Chapter 6 Review and Practice Test

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.

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Worked example

commission

A commission is a percentage of total sales as determined by the rate of commission.

discount

An amount of discount is a percent off the original price, determined by the discount rate.

mark-up

The mark-up is the amount added to the wholesale price, determined by the mark-up rate.

percent

percent decrease

The percent decrease is the percent the amount of decrease is of the original amount.

percent increase

The percent increase is the percent the amount of increase is of the original amount.

proportion

sales tax

The sales tax is a percent of the purchase price.

simple interest

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Key Concepts

• Convert a percent to a fraction. Step 1. Write the percent as a ratio with the denominator 100. Step 2. Simplify the fraction if possible.

• Convert a percent to a decimal. Step 1. Write the percent as a ratio with the denominator 100. Step 2. Convert the fraction to a decimal by dividing the numerator by the denominator.

• Convert a decimal to a percent. Step 1. Write the decimal as a fraction. Step 2. If the denominator of the fraction is not 100, rewrite it as an equivalent fraction with denominator 100. Step 3. Write this ratio as a percent.

• Convert a fraction to a percent. Step 1. Convert the fraction to a decimal. Step 2. Convert the decimal to a percent.

• Solve an application. Step 1. Identify what you are asked to find and choose a variable to represent it. Step 2. Write a sentence that gives the information to find it. Step 3. Translate the sentence into an equation. Step 4. Solve the equation using good algebra techniques. Step 5. Write a complete sentence that answers the question. Step 6. Check the answer in the problem and make sure it makes sense.

• Find percent increase. Step 1. Find the amount of increase: $\mathrm{increase}=\mathrm{new amount}-\mathrm{original amount}$ Step 2. Find the percent increase as a percent of the original amount.

• Find percent decrease. Step 1. Find the amount of decrease. $\mathrm{decrease}=\mathrm{original amount}-\mathrm{new amount}$ Step 2. Find the percent decrease as a percent of the original amount.

• Sales Tax The sales tax is a percent of the purchase price. $\mathrm{sales tax}=\mathrm{tax rate}\cdot\mathrm{purchase price}$$\mathrm{total cost}=\mathrm{purchase price}+\mathrm{sales tax}$

• Commission A commission is a percentage of total sales as determined by the rate of commission. $\mathrm{commission}=\mathrm{rate of commission}\cdot\mathrm{original price}$

• Discount An amount of discount is a percent off the original price, determined by the discount rate. $\mathrm{amount of discount}=\mathrm{discount rate}\cdot\mathrm{original price}$$\mathrm{sale price}=\mathrm{original price}-\mathrm{discount}$

• Mark-up The mark-up is the amount added to the wholesale price, determined by the mark-up rate. $\mathrm{amount of mark-up}=\mathrm{mark-up rate wholesale price}$$\mathrm{list price}=\mathrm{wholesale price}+\mathrm{mark up}$

• Simple interest If an amount of money, $P$, the principal, is invested for a period of $t$ years at an annual interest rate $r$, the amount of interest, $I$, earned is $I=Prt$ Interest earned according to this formula is called simple interest.

• Proportion A proportion is an equation of the form $\frac{a}{b}=\frac{c}{d}$, where $b\ne{}0$, $d\ne{}0$.The proportion states two ratios or rates are equal. The proportion is read “$a$ is to $b$, as $c$ is to $d$”.

• Cross Products of a Proportion For any proportion of the form $\frac{a}{b}=\frac{c}{d}$, where $b\ne{}0$, its cross products are equal: $a\cdot{}d=b\cdot{}c$.

• Percent Proportion The amount is to the base as the percent is to 100. $\frac{\mathrm{amount}}{\mathrm{base}}=\frac{\mathrm{percent}}{100}$

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Review Exercises

In the following exercises, write each percent as a ratio.

In the following exercises, write as a ratio and then as a percent.

In the following exercises, convert each percent to a fraction.

In the following exercises, convert each percent to a decimal.

In the following exercises, convert each percent to ⓐ a simplified fraction and ⓑ a decimal.

In the following exercises, convert each decimal to a percent.

In the following exercises, convert each fraction to a percent.

In the following exercises, translate and solve.

In the following exercises, solve.

In the following exercises, solve.

In the following exercises, find ⓐ the sales tax ⓑ the total cost.

In the following exercises, find the sales tax rate.

In the following exercises, find the commission.

In the following exercises, find the rate of commission.

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 ⓑ the discount rate. (Round to the nearest tenth of a percent if needed.)

In the following exercises, find

1. ⓐ the amount of the mark-up

2. ⓑ the list price

In the following exercises, solve the simple interest problem.

In the following exercises, write each sentence as a proportion.

In the following exercises, determine whether each equation is a proportion.

In the following exercises, solve each proportion.

In the following exercises, solve the proportion problem.

In the following exercises, translate to a proportion.

In the following exercises, translate and solve using proportions.

##

In the following exercises, convert each percent to ⓐ a decimal ⓑ a simplified fraction.

In the following exercises, convert each fraction to a percent. (Round to $3$ decimal places if needed.)

In the following exercises, solve the percent problem.

Key Terms

commission

A commission is a percentage of total sales as determined by the rate of commission.

discount

An amount of discount is a percent off the original price, determined by the discount rate.

mark-up

The mark-up is the amount added to the wholesale price, determined by the mark-up rate.

percent

percent decrease

The percent decrease is the percent the amount of decrease is of the original amount.

percent increase

The percent increase is the percent the amount of increase is of the original amount.

proportion

sales tax

The sales tax is a percent of the purchase price.

simple interest

##

Key Concepts

• Convert a percent to a fraction. Step 1. Write the percent as a ratio with the denominator 100. Step 2. Simplify the fraction if possible.

• Convert a percent to a decimal. Step 1. Write the percent as a ratio with the denominator 100. Step 2. Convert the fraction to a decimal by dividing the numerator by the denominator.

• Convert a decimal to a percent. Step 1. Write the decimal as a fraction. Step 2. If the denominator of the fraction is not 100, rewrite it as an equivalent fraction with denominator 100. Step 3. Write this ratio as a percent.

• Convert a fraction to a percent. Step 1. Convert the fraction to a decimal. Step 2. Convert the decimal to a percent.

• Solve an application. Step 1. Identify what you are asked to find and choose a variable to represent it. Step 2. Write a sentence that gives the information to find it. Step 3. Translate the sentence into an equation. Step 4. Solve the equation using good algebra techniques. Step 5. Write a complete sentence that answers the question. Step 6. Check the answer in the problem and make sure it makes sense.

• Find percent increase. Step 1. Find the amount of increase: $\mathrm{increase}=\mathrm{new amount}-\mathrm{original amount}$ Step 2. Find the percent increase as a percent of the original amount.

• Find percent decrease. Step 1. Find the amount of decrease. $\mathrm{decrease}=\mathrm{original amount}-\mathrm{new amount}$ Step 2. Find the percent decrease as a percent of the original amount.

• Sales Tax The sales tax is a percent of the purchase price. $\mathrm{sales tax}=\mathrm{tax rate}\cdot\mathrm{purchase price}$$\mathrm{total cost}=\mathrm{purchase price}+\mathrm{sales tax}$

• Commission A commission is a percentage of total sales as determined by the rate of commission. $\mathrm{commission}=\mathrm{rate of commission}\cdot\mathrm{original price}$

• Discount An amount of discount is a percent off the original price, determined by the discount rate. $\mathrm{amount of discount}=\mathrm{discount rate}\cdot\mathrm{original price}$$\mathrm{sale price}=\mathrm{original price}-\mathrm{discount}$

• Mark-up The mark-up is the amount added to the wholesale price, determined by the mark-up rate. $\mathrm{amount of mark-up}=\mathrm{mark-up rate wholesale price}$$\mathrm{list price}=\mathrm{wholesale price}+\mathrm{mark up}$

• Simple interest If an amount of money, $P$, the principal, is invested for a period of $t$ years at an annual interest rate $r$, the amount of interest, $I$, earned is $I=Prt$ Interest earned according to this formula is called simple interest.

• Proportion A proportion is an equation of the form $\frac{a}{b}=\frac{c}{d}$, where $b\ne{}0$, $d\ne{}0$.The proportion states two ratios or rates are equal. The proportion is read “$a$ is to $b$, as $c$ is to $d$”.

• Cross Products of a Proportion For any proportion of the form $\frac{a}{b}=\frac{c}{d}$, where $b\ne{}0$, its cross products are equal: $a\cdot{}d=b\cdot{}c$.

• Percent Proportion The amount is to the base as the percent is to 100. $\frac{\mathrm{amount}}{\mathrm{base}}=\frac{\mathrm{percent}}{100}$

##

Review Exercises

In the following exercises, write each percent as a ratio.

In the following exercises, write as a ratio and then as a percent.

In the following exercises, convert each percent to a fraction.

In the following exercises, convert each percent to a decimal.

In the following exercises, convert each percent to ⓐ a simplified fraction and ⓑ a decimal.

In the following exercises, convert each decimal to a percent.

In the following exercises, convert each fraction to a percent.

In the following exercises, translate and solve.

In the following exercises, solve.

In the following exercises, solve.

In the following exercises, find ⓐ the sales tax ⓑ the total cost.

In the following exercises, find the sales tax rate.

In the following exercises, find the commission.

In the following exercises, find the rate of commission.

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 ⓑ the discount rate. (Round to the nearest tenth of a percent if needed.)

In the following exercises, find

1. ⓐ the amount of the mark-up

2. ⓑ the list price

In the following exercises, solve the simple interest problem.

In the following exercises, write each sentence as a proportion.

In the following exercises, determine whether each equation is a proportion.

In the following exercises, solve each proportion.

In the following exercises, solve the proportion problem.

In the following exercises, translate to a proportion.

In the following exercises, translate and solve using proportions.

##

In the following exercises, convert each percent to ⓐ a decimal ⓑ a simplified fraction.

In the following exercises, convert each fraction to a percent. (Round to $3$ decimal places if needed.)

In the following exercises, solve the percent problem.

1. $\mathrm{32\%}$ admission rate for the university

2. $13$ out of $100$ architects are women.

3. $\mathrm{48\%}$

4. $\mathrm{64.1\%}$

1. In $2012,\mathrm{13.5\%}$ of the United States population was age $65$ or over. (Source: www.census.gov)

2. When a die is tossed, the probability it will land with an even number of dots on the top side is $\mathrm{50\%}.$

3. $0.04$

4. $2.82$

1. $0.003$

2. $\frac{3}{4}$

3. $3\frac{5}{8}$

4. According to the Centers for Disease Control, $\frac{2}{5}$ of adults do not take a vitamin or supplement.

1. What percent of $120$ is $81.6?$

2. When Aurelio and his family ate dinner at a restaurant, the bill was $\text{\textdollar 83.50}.$ Aurelio wants to leave $\mathrm{20\%}$ of the total bill as a tip. How much should the tip be?

3. The nutrition label on a package of granola bars says that each granola bar has $190$ calories, and $54$ calories are from fat. What percent of the total calories is from fat?

4. Jorge got a raise in his hourly pay, from $\text{\textdollar 19.00}$ to $\text{\textdollar 19.76}.$ Find the percent increase.

1. The cost of a lawn mower was $\text{\textdollar 750}.$ The sales tax rate is $\mathrm{6\%}$ of the purchase price.

2. Andy bought a piano for $\text{\textdollar 4,600}.$ The sales tax on the purchase was $\text{\textdollar 333.50}.$

3. Ginny is a realtor. She receives $\mathrm{3\%}$ commission when she sells a house. How much commission will she receive for selling a house for $\text{\textdollar 380,000}?$

4. Ruben received $\text{\textdollar 675}$ commission when he sold a $\text{\textdollar 4,500}$ painting at the art gallery where he works. What was the rate of commission?

1. Aya bought a pair of shoes that was on sale for $\text{\textdollar 30}$ off. The original price of the shoes was $\text{\textdollar 75}.$

2. Nga bought a microwave for her office. The microwave was discounted $\mathrm{30\%}$ from an original price of $\text{\textdollar 84.90}.$

3. Hilda bought a bedspread on sale for $\text{\textdollar 37}.$ The original price of the bedspread was $\text{\textdollar 50}.$

4. Manny paid $\text{\textdollar 0.80}$ a pound for apples. He added $\mathrm{60\%}$ mark-up before selling them at his produce stand. What price did he charge for the apples?

1. Find the simple interest earned after $4$ years on $\text{\textdollar 2,250}$ invested at an interest rate of $\mathrm{5\%}.$

2. Find the principal invested if $\text{\textdollar 660}$ interest was earned in $5$ years at an interest rate of $\mathrm{3\%}.$

3. Kazuo deposited $\text{\textdollar 10,000}$ in a bank account with interest rate $\mathrm{4.5\%}.$ How much interest was earned in $2$ years?

4. Fresia lent her son $\text{\textdollar 5,000}$ for college expenses. Three years later he repaid her the $\text{\textdollar 5,000}$ plus $\text{\textdollar 375}$ interest. What was the rate of interest?

1. $3$ is to $8$ as $12$ is to $32.$

2. $1$ teacher to $18$ students is the same as $23$ teachers to $414$ students.

3. $\text{\textdollar 7.35}$ for $15$ ounces is the same as $\text{\textdollar 2.94}$ for $6$ ounces.

4. $\frac{5}{13}=\frac{30}{78}$

1. $\frac{12}{18}=\frac{6.99}{10.99}$

2. The children’s dosage of acetaminophen is $5$ milliliters (ml) for every $25$ pounds of a child’s weight. How many milliliters of acetaminophen will be prescribed for a $60$ pound child?

3. An $8$ ounce serving of ice cream has $272$ calories. If Lavonne eats $10$ ounces of ice cream, how many calories does she get?

4. Zack wants to drive from Omaha to Denver, a distance of $494$ miles. If his car gets $38$ miles to the gallon, how many gallons of gas will Zack need to get to Denver?

1. What number is $\mathrm{62\%}$ of $395?$

2. What percent of $1,000$ is $15?$

3. What number is $\mathrm{85\%}$ of $900?$

4. $\text{\textdollar 3.51}$ is $\mathrm{4.5\%}$ of what number?

1. $\mathrm{24\%}$

2. $\mathrm{350\%}$

3. $\frac{1}{3}$

4. $65$ is what percent of $260?$

1. The total number of vehicles on one freeway dropped from $\mathrm{84,000}$ to $\mathrm{74,000}.$ Find the percent decrease (round to the nearest tenth of a percent).

2. Mara received $\text{\textdollar 31.80}$ commission when she sold a $\text{\textdollar 795}$ suit. What was her rate of commission?

3. Oxana bought a dresser at a garage sale for $\text{\textdollar 20}.$ She refinished it, then added a $\mathrm{250\%}$ markup before advertising it for sale. What price did she ask for the dresser?

4. Brenda borrowed $\text{\textdollar 400}$ from her brother. Two years later, she repaid the $\text{\textdollar 400}$ plus $\text{\textdollar 50}$ interest. What was the rate of interest?

1. Solve for a: $\frac{12}{a}=\frac{-15}{65}$