. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the sentence into an equation. Write the equation by adding the total values of all the types of coins. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

Problem-Solving Strategy for Geometry Applications Step 1. Read the problem and make all the words and ideas are understood. Draw the figure and label it with the given information. Step 2. Identify what we are looking for. Step 3. Name what we are looking for by choosing a variable to represent it. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

Triangle Properties For $\triangle{}ABC$ Angle measures: $m∠A+m∠B+m∠C=180$ Perimeter: $P=a+b+c$ Area: $A=\frac{1}{2}bh,\,b=\mathrm{base},h=\mathrm{height}$ A right triangle has one $90^{\circ}$ angle.

The Pythagorean Theorem In any right triangle, ${a}^{2}+{b}^{2}={c}^{2}$ where c is the length of the hypotenuse and a and b are the lengths of the legs.

Properties of Rectangles Rectangles have four sides and four right (90°) angles. The lengths of opposite sides are equal. The perimeter of a rectangle is the sum of twice the length and twice the width: $P=2L+2W.$ The area of a rectangle is the length times the width: $A=LW.$

Distance, Rate, and Time D = rt where D = distance, r = rate, t = time

Problem-Solving Strategy—Distance, Rate, and Time Applications Step 1. Read the problem. Make sure all the words and ideas are understood. Draw a diagram to illustrate what it happening. Create a table to organize the information: Label the columns rate, time, distance. List the two scenarios. Write in the information you know. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Complete the chart. Use variable expressions to represent that quantity in each row. Multiply the rate times the time to get the distance. Step 4. Translate into an equation. Restate the problem in one sentence with all the important information. Then, translate the sentence into an equation. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

Solving inequalities Step 1. Read the problem. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Step 4. Translate. Write a sentence that gives the information to find it. Translate into an inequality. Step 5. Solve the inequality. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

Review Exercises

Approach Word Problems with a Positive Attitude

In the following exercises, reflect on your approach to word problems.

Use a Problem-Solving Strategy for Word Problems

In the following exercises, solve using the problem-solving strategy for word problems. Remember to write a complete sentence to answer each question.

Solve Number Problems

In the following exercises, solve each number word problem.

Translate and Solve Basic Percent Equations

In the following exercises, translate and solve.

Solve Percent Applications

In the following exercises, solve.

Find Percent Increase and Percent Decrease

In the following exercises, solve.

Solve Simple Interest Applications

In the following exercises, solve.

Solve Applications with Discount or Mark-up

In the following exercises, find the sale price.

In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.

In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent, if needed.)

In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.

Solve Coin Word Problems

In the following exercises, solve each coin word problem.

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

Solve Mixture Word Problems

In the following exercises, solve each mixture word problem.

Solve Applications Using Triangle Properties

In the following exercises, solve using triangle properties.

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

Solve Applications Using Rectangle Properties

In the following exercises, solve using rectangle properties.

Solve Uniform Motion Applications

In the following exercises, solve.

Solve Applications with Linear Inequalities

In the following exercises, solve.

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

3. Key Terms

Key Terms

amount of discount

The amount of discount is the amount resulting when a discount rate is multiplied by the original price of an item.

discount rate

The discount rate is the percent used to determine the amount of a discount, common in retail settings.

interest

Interest is the money that a bank pays its customers for keeping their money in the bank.

list price

The list price is the price a retailer sells an item for.

mark-up

A mark-up is a percentage of the original cost used to increase the price of an item.

mixture problems

Mixture problems combine two or more items with different values together.

original cost

The original cost in a retail setting, is the price that a retailer pays for an item.

principal

The principal is the original amount of money invested or borrowed for a period of time at a specific interest rate.

rate of interest

The rate of interest is a percent of the principal, usually expressed as a percent per year.

simple interest

Key Concepts

Problem-Solving Strategy Step 1. Read the problem. Make sure all the words and ideas are understood. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the English sentence into an algebra equation. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

Consecutive Integers Consecutive integers are integers that immediately follow each other.

\begin{array}{cccl} n & & & {1}^{\mathrm{st}}\,\mathrm{integer} \\ n+1 & & & {2}^{\mathrm{nd}}\,\mathrm{integer consecutive integer} \\ n+2 & & & {3}^{\mathrm{rd}}\,\mathrm{consecutive integer . . . etc.} \\ \end{array}

Consecutive even integers are even integers that immediately follow one another.

\begin{array}{cccl} n & & & {1}^{\mathrm{st}}\,\mathrm{integer} \\ n+2 & & & {2}^{\mathrm{nd}}\,\mathrm{integer consecutive integer} \\ n+4 & & & {3}^{\mathrm{rd}}\,\mathrm{consecutive integer . . . etc.} \\ \end{array}

Consecutive odd integers are odd integers that immediately follow one another.

\begin{array}{cccl} n & & & {1}^{\mathrm{st}}\,\mathrm{integer} \\ n+2 & & & {2}^{\mathrm{nd}}\,\mathrm{integer consecutive integer} \\ n+4 & & & {3}^{\mathrm{rd}}\,\mathrm{consecutive integer . . . etc.} \\ \end{array}

Percent Increase To find the percent increase: Step 1. Find the amount of increase. $\mathrm{increase}=\mathrm{new amount}-\mathrm{original}\,\mathrm{amount}$ Step 2. Find the percent increase. Increase is what percent of the original amount?

Percent Decrease To find the percent decrease: Step 1. Find the amount of decrease. $\mathrm{decrease}=\mathrm{original amount}-\mathrm{new}\,\mathrm{amount}$ Step 2. Find the percent decrease. Decrease is what percent of the original amount?

Simple Interest If an amount of money, P, called the principal, is invested for a period of t years at an annual interest rate r, the amount of interest, I, earned is

\begin{array}{rll} I & = & Prt \\ \mathrm{where}\,I & = & \mathrm{interest} \\ P & = & \mathrm{principal} \\ r & = & \mathrm{rate} \\ t & = & \mathrm{time} \\ \end{array}

Discount amount of discount is discount rate $\cdot$ original price sale price is original price – discount

Mark-up amount of mark-up is mark-up rate $\cdot$ original cost list price is original cost + mark up

Total Value of Coins For the same type of coin, the total value of a number of coins is found by using the model. $number\cdot{}value=total\,value$ where number is the number of coins and value is the value of each coin; total value is the total value of all the coins

Problem-Solving Strategy—Coin Word Problems Step 1. Read the problem. Make all the words and ideas are understood. Determine the types of coins involved. Create a table to organize the information. Label the columns type, number, value, total value. List the types of coins. Write in the value of each type of coin. Write in the total value of all the coins. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Use variable expressions to represent the number of each type of coin and write them in the table. Multiply the number times the value to get the total value of each type of coin. Step 4. Translate into an equation. It may be helpful to restate the problem in one sentence with all the important information. Then, translate the sentence into an equation. Write the equation by adding the total values of all the types of coins. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

Problem-Solving Strategy for Geometry Applications Step 1. Read the problem and make all the words and ideas are understood. Draw the figure and label it with the given information. Step 2. Identify what we are looking for. Step 3. Name what we are looking for by choosing a variable to represent it. Step 4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

Triangle Properties For $\triangle{}ABC$ Angle measures: $m∠A+m∠B+m∠C=180$ Perimeter: $P=a+b+c$ Area: $A=\frac{1}{2}bh,\,b=\mathrm{base},h=\mathrm{height}$ A right triangle has one $90^{\circ}$ angle.

The Pythagorean Theorem In any right triangle, ${a}^{2}+{b}^{2}={c}^{2}$ where c is the length of the hypotenuse and a and b are the lengths of the legs.

Properties of Rectangles Rectangles have four sides and four right (90°) angles. The lengths of opposite sides are equal. The perimeter of a rectangle is the sum of twice the length and twice the width: $P=2L+2W.$ The area of a rectangle is the length times the width: $A=LW.$

Distance, Rate, and Time D = rt where D = distance, r = rate, t = time

Problem-Solving Strategy—Distance, Rate, and Time Applications Step 1. Read the problem. Make sure all the words and ideas are understood. Draw a diagram to illustrate what it happening. Create a table to organize the information: Label the columns rate, time, distance. List the two scenarios. Write in the information you know. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Complete the chart. Use variable expressions to represent that quantity in each row. Multiply the rate times the time to get the distance. Step 4. Translate into an equation. Restate the problem in one sentence with all the important information. Then, translate the sentence into an equation. Step 5. Solve the equation using good algebra techniques. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

Solving inequalities Step 1. Read the problem. Step 2. Identify what we are looking for. Step 3. Name what we are looking for. Choose a variable to represent that quantity. Step 4. Translate. Write a sentence that gives the information to find it. Translate into an inequality. Step 5. Solve the inequality. Step 6. Check the answer in the problem and make sure it makes sense. Step 7. Answer the question with a complete sentence.

Review Exercises

Approach Word Problems with a Positive Attitude

In the following exercises, reflect on your approach to word problems.

Use a Problem-Solving Strategy for Word Problems

In the following exercises, solve using the problem-solving strategy for word problems. Remember to write a complete sentence to answer each question.

Solve Number Problems

In the following exercises, solve each number word problem.

Translate and Solve Basic Percent Equations

In the following exercises, translate and solve.

Solve Percent Applications

In the following exercises, solve.

Find Percent Increase and Percent Decrease

In the following exercises, solve.

Solve Simple Interest Applications

In the following exercises, solve.

Solve Applications with Discount or Mark-up

In the following exercises, find the sale price.

In the following exercises, find ⓐ the amount of discount and ⓑ the sale price.

In the following exercises, find ⓐ the amount of discount and ⓑ the discount rate. (Round to the nearest tenth of a percent, if needed.)

In the following exercises, find ⓐ the amount of the mark-up and ⓑ the list price.

Solve Coin Word Problems

In the following exercises, solve each coin word problem.

Solve Ticket and Stamp Word Problems

In the following exercises, solve each ticket or stamp word problem.

Solve Mixture Word Problems

In the following exercises, solve each mixture word problem.

Solve Applications Using Triangle Properties

In the following exercises, solve using triangle properties.

Use the Pythagorean Theorem

In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse.

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

In the following exercises, solve. Approximate to the nearest tenth, if necessary.

Solve Applications Using Rectangle Properties

In the following exercises, solve using rectangle properties.

Solve Uniform Motion Applications

In the following exercises, solve.

Solve Applications with Linear Inequalities

In the following exercises, solve.

In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary.

4. Practice 2

Find three consecutive integers whose sum is $-141.$

Find three consecutive odd integers whose sum is 51.

What percent of 125 is 150?

5. Practice 3

The bill for Dino’s lunch was $19.45. He wanted to leave 20% of the total bill as a tip. How much should the tip be?

Dolores bought a crib on sale for $350. The sale price was 40% of the original price. What was the original price of the crib?

Angel’s got a raise in his annual salary from $55,400 to$ 56,785. Find the percent increase.

6. Practice 4

Winston deposited $3,294 in a bank account with interest rate 2.6%. How much interest was earned in 5 years?

Jaime’s refrigerator loan statement said he would pay $1,026 in interest for a 4-year loan at 13.5%. How much did Jaime borrow to buy the refrigerator?

The original price of a handbag was $84. Carole bought it on sale for$ 21 off.

7. Practice 6

Francie has $4.35 in dimes and quarters. The number of dimes is five more than the number of quarters. How many of each coin does she have?

Paulette has $140 in$ 5 and $10 bills. The number of$ 10 bills is one less than twice the number of $5 bills. How many of each does she have?

A church luncheon made $842. Adult tickets cost$ 10 each and children’s tickets cost $6 each. The number of children was 12 more than twice the number of adults. How many of each ticket were sold?

8. Practice 7

125 tickets were sold for the jazz band concert for a total of $1,022. Student tickets cost$ 6 each and general admission tickets cost $10 each. How many of each kind of ticket were sold?

Ana spent $4.06 buying stamps. The number of$ 0.41 stamps she bought was five more than the number of $0.26 stamps. How many of each did she buy?

Marquese is making 10 pounds of trail mix from raisins and nuts. Raisins cost $3.45 per pound and nuts cost$ 7.95 per pound. How many pounds of raisins and how many pounds of nuts should Marquese use for the trail mix to cost him $6.96 per pound?

9. Practice 8

Shawn has $15,000 to invest. She will put some of it into a fund that pays 4.5% annual interest and the rest in a certificate of deposit that pays 1.8% annual interest. How much should she invest in each account if she wants to earn 4.05% annual interest on the total amount?

The measures of two angles of a triangle are 22 and 85 degrees. Find the measure of the third angle.

A triangular road sign has base 30 inches and height 40 inches. What is its area?

10. Practice 9

One angle of a triangle is $30^{\circ}$ more than the smallest angle. The largest angle is the sum of the other angles. Find the measures of all three angles.

The measure of the smallest angle in a right triangle is $45^{\circ}$ less than the measure of the next larger angle. Find the measures of all three angles.

Sergio needs to attach a wire to hold the antenna to the roof of his house, as shown in the figure. The antenna is 8 feet tall and Sergio has 10 feet of wire. How far from the base of the antenna can he attach the wire?

11. Practice 10

The length of a rectangle is 36 feet and the width is 19 feet. Find the ⓐ perimeter ⓑ area.

The area of a rectangle is 2356 square meters. The length is 38 meters. What is the width?

The length of a rectangle is 12 cm more than the width. The perimeter is 74 cm. Find the length and the width.

12. Practice 12

Julianne has a weekly food budget of $231 for her family. If she plans to budget the same amount for each of the seven days of the week, what is the maximum amount she can spend on food each day?

Briana has been offered a sales job in another city. The offer was for $42,500 plus 8% of her total sales. In order to make it worth the move, Briana needs to have an annual salary of at least$ 66,500. What would her total sales need to be for her to move?

Costa is an accountant. During tax season, he charges $125 to do a simple tax return. His expenses for buying software, renting an office, and advertising are$ 6,000. How many tax returns must he do if he wants to make a profit of at least $8,000?

13. Practice 13

The sum of two consecutive odd integers is $-96.$ Find the numbers.

Humberto’s hourly pay increased from $16.25 to$ 17.55. Find the percent increase.

Dotty bought a freezer on sale for $486.50. The original price of the freezer was$ 695. Find ⓐ the amount of discount and ⓑ the discount rate.

14. Practice 14

At a concert, $1,600 in tickets were sold. Adult tickets were$ 9 each and children’s tickets were $4 each. If the number of adult tickets was 30 less than twice the number of children’s tickets, how many of each kind were sold?

The measure of one angle of a triangle is twice the measure of the smallest angle. The measure of the third angle is 14 more than the measure of the smallest angle. Find the measures of all three angles.

A baseball diamond is really a square with sides of 90 feet. How far is it from home plate to second base, as shown?

15. Practice 1

How has your attitude towards solving word problems changed as a result of working through this chapter? Explain.

One number is nine less than another. Their sum is negative twenty-seven. Find the numbers.

One number is two more than four times another. Their sum is $-13.$ Find the numbers.

16. Practice 5

Emmett bought a pair of shoes on sale at 40% off from an original price of $138.

Zack bought a printer for his office that was on sale for $380. The original price of the printer was$ 450.

Nga and Lauren bought a chest at a flea market for $50. They re-finished it and then added a 350% mark-up.

17. Practice 11

When Gabe drives from Sacramento to Redding it takes him 2.2 hours. It takes Elsa 2 hours to drive the same distance. Elsa’s speed is seven miles per hour faster than Gabe’s speed. Find Gabe’s speed and Elsa’s speed.

Two busses leave Amarillo at the same time. The Albuquerque bus heads west on the I-40 at a speed of 72 miles per hour, and the Oklahoma City bus heads east on the I-40 at a speed of 78 miles per hour. How many hours will it take them to be 375 miles apart?

At 6:30, Devon left her house and rode her bike on the flat road until 7:30. Then she started riding uphill and rode until 8:00. She rode a total of 15 miles. Her speed on the flat road was three miles per hour faster than her speed going uphill. Find Devon’s speed on the flat road and riding uphill.

18. Practice 15

Two planes leave Dallas at the same time. One heads east at a speed of 428 miles per hour. The other plane heads west at a speed of 382 miles per hour. How many hours will it take them to be 2,025 miles apart?

Chloe has a budget of $800 for costumes for the 18 members of her musical theater group. If all the costumes are the same price, what is the maximum she can spend for each costume?

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