6.5 Solve Proportions and their Applications

Before you get started, take this readiness quiz.

Simplify: $\frac{\frac{1}{3}}{4} .$
If you missed this problem, review the reference.

Solve: $\frac{x}{4} = 20 .$
If you missed this problem, review the reference.

Write as a rate: Sale rode his bike $24$ miles in $2$ hours.
If you missed this problem, review the reference.

Write each sentence as a proportion:

1. ⓐ $3$ is to $7$ as $15$ is to $35 .$
2. ⓑ $5$ hits in $8$ at bats is the same as $30$ hits in $48$ at-bats.
3. ⓒ $\$\text{1}.\text{50}$ for $6$ ounces is equivalent to $\$\text{2}.\text{25}$ for $9$ ounces.

Write each sentence as a proportion:

1. ⓐ $5$ is to $9$ as $20$ is to $36 .$
2. ⓑ $7$ hits in $11$ at-bats is the same as $28$ hits in $44$ at-bats.
3. ⓒ $\$\text{2}.\text{50}$ for $8$ ounces is equivalent to $\$\text{3}.\text{75}$ for $12$ ounces.

Write each sentence as a proportion:

1. ⓐ $6$ is to $7$ as $36$ is to $42 .$
2. ⓑ $8$ adults for $36$ children is the same as $12$ adults for $54$ children.
3. ⓒ $\$\text{3}.\text{75}$ for $6$ ounces is equivalent to $\$\text{2}.\text{50}$ for $4$ ounces.

Determine whether each equation is a proportion:

1. ⓐ $\frac{7}{9} = \frac{54}{72}$
2. ⓑ $\frac{24.5}{45.5} = \frac{7}{13}$

Determine whether each equation is a proportion:

1. ⓐ $\frac{8}{9} = \frac{56}{73}$
2. ⓑ $\frac{28.5}{52.5} = \frac{8}{15}$

Solve the proportion: $\frac{n}{84} = \frac{11}{12} .$

Solve the proportion: $\frac{y}{96} = \frac{13}{12} .$

Solve the proportion: $\frac{91}{b} = \frac{7}{5} .$

Solve the proportion: $\frac{39}{c} = \frac{13}{8} .$

Solve the proportion: $\frac{84}{98} = \frac{−6}{x} .$

Solve the proportion: $\frac{−7}{y} = \frac{105}{135} .$

Pediatricians prescribe $5$ milliliters (ml) of acetaminophen for every $25$ pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Emilia, who weighs $60$ pounds?

For every $1$ kilogram (kg) of a child’s weight, pediatricians prescribe $15$ milligrams (mg) of a fever reducer. If Isabella weighs $12$ kg, how many milligrams of the fever reducer will the pediatrician prescribe?

Marissa loves the Caramel Macchiato at the coffee shop. The $16$ oz. medium size has $240$ calories. How many calories will she get if she drinks the large $20$ oz. size?

Yaneli loves Starburst candies, but wants to keep her snacks to $100$ calories. If the candies have $160$ calories for $8$ pieces, how many pieces can she have in her snack?

Yurianna is going to Europe and wants to change $\$\text{800}$ dollars into Euros. At the current exchange rate, $\$\text{1}$ US is equal to $0.738$ Euro. How many Euros will she have for her trip?

Corey and Nicole are traveling to Japan and need to exchange $\$\text{600}$ into Japanese yen. If each dollar is $94.1$ yen, how many yen will they get?

Translate to a proportion: What number is $\text{60}\%$ of $105 ?$

Translate to a proportion: What number is $\text{40}\%$ of $85 ?$

Translate to a proportion: $36$ is $\text{25}\%$ of what number?

Translate to a proportion: $27$ is $\text{36}\%$ of what number?

Translate to a proportion: What percent of $52$ is $39 ?$

Translate to a proportion: What percent of $92$ is $23 ?$

Translate and solve using proportions: What number is $\text{65}\%$ of $40 ?$

Translate and solve using proportions: What number is $\text{85}\%$ of $40 ?$

Translate and solve using proportions: $\text{125}\%$ of $64$ is what number?

Translate and solve using proportions: $\text{175}\%$ of $84$ is what number?

Translate and solve using proportions: $\text{8}.\text{5}\%$ of what number is $\$\text{3}.\text{23} ?$

Translate and solve using proportions: $\text{7}.\text{25}\%$ of what number is $\$\text{4}.\text{64} ?$

Translate and solve using proportions: What percent of $72$ is $27 ?$

Translate and solve using proportions: What percent of $92$ is $23 ?$

Write each sentence as a proportion.

$4$ is to $15$ as $36$ is to $135 .$

Write each sentence as a proportion.

$12$ is to $5$ as $96$ is to $40 .$

Write each sentence as a proportion.

$5$ wins in $7$ games is the same as $115$ wins in $161$ games.

Write each sentence as a proportion.

$8$ campers to $1$ counselor is the same as $48$ campers to $6$ counselors.

Write each sentence as a proportion.

$\$\text{9}.\text{36}$ for $18$ ounces is the same as $\$\text{2}.\text{60}$ for $5$ ounces.

Write each sentence as a proportion.

$\$\text{18}.\text{04}$ for $11$ pounds is the same as $\$\text{4}.\text{92}$ for $3$ pounds.

Determine whether each equation is a proportion.

$\frac{7}{15} = \frac{56}{120}$

Determine whether each equation is a proportion.

$\frac{11}{6} = \frac{21}{16}$

Determine whether each equation is a proportion.

$\frac{12}{18} = \frac{4.99}{7.56}$

Determine whether each equation is a proportion.

$\frac{13.5}{8.5} = \frac{31.05}{19.55}$

Solve each proportion.

$\frac{x}{56} = \frac{7}{8}$

Solve each proportion.

$\frac{49}{63} = \frac{z}{9}$

Solve each proportion.

$\frac{5}{a} = \frac{65}{117}$

Solve each proportion.

$\frac{98}{154} = \frac{−7}{p}$

Solve each proportion.

$\frac{a}{−8} = \frac{−42}{48}$

Solve each proportion.

$\frac{2.6}{3.9} = \frac{c}{3}$

Solve each proportion.

$\frac{2.7}{j} = \frac{0.9}{0.2}$

Solve each proportion.

$\frac{\frac{1}{2}}{1} = \frac{m}{8}$

Solve the proportion problem.

Pediatricians prescribe $5$ milliliters (ml) of acetaminophen for every $25$ pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs $45$ pounds?

Solve the proportion problem.

At the gym, Carol takes her pulse for $10$ sec and counts $19$ beats. How many beats per minute is this? Has Carol met her target heart rate of $140$ beats per minute?

Solve the proportion problem.

A new energy drink advertises $106$ calories for $8$ ounces. How many calories are in $12$ ounces of the drink?

Solve the proportion problem.

Karen eats $\frac{1}{2}$ cup of oatmeal that counts for $2$ points on her weight loss program. Her husband, Joe, can have $3$ points of oatmeal for breakfast. How much oatmeal can he have?

Solve the proportion problem.

Janice is traveling to Canada and will change $\$\text{250}$ US dollars into Canadian dollars. At the current exchange rate, $\$\text{1}$ US is equal to $\$\text{1}.\text{01}$ Canadian. How many Canadian dollars will she get for her trip?

Solve the proportion problem.

Steve changed $\$\text{600}$ into $480$ Euros. How many Euros did he receive per US dollar?

Solve the proportion problem.

At the laundromat, Lucy changed $\$\text{12}.\text{00}$ into quarters. How many quarters did she get?

Solve the proportion problem.

Jesse’s car gets $30$ miles per gallon of gas. If Las Vegas is $285$ miles away, how many gallons of gas are needed to get there and then home? If gas is $\$\text{3}.\text{09}$ per gallon, what is the total cost of the gas for the trip?

Solve the proportion problem.

Hugh leaves early one morning to drive from his home in Chicago to go to Mount Rushmore, $812$ miles away. After $3$ hours, he has gone $190$ miles. At that rate, how long will the whole drive take?

Solve the proportion problem.

Phil wants to fertilize his lawn. Each bag of fertilizer covers about $4,000$ square feet of lawn. Phil’s lawn is approximately $13,500$ square feet. How many bags of fertilizer will he have to buy?

Translate to a proportion.

What number is $\text{35}\%$ of $250 ?$

Translate to a proportion.

What number is $\text{110}\%$ of $47 ?$

Translate to a proportion.

$45$ is $\text{30}\%$ of what number?

Translate to a proportion.

$90$ is $\text{150}\%$ of what number?

Translate to a proportion.

What percent of $85$ is $17 ?$

Translate to a proportion.

What percent of $260$ is $340 ?$

Translate and solve using proportions.

What number is $\text{65}\%$ of $180 ?$

Translate and solve using proportions.

$\text{18}\%$ of $92$ is what number?

Translate and solve using proportions.

$\text{175}\%$ of $26$ is what number?

Translate and solve using proportions.

What is $\text{300}\%$ of $488 ?$

Translate and solve using proportions.

$\text{17}\%$ of what number is $\$\text{7}.\text{65} ?$

Translate and solve using proportions.

$\$\text{13}.\text{53}$ is $\text{8}.\text{25}\%$ of what number?

Translate and solve using proportions.

What percent of $56$ is $14 ?$

Translate and solve using proportions.

What percent of $96$ is $12 ?$

Mixing a concentrate Sam bought a large bottle of concentrated cleaning solution at the warehouse store. He must mix the concentrate with water to make a solution for washing his windows. The directions tell him to mix $3$ ounces of concentrate with $5$ ounces of water. If he puts $12$ ounces of concentrate in a bucket, how many ounces of water should he add? How many ounces of the solution will he have…

To solve “what number is $\text{45}\%$ of $350 \"$ do you prefer to use an equation like you did in the section on Decimal Operations or a proportion like you did in this section? Explain your reason.

Write each percent as a ratio.

$\text{32}\%$ admission rate for the university

Write as a ratio and then as a percent.

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

Convert each percent to a fraction.

$\text{48}\%$

Convert each percent to a fraction.

$\text{64}.\text{1}\%$

Convert each percent to a decimal.

$\text{6}\%$

Convert each percent to a decimal.

$\text{128}\%$

Convert each percent to ⓐ a simplified fraction and ⓑ a decimal.

In $2012 , \text{13}.\text{5}\%$ of the United States population was age $65$ or over. (Source: www.census.gov)

Convert each percent to ⓐ a simplified fraction and ⓑ a decimal.

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

Convert each decimal to a percent.

$0.04$

Convert each decimal to a percent.

$2.82$

Convert each decimal to a percent.

$0.003$

Convert each fraction to a percent.

$\frac{3}{4}$

Convert each fraction to a percent.

$3 \frac{5}{8}$

Convert each fraction to a percent.

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

Translate and solve.

What number is $\text{46}\%$ of $350 ?$

Translate and solve.

$84$ is $\text{35}\%$ of what number?

Translate and solve.

$\text{200}\%$ of what number is $50 ?$

Translate and solve.

What percent of $120$ is $81.6 ?$

Solve.

When Aurelio and his family ate dinner at a restaurant, the bill was $\$\text{83}.\text{50} .$ Aurelio wants to leave $\text{20}\%$ of the total bill as a tip. How much should the tip be?

Solve.

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?

Solve.

Jorge got a raise in his hourly pay, from $\$\text{19}.\text{00}$ to $\$\text{19}.\text{76} .$ Find the percent increase.

Find ⓐ the sales tax ⓑ the total cost.

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

Find the sales tax rate.

Andy bought a piano for $\$\text{4},\text{600} .$ The sales tax on the purchase was $\$\text{333}.\text{50} .$

Find the commission.

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

Find the rate of commission.

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

Find the sale price.

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

Find ⓐ the amount of discount and ⓑ the sale price.

Nga bought a microwave for her office. The microwave was discounted $\text{30}\%$ from an original price of $\$\text{84}.\text{90} .$

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

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

Find

1. ⓐ the amount of the mark-up
2. ⓑ the list price

Manny paid $\$\text{0}.\text{80}$ a pound for apples. He added $\text{60}\%$ mark-up before selling them at his produce stand. What price did he charge for the apples?

Solve the simple interest problem.

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

Solve the simple interest problem.

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

Solve the simple interest problem.

Kazuo deposited $\$\text{10},\text{000}$ in a bank account with interest rate $\text{4}.\text{5}\% .$ How much interest was earned in $2$ years?

Solve the simple interest problem.

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

Write each sentence as a proportion.

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

Write each sentence as a proportion.

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

Determine whether each equation is a proportion.

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

Determine whether each equation is a proportion.

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

Solve each proportion.

$\frac{x}{36} = \frac{5}{9}$

Solve each proportion.

$\frac{1.2}{1.8} = \frac{d}{6}$

Solve the proportion problem.

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?

Solve the proportion problem.

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

Solve the proportion problem.

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?

Translate to a proportion.

What number is $\text{62}\%$ of $395 ?$

Translate to a proportion.

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

Translate and solve using proportions.

What number is $\text{85}\%$ of $900 ?$

Translate and solve using proportions.

$\$\text{3}.\text{51}$ is $\text{4}.\text{5}\%$ of what number?

Convert each percent to ⓐ a decimal ⓑ a simplified fraction.

$\text{24}\%$

Convert each percent to ⓐ a decimal ⓑ a simplified fraction.

$\text{350}\%$

Convert each fraction to a percent. (Round to $3$ decimal places if needed.)

$\frac{1}{3}$

Solve the percent problem.

$65$ is what percent of $260 ?$

Solve the percent problem.

$\text{150}\%$ of what number is $60 ?$

Solve the percent problem.

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

Solve the percent problem.

Mara received $\$\text{31}.\text{80}$ commission when she sold a $\$\text{795}$ suit. What was her rate of commission?

Solve the percent problem.

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

Solve the percent problem.

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

Solve the percent problem.

Solve for a: $\frac{12}{a} = \frac{−15}{65}$