Percentages and Proportions

$\displaystyle{\begin{aligned}[t] \frac{90}{100} \amp = \frac{x}{270} \\ 100x \amp = 90 \cdot 270 \\ 100x \amp = 24300 \\ \frac{100x}{100} \amp = \frac{24300}{100} \\ x \amp = 243 \end{aligned} }$

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Ivan made $243$ 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 $90\%$ of $270\text{?}$ Assume $x$ is $90\%$ of $270\text{.}$ We have:

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$\displaystyle{ \begin{aligned} x \amp = 0.9 \cdot 270 \\ \amp = 243 \end{aligned} }$

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Ivan made $243$ free throws last season.

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Method 3

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In the sentence “What is $90\%$ of $270\text{,}$”

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“what” is the percentage,
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$90\%$ is the rate,
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$270$ 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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$\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 90\% \cdot 270 = 0.9 \cdot 270 = 243 }$

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Ivan made $243$ free throws last season.

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5.

A county has $49500$ residents. In the last election, $54\%$ 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.

$26730$

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Solution.

This problem can be boiled down to this question: What is $54\%$ of $49500\text{?}$ 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 $54\%$ of $49500$ is $x\text{,}$ so “$54$ out of $100$” corresponds to “$x$ out of $49500$”.

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We will write and solve the proportion:

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$\displaystyle{\begin{aligned}[t] \frac{54}{100} \amp = \frac{x}{49500} \\ 100x \amp = 54 \cdot 49500 \\ 100x \amp = 2673000 \\ \frac{100x}{100} \amp = \frac{2673000}{100} \\ x \amp = 26730 \end{aligned} }$

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In the last election, $26730$ 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 $54\%$ of $49500\text{?}$ Assume $x$ is $54\%$ of $49500\text{.}$ We have:

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$\displaystyle{ \begin{aligned} x \amp = 0.54 \cdot 49500 \\ \amp = 26730 \end{aligned} }$

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In the last election, $26730$ residents in the county. turned out to vote.

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Method 3

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In the sentence “What is $54\%$ of $49500\text{,}$”

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“what” is the percentage,
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$54\%$ is the rate,
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$49500$ 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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$\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 54\% \cdot 49500 = 0.54 \cdot 49500 = 26730 }$

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In the last election, $26730$ residents in the county. turned out to vote.

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6.

A painting is on sale with $35\%$ off. Its original price was ${\$400.00}\text{.}$ What is its price on sale?

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The painting sells for on sale.

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Answer.

$\$260.00$

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Solution.

The painting is $35\%$ off, implying that its current price is $65\%$ of its original price.

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This problem can be boiled down to this question: What is $65\%$ of $400\text{?}$ 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 $65\%$ of $400$ is $x\text{,}$ so “$65$ out of $100$” corresponds to “$x$ out of $400$”.

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We will write and solve the proportion:

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$\displaystyle{\begin{aligned}[t] \frac{65}{100} \amp = \frac{x}{400} \\ 100x \amp = 65 \cdot 400 \\ 100x \amp = 26000 \\ \frac{100x}{100} \amp = \frac{26000}{100} \\ x \amp = 260 \end{aligned} }$

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The painting sells for ${\$260.00}$ 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 $65\%$ of $400\text{?}$ Assume $x$ is $65\%$ of $400\text{.}$ We have:

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$\displaystyle{ \begin{aligned} x \amp = 0.65 \cdot 400 \\ \amp = 260 \end{aligned} }$

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The painting sells for ${\$260.00}$ on sale.

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Method 3

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In the sentence “What is $65\%$ of $400\text{,}$”

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“what” is the percentage,
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$65\%$ is the rate,
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$400$ 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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$\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 65\% \cdot 400 = 0.65 \cdot 400 = 260 }$

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The painting sells for ${\$260.00}$ on sale.

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7.

A watch’s wholesale price was ${\$260.00}\text{.}$ The retailer marked up the price by $40\%\text{.}$ What’s the watch’s new price (markup price)?

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The watch’s markup price is .

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Answer.

$\$364.00$

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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 $40\%$ of its original price, ${\$260.00}\text{.}$

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The problem can be boiled down to this question: What is $40\%$ of $260\text{?}$ 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 $40\%$ of $260$ is $x\text{,}$ so “$40$ out of $100$” corresponds to “$x$ out of $260$”.

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We will write and solve the proportion:

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$\displaystyle{\begin{aligned}[t] \frac{40}{100} \amp = \frac{x}{260} \\ 100x \amp = 40 \cdot 260 \\ 100x \amp = 10400 \\ \frac{100x}{100} \amp = \frac{10400}{100} \\ x \amp = 104 \end{aligned} }$

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The amount of price increase was ${\$104.00}\text{,}$ so the new price is ${\$260.00}+{\$104.00}={\$364.00}\text{.}$

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So the watch’s markup price is ${\$364.00}\text{.}$

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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 $40\%$ of $260\text{?}$ Assume $x$ is $40\%$ of $260\text{.}$ We have:

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$\displaystyle{ \begin{aligned} x \amp = 0.4 \cdot 260 \\ \amp = 104 \end{aligned} }$

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The amount of price increase was ${\$104.00}\text{,}$ so the new price is ${\$260.00}+{\$104.00}={\$364.00}\text{.}$

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So the watch’s markup price is ${\$364.00}\text{.}$

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Method 3

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In the sentence “What is $40\%$ of $260\text{,}$”

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“what” is the percentage,
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🔗
$40\%$ is the rate,
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🔗
$260$ 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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$\displaystyle{ \text{percentage } = \text{rate} \cdot \text{base} = 40\% \cdot 260 = 0.4 \cdot 260 = 104 }$

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The amount of price increase was ${\$104.00}\text{,}$ so the new price is ${\$260.00}+{\$104.00}={\$364.00}\text{.}$

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So the watch’s markup price is ${\$364.00}\text{.}$

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8.

Write the given ratio as a fraction in simplest form.

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`8` to `20=`

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Reduced Fraction: numeric

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Answer.

${\frac{2}{5}}$

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Solution.

We can write the ratio as the fraction:

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$\displaystyle{\frac{8}{20}}$

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To reduce, look for common factors.

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$8 = 2\cdot2\cdot2$

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$20 = 5\cdot2\cdot2$

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$\displaystyle{\frac{2\cdot2\cdot2}{5\cdot2\cdot2}=\frac{2}{5}}$

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9.

Solve the following proportion.

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$\displaystyle{ \frac{3}{5}= \frac{y}{15}}$

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`y=` numeric

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Answer.

$9$

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Solution.

To solve a proportion, set the cross products equal:

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$5y = 15\cdot3$

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$5y = 45$

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Divide both sides by 5 to solve:

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$y = 9$

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10.

Solve the following proportion.

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$\displaystyle{ \frac{2}{-6}= \frac{4}{x}}$

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`x=` numeric

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Answer.

$-12$

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Solution.

To solve a proportion, set the cross products equal:

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$2x = -6\cdot4$

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$2x = -24$

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Divide both sides by 2 to solve:

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$x = -12$

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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 60 milliliters (ml) of acetaminophen for every 20 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs 65 pounds?

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Solution: ml (rounded to the nearest ml)

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Answer.

$195$

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Solution.

The ratio given is:

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$\displaystyle{\frac{60 \; \textrm{ml}}{20 \; \textrm{lbs}}}$

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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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$\displaystyle{\frac{60 \; \textrm{ml}}{20 \; \textrm{lbs}}=\frac{x \; \textrm{ml}}{65 \; \textrm{lbs}}}$

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$\displaystyle{\frac{60}{20}=\frac{x}{65}}$

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Set the cross products equal:

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$20x = 65\cdot60$

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$20x = 3900$

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Divide both sides by 20.

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$x = 195$

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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 $\frac{1}{4}$ cup of butter to make 6 cookies. Hilda needs to make 42 cookies for the bake sale. How many cups of butter will she need?

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Solution: cups

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Answer.

${\frac{7}{4}}$

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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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$\displaystyle{\frac{\frac{1}{4}}{6}=\frac{x}{42}}$