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Proportions

Multiplication to solve for an unknown given two equal ratios.

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Proportions

Blake volunteered to be part of a "living flag" at the state fairgrounds on Independence Day. He had a small flag and wondered how the organizers would convert the dimensions to accommodate hundreds of people. Blake's flag measured 4 inches by 6 inches. If the length of the living flag was to be 120 feet, what should the width be?

In this concept, you will learn how to write and work with proportions.

Working with Proportions

A ratio represents a comparison between two quantities. Equivalent ratios are ratios that are equal. A proportion is made up of two equivalent ratios.

An example of a proportion is  12=24\begin{align*}\frac{1}{2} = \frac{2}{4}\end{align*}

12\begin{align*}\frac{1}{2}\end{align*}and 24\begin{align*}\frac{2}{4}\end{align*}are in proportion to one another. They are proportional.

When given three parts of a proportion, the fourth can be determined.

Here is an example.

12=x12\begin{align*}\frac{1}{2} = \frac{x}{12}\end{align*}

In this proportion, x is unknown.

Proportional reasoning, or examining the relationship between two numbers, can be used to determine the value of x.

First, since both denominators are given, begin with those two numbers.

2 and 12

Next, recognize the ratio that is given. In this problem it is 12\begin{align*}\frac{1}{2}\end{align*}. Think about what can be done to the denominator of the given ratio to make it equal to the denominator of the ratio with the unknown in it. In this case, what can be done to 2 to make it equal to 12.

2 x 6 = 12

2 can be multiplied times 6 to equal 12.

Then, multiply both the numerator and denominator of the given ratio times 6.

12=1×62×6=612=x12\begin{align*}\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12} = \frac{x}{12}\end{align*}

12=612\begin{align*}\frac{1}{2} = \frac{6}{12}\end{align*}

The ratio 12\begin{align*}\frac{1}{2}\end{align*} is equal to the ratio 612\begin{align*}\frac{6}{12}\end{align*}

The two ratios are proportional to one another. They have the same proportions.

Examples

Example 1

Earlier, you were given a problem about Blake and the living flag.

He wondered how his little 4" by 6" parade flag  could be used to calculate the width of a giant human flag that has a length of 120 feet.

First, write an equation for the proportion.

46=x120\begin{align*}\frac{4}{6}=\frac{x}{120}\end{align*}

Next, work with the denominators.

6 and 120

Then, using the given ratio of 46\begin{align*}\frac{4}{6}\end{align*}, determine what can be done to 6 to make it equal to 120.

6 x 20 = 120

Next, multiply both the numerator and denominator of the given ratio times 20.

4×206×20=80120\begin{align*}\frac{4\times 20}{6\times 20}=\frac{80}{120}\end{align*}

The answer is y = 80 feet.

46=80120\begin{align*}\frac{4}{6}=\frac{80}{120}\end{align*}

Example 2

Use proportional reasoning to solve for x.

1535=x7\begin{align*}\frac{15}{35}=\frac{x}{7}\end{align*}

35 and 7

Next, recognize the given ratio, 1535\begin{align*}\frac{15}{35}\end{align*}, and determine what can be done to 35 to make it equal to 7.

35÷5=7\begin{align*}35\div 5=7\end{align*}

Dividing 35 by 5 equals 7.

Then, divide both the numerator and denominator of the given ratio by 5.

15÷535÷5=37\begin{align*}\frac{15\div 5}{35\div 5}=\frac{3}{7}\end{align*}

The answer is x = 3

1535=37\begin{align*}\frac{15}{35}=\frac{3}{7}\end{align*}

Example 3

3236=x9\begin{align*}\frac{32}{36}=\frac{x}{9}\end{align*}

Solve for x:

First, begin with the given denominators.

36 and 9

Next, using the given ratio of 3236\begin{align*}\frac{32}{36}\end{align*}, determine what can be done to 36 to make it equal to 9.

36÷4=9\begin{align*}36\div 4=9\end{align*}

Then, divide both the numerator and denominator of the given ratio by 4.

32÷436÷4=89\begin{align*}\frac{32\div 4}{36\div 4}=\frac{8}{9}\end{align*}

The answer is x = 8

3236=89\begin{align*}\frac{32}{36}=\frac{8}{9}\end{align*}

Example 4

1224=4z\begin{align*}\frac{12}{24} = \frac{4}{z}\end{align*}

Solve for z.

First, begin with the numerators as both are given.

12 and 4

Next, determine what can be done to the numerator 12 in the given ratio of 1224\begin{align*}\frac{12}{24}\end{align*}to make it equal to 4 in the unknown ratio.

12÷3=4\begin{align*}12\div 3=4\end{align*}

Then, divide both sides of the given ratio by 3.

12÷324÷3=48\begin{align*}\frac{12\div 3}{24\div 3}=\frac{4}{8}\end{align*}

The answer is z = 8.

1224=48\begin{align*}\frac{12}{24}=\frac{4}{8}\end{align*}

Example 5

Solve for y.

y5=1220\begin{align*}\frac{y}{5} = \frac{12}{20}\end{align*}

First, work with the denominators.

5 and 20

Next, use the denominator, 20, in the given ratio of 1220\begin{align*}\frac{12}{20}\end{align*} and determine what can be done to 20 to make it equal to the denominator, 5, in the unknown ratio.

20÷4=5\begin{align*}20\div 4=5\end{align*}

Then, divide both sides of the given ratio by 4.

12÷420÷4=35\begin{align*}\frac{12\div 4}{20\div 4}=\frac{3}{5}\end{align*}

The answer is y = 3.

35=1220\begin{align*}\frac{3}{5}=\frac{12}{20}\end{align*}

Review

Tell whether or not each pair of ratios form a proportion.

1. 12 and 48\begin{align*}\frac{1}{2} \text{ and } \frac{4}{8}\end{align*}
2. 37 and 614\begin{align*}\frac{3}{7} \text{ and } \frac{6}{14}\end{align*}
3. 52 and 106\begin{align*}\frac{5}{2} \text{ and } \frac{10}{6}\end{align*}
4. 31 and 93\begin{align*}\frac{3}{1} \text{ and } \frac{9}{3}\end{align*}
5. 29 and 1.54.5\begin{align*}\frac{2}{9} \text{ and } \frac{1.5}{4.5}\end{align*}
6. 49 and 810\begin{align*}\frac{4}{9} \text{ and } \frac{8}{10}\end{align*}
7. 14 and 520\begin{align*}\frac{1}{4} \text{ and } \frac{5}{20}\end{align*}
8. 34 and 910\begin{align*}\frac{3}{4} \text{ and } \frac{9}{10}\end{align*}

Use proportional reasoning to find the value of the variable in each proportion.

1. 14=a20\begin{align*}\frac{1}{4} = \frac{a}{20}\end{align*}
2. 1530=x2\begin{align*}\frac{15}{30} = \frac{x}{2}\end{align*}
3. 29=n63\begin{align*}\frac{2}{9} = \frac{n}{63}\end{align*}
4. z7=1221\begin{align*}\frac{z}{7} = \frac{12}{21}\end{align*}
5. 35=t60\begin{align*}\frac{3}{5} = \frac{t}{60}\end{align*}
6. k72=512\begin{align*}\frac{k}{72} = \frac{5}{12}\end{align*}
7. x32=48\begin{align*}\frac{x}{32} = \frac{4}{8}\end{align*}

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Color Highlighted Text Notes

Vocabulary Language: English

Equivalent Ratios

Equivalent ratios are ratios that can each be simplified to the same ratio.

Proportion

A proportion is an equation that shows two equivalent ratios.

Proportional Reasoning

Proportional reasoning involves deducing the relationship between the numerators or the denominators of a proportion. Anytime you have a proportion, there is some kind of relationship between the values.