Percentages and Proportions

In last season’s basketball games, Anthony made $70\%$ in free throws. If he attempted a total of $190$ free throws, how many free throws did he make?

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Anthony made free throws last season.

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Answer.

$133$

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Solution.

This problem can be boiled down to this question: What is $70\%$ of $190\text{?}$ 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 $70\%$ of $190$ is $x\text{,}$ so “$70$ out of $100$” corresponds to “$x$ out of $190$”.

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We will write and solve the proportion:

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$\displaystyle{\begin{aligned}[t] \frac{70}{100} \amp = \frac{x}{190} \\ 100x \amp = 70 \cdot 190 \\ 100x \amp = 13300 \\ \frac{100x}{100} \amp = \frac{13300}{100} \\ x \amp = 133 \end{aligned} }$

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Anthony made $133$ 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 $70\%$ of $190\text{?}$ Assume $x$ is $70\%$ of $190\text{.}$ We have:

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$\displaystyle{ \begin{aligned} x \amp = 0.7 \cdot 190 \\ \amp = 133 \end{aligned} }$

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Anthony made $133$ free throws last season.

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Method 3

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In the sentence “What is $70\%$ of $190\text{,}$”

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“what” is the percentage,
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$70\%$ is the rate,
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$190$ 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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$\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 70\% \cdot 190 = 0.7 \cdot 190 = 133 }$

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Anthony made $133$ free throws last season.

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5.

A county has $32200$ residents. In the last election, $45\%$ 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.

$14490$

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Solution.

This problem can be boiled down to this question: What is $45\%$ of $32200\text{?}$ 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 $45\%$ of $32200$ is $x\text{,}$ so “$45$ out of $100$” corresponds to “$x$ out of $32200$”.

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We will write and solve the proportion:

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$\displaystyle{\begin{aligned}[t] \frac{45}{100} \amp = \frac{x}{32200} \\ 100x \amp = 45 \cdot 32200 \\ 100x \amp = 1449000 \\ \frac{100x}{100} \amp = \frac{1449000}{100} \\ x \amp = 14490 \end{aligned} }$

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In the last election, $14490$ 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 $45\%$ of $32200\text{?}$ Assume $x$ is $45\%$ of $32200\text{.}$ We have:

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$\displaystyle{ \begin{aligned} x \amp = 0.45 \cdot 32200 \\ \amp = 14490 \end{aligned} }$

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In the last election, $14490$ residents in the county. turned out to vote.

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Method 3

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In the sentence “What is $45\%$ of $32200\text{,}$”

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“what” is the percentage,
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$45\%$ is the rate,
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$32200$ 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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$\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 45\% \cdot 32200 = 0.45 \cdot 32200 = 14490 }$

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In the last election, $14490$ residents in the county. turned out to vote.

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6.

A painting is on sale with $40\%$ off. Its original price was ${\$650.00}\text{.}$ What is its price on sale?

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The painting sells for on sale.

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Answer.

$\$390.00$

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Solution.

The painting is $40\%$ off, implying that its current price is $60\%$ of its original price.

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This problem can be boiled down to this question: What is $60\%$ of $650\text{?}$ 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 $650$ is $x\text{,}$ so “$60$ out of $100$” corresponds to “$x$ out of $650$”.

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We will write and solve the proportion:

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$\displaystyle{\begin{aligned}[t] \frac{60}{100} \amp = \frac{x}{650} \\ 100x \amp = 60 \cdot 650 \\ 100x \amp = 39000 \\ \frac{100x}{100} \amp = \frac{39000}{100} \\ x \amp = 390 \end{aligned} }$

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The painting sells for ${\$390.00}$ 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 $60\%$ of $650\text{?}$ Assume $x$ is $60\%$ of $650\text{.}$ We have:

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$\displaystyle{ \begin{aligned} x \amp = 0.6 \cdot 650 \\ \amp = 390 \end{aligned} }$

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The painting sells for ${\$390.00}$ on sale.

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Method 3

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In the sentence “What is $60\%$ of $650\text{,}$”

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“what” is the percentage,
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$60\%$ is the rate,
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$650$ 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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$\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 60\% \cdot 650 = 0.6 \cdot 650 = 390 }$

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The painting sells for ${\$390.00}$ on sale.

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7.

A watch’s wholesale price was ${\$440.00}\text{.}$ The retailer marked up the price by $35\%\text{.}$ What’s the watch’s new price (markup price)?

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The watch’s markup price is .

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Answer.

$\$594.00$

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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 $35\%$ of its original price, ${\$440.00}\text{.}$

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The problem can be boiled down to this question: What is $35\%$ of $440\text{?}$ 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 $35\%$ of $440$ is $x\text{,}$ so “$35$ out of $100$” corresponds to “$x$ out of $440$”.

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We will write and solve the proportion:

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$\displaystyle{\begin{aligned}[t] \frac{35}{100} \amp = \frac{x}{440} \\ 100x \amp = 35 \cdot 440 \\ 100x \amp = 15400 \\ \frac{100x}{100} \amp = \frac{15400}{100} \\ x \amp = 154 \end{aligned} }$

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The amount of price increase was ${\$154.00}\text{,}$ so the new price is ${\$440.00}+{\$154.00}={\$594.00}\text{.}$

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So the watch’s markup price is ${\$594.00}\text{.}$

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

🔗

The question is: What is $35\%$ of $440\text{?}$ Assume $x$ is $35\%$ of $440\text{.}$ We have:

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$\displaystyle{ \begin{aligned} x \amp = 0.35 \cdot 440 \\ \amp = 154 \end{aligned} }$

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The amount of price increase was ${\$154.00}\text{,}$ so the new price is ${\$440.00}+{\$154.00}={\$594.00}\text{.}$

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So the watch’s markup price is ${\$594.00}\text{.}$

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Method 3

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In the sentence “What is $35\%$ of $440\text{,}$”

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“what” is the percentage,
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$35\%$ is the rate,
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$440$ 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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$\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 35\% \cdot 440 = 0.35 \cdot 440 = 154 }$

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The amount of price increase was ${\$154.00}\text{,}$ so the new price is ${\$440.00}+{\$154.00}={\$594.00}\text{.}$

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So the watch’s markup price is ${\$594.00}\text{.}$

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8.

Write the given ratio as a fraction in simplest form.

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`165` to `30=`

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Reduced Fraction: numeric

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Answer.

${\frac{11}{2}}$

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Solution.

We can write the ratio as the fraction:

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$\displaystyle{\frac{165}{30}}$

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To reduce, look for common factors.

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$165 = 11\cdot5\cdot3$

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$30 = 2\cdot5\cdot3$

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$\displaystyle{\frac{11\cdot5\cdot3}{2\cdot5\cdot3}=\frac{11}{2}}$

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9.

Solve the following proportion.

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$\displaystyle{ \frac{11}{5}= \frac{y}{30}}$

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`y=` numeric

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Answer.

$66$

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Solution.

To solve a proportion, set the cross products equal:

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$5y = 30\cdot11$

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$5y = 330$

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Divide both sides by 5 to solve:

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$y = 66$

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10.

Solve the following proportion.

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$\displaystyle{ \frac{3}{-12}= \frac{9}{x}}$

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`x=` numeric

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Answer.

$-36$

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Solution.

To solve a proportion, set the cross products equal:

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$3x = -12\cdot9$

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$3x = -108$

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Divide both sides by 3 to solve:

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$x = -36$

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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 30 milliliters (ml) of acetaminophen for every 15 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs 70 pounds?

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Solution: ml (rounded to the nearest ml)

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Answer.

$140$

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Solution.

The ratio given is:

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$\displaystyle{\frac{30 \; \textrm{ml}}{15 \; \textrm{lbs}}}$

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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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$\displaystyle{\frac{30 \; \textrm{ml}}{15 \; \textrm{lbs}}=\frac{x \; \textrm{ml}}{70 \; \textrm{lbs}}}$

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$\displaystyle{\frac{30}{15}=\frac{x}{70}}$

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Set the cross products equal:

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$15x = 70\cdot30$

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$15x = 2100$

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Divide both sides by 15.

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$x = 140$

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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 24 cookies. Hilda needs to make 48 cookies for the bake sale. How many cups of butter will she need?

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Solution: cups

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Answer.

$1$

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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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$\displaystyle{\frac{\frac{1}{2}}{24}=\frac{x}{48}}$