Percentages and Proportions

In last season’s basketball games, Charity made $60\%$ in free throws. If he attempted a total of $140$ free throws, how many free throws did she make?

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Charity made free throws last season.

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Answer.

$84$

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Solution.

This problem can be boiled down to this question: What is $60\%$ of $140\text{?}$ We will show multiple methods to solve this problem.

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Method 1

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We will use proportion to solve this problem. Assume $60\%$ of $140$ is $x\text{,}$ so “$60$ out of $100$” corresponds to “$x$ out of $140$”.

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We will write and solve the proportion:

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$\displaystyle{\begin{aligned}[t] \frac{60}{100} \amp = \frac{x}{140} \\ 100x \amp = 60 \cdot 140 \\ 100x \amp = 8400 \\ \frac{100x}{100} \amp = \frac{8400}{100} \\ x \amp = 84 \end{aligned} }$

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Charity made $84$ free throws last season.

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Method 2

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We will use the percentage formula to solve this problem. This translation from English to math may help you remember the percentage formula.

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$2 \text{ is } 50\% \text{ of } 4 \iff 2 = 0.5 \cdot 4$

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The question is: What is $60\%$ of $140\text{?}$ Assume $x$ is $60\%$ of $140\text{.}$ We have:

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$\displaystyle{ \begin{aligned} x \amp = 0.6 \cdot 140 \\ \amp = 84 \end{aligned} }$

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Charity made $84$ free throws last season.

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Method 3

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In the sentence “What is $60\%$ of $140\text{,}$”

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“what” is the percentage,
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$60\%$ is the rate,
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$140$ is the base (following the word “of”).
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By the formula $\text{percentage} = \text{rate} \cdot \text{base}\text{,}$ we do a multiplication to solve the problem:

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$\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 60\% \cdot 140 = 0.6 \cdot 140 = 84 }$

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Charity made $84$ free throws last season.

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5.

A county has $22300$ residents. In the last election, $37\%$ turned out to vote. How many residents voted?

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In the last election, residents in the county turned out to vote.

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Answer.

$8251$

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Solution.

This problem can be boiled down to this question: What is $37\%$ of $22300\text{?}$ We will show multiple methods to solve this problem.

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Method 1

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We will use proportion to solve this problem. Assume $37\%$ of $22300$ is $x\text{,}$ so “$37$ out of $100$” corresponds to “$x$ out of $22300$”.

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We will write and solve the proportion:

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$\displaystyle{\begin{aligned}[t] \frac{37}{100} \amp = \frac{x}{22300} \\ 100x \amp = 37 \cdot 22300 \\ 100x \amp = 825100 \\ \frac{100x}{100} \amp = \frac{825100}{100} \\ x \amp = 8251 \end{aligned} }$

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In the last election, $8251$ residents in the county. turned out to vote.

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Method 2

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We will use the percentage formula to solve this problem. This translation from English to math may help you remember the percentage formula.

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$2 \text{ is } 50\% \text{ of } 4 \iff 2 = 0.5 \cdot 4$

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The question is: What is $37\%$ of $22300\text{?}$ Assume $x$ is $37\%$ of $22300\text{.}$ We have:

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$\displaystyle{ \begin{aligned} x \amp = 0.37 \cdot 22300 \\ \amp = 8251 \end{aligned} }$

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In the last election, $8251$ residents in the county. turned out to vote.

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Method 3

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In the sentence “What is $37\%$ of $22300\text{,}$”

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“what” is the percentage,
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$37\%$ is the rate,
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$22300$ is the base (following the word “of”).
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By the formula $\text{percentage} = \text{rate} \cdot \text{base}\text{,}$ we do a multiplication to solve the problem:

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$\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 37\% \cdot 22300 = 0.37 \cdot 22300 = 8251 }$

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In the last election, $8251$ residents in the county. turned out to vote.

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6.

A painting is on sale with $15\%$ off. Its original price was ${\$550.00}\text{.}$ What is its price on sale?

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The painting sells for on sale.

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Answer.

$\$467.50$

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Solution.

The painting is $15\%$ off, implying that its current price is $85\%$ of its original price.

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This problem can be boiled down to this question: What is $85\%$ of $550\text{?}$ We will show multiple methods to solve this problem.

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Method 1

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We will use proportion to solve this problem. Assume $85\%$ of $550$ is $x\text{,}$ so “$85$ out of $100$” corresponds to “$x$ out of $550$”.

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We will write and solve the proportion:

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$\displaystyle{\begin{aligned}[t] \frac{85}{100} \amp = \frac{x}{550} \\ 100x \amp = 85 \cdot 550 \\ 100x \amp = 46750 \\ \frac{100x}{100} \amp = \frac{46750}{100} \\ x \amp = 467.5 \end{aligned} }$

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The painting sells for ${\$467.50}$ on sale.

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Method 2

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We will use the percentage formula to solve this problem. This translation from English to math may help you remember the percentage formula.

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$2 \text{ is } 50\% \text{ of } 4 \iff 2 = 0.5 \cdot 4$

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The question is: What is $85\%$ of $550\text{?}$ Assume $x$ is $85\%$ of $550\text{.}$ We have:

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$\displaystyle{ \begin{aligned} x \amp = 0.85 \cdot 550 \\ \amp = 467.5 \end{aligned} }$

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The painting sells for ${\$467.50}$ on sale.

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Method 3

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In the sentence “What is $85\%$ of $550\text{,}$”

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“what” is the percentage,
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$85\%$ is the rate,
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$550$ is the base (following the word “of”).
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By the formula $\text{percentage} = \text{rate} \cdot \text{base}\text{,}$ we do a multiplication to solve the problem:

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$\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 85\% \cdot 550 = 0.85 \cdot 550 = 467.5 }$

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The painting sells for ${\$467.50}$ on sale.

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7.

A watch’s wholesale price was ${\$360.00}\text{.}$ The retailer marked up the price by $30\%\text{.}$ What’s the watch’s new price (markup price)?

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The watch’s markup price is .

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Answer.

$\$468.00$

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Solution.

First, we need to find the amount of increase in price. It’s given that the watch’s price was marked up by $30\%$ of its original price, ${\$360.00}\text{.}$

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The problem can be boiled down to this question: What is $30\%$ of $360\text{?}$ We will show multiple methods to solve this problem.

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Method 1

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We will use proportion to solve this problem. Assume $30\%$ of $360$ is $x\text{,}$ so “$30$ out of $100$” corresponds to “$x$ out of $360$”.

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We will write and solve the proportion:

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$\displaystyle{\begin{aligned}[t] \frac{30}{100} \amp = \frac{x}{360} \\ 100x \amp = 30 \cdot 360 \\ 100x \amp = 10800 \\ \frac{100x}{100} \amp = \frac{10800}{100} \\ x \amp = 108 \end{aligned} }$

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The amount of price increase was ${\$108.00}\text{,}$ so the new price is ${\$360.00}+{\$108.00}={\$468.00}\text{.}$

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So the watch’s markup price is ${\$468.00}\text{.}$

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Method 2

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We will use the percentage formula to solve this problem. This translation from English to math may help you remember the percentage formula.

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$2 \text{ is } 50\% \text{ of } 4 \iff 2 = 0.5 \cdot 4$

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The question is: What is $30\%$ of $360\text{?}$ Assume $x$ is $30\%$ of $360\text{.}$ We have:

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$\displaystyle{ \begin{aligned} x \amp = 0.3 \cdot 360 \\ \amp = 108 \end{aligned} }$

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The amount of price increase was ${\$108.00}\text{,}$ so the new price is ${\$360.00}+{\$108.00}={\$468.00}\text{.}$

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So the watch’s markup price is ${\$468.00}\text{.}$

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Method 3

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In the sentence “What is $30\%$ of $360\text{,}$”

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“what” is the percentage,
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$30\%$ is the rate,
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$360$ is the base (following the word “of”).
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By the formula $\text{percentage} = \text{rate} \cdot \text{base}\text{,}$ we do a multiplication to solve the problem:

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$\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 30\% \cdot 360 = 0.3 \cdot 360 = 108 }$

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The amount of price increase was ${\$108.00}\text{,}$ so the new price is ${\$360.00}+{\$108.00}={\$468.00}\text{.}$

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So the watch’s markup price is ${\$468.00}\text{.}$

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8.

Write the given ratio as a fraction in simplest form.

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`15` to `3=`

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Reduced Fraction: numeric

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Answer.

$5$

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Solution.

We can write the ratio as the fraction:

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$\displaystyle{\frac{15}{3}}$

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To reduce, look for common factors.

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$15 = 5\cdot3\cdot1$

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$3 = 1\cdot3\cdot1$

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$\displaystyle{\frac{5\cdot3\cdot1}{1\cdot3\cdot1}=\frac{5}{1}}$

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9.

Solve the following proportion.

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$\displaystyle{ \frac{7}{11}= \frac{y}{66}}$

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`y=` numeric

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Answer.

$42$

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Solution.

To solve a proportion, set the cross products equal:

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$11y = 66\cdot7$

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$11y = 462$

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Divide both sides by 11 to solve:

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$y = 42$

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10.

Solve the following proportion.

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$\displaystyle{ \frac{-3}{9}= \frac{-6}{x}}$

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`x=` numeric

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Answer.

$18$

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Solution.

To solve a proportion, set the cross products equal:

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$-3x = 9\cdot-6$

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$-3x = -54$

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Divide both sides by -3 to solve:

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$x = 18$

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11.

Set up a proportion to solve the application problem. Round your answer to the nearest milliliter:

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Pediatricians prescribe 80 milliliters (ml) of acetaminophen for every 10 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs 105 pounds?

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Solution: ml (rounded to the nearest ml)

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Answer.

$840$

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Solution.

The ratio given is:

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$\displaystyle{\frac{80 \; \textrm{ml}}{10 \; \textrm{lbs}}}$

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We can set up a ratio, making sure that the units are the same on each side.

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$\displaystyle{\frac{80 \; \textrm{ml}}{10 \; \textrm{lbs}}=\frac{x \; \textrm{ml}}{105 \; \textrm{lbs}}}$

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$\displaystyle{\frac{80}{10}=\frac{x}{105}}$

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Set the cross products equal:

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$10x = 105\cdot80$

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$10x = 8400$

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Divide both sides by 10.

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$x = 840$

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12.

Set up a proportion to solve the application problem. Enter a reduced fraction or integer as your final answer.

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An oatmeal cookie recipe calls for $\frac{1}{2}$ cup of butter to make 18 cookies. Hilda needs to make 30 cookies for the bake sale. How many cups of butter will she need?

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Solution: cups

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Answer.

${\frac{5}{6}}$

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Solution.

Let x = the number of cups of butter that Hilda needs.

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We can use the proportion:

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$\displaystyle{\frac{\frac{1}{2}}{18}=\frac{x}{30}}$