A proportional relationship shows the relationship between two quantities; it compares how much there is of one thing compared to another. In other words, it is a collection of equivalent ratios.

We can use this relationship to help us write equations to solve problems. A proportion is an equality between 2 ratios, or an equation that states that two ratios are equal.

\displaystyle \dfrac{a}{b} = \dfrac{c}{d}

\bm{a,\,c}

represent same part or type

\bm{b,\,d}

represent same type, sum of parts, or whole

When writing a proportion we need to make sure that both numerators represent the same thing and both denominators represent the same thing.

Not only can we write proportions from equivalent ratios, but we can also solve these equations to help answer questions and discover missing values.

Consider this ratio table that shows how far Jenny can run in miles per minute:

time (min.)	0	9	18	27	36	63
distance (mi.)	0	1	2	3	4	⬚

We can solve for the missing distance by using equivalent ratios:

\displaystyle \dfrac{1}{9}

\displaystyle =

\displaystyle \dfrac{d}{63}

Let d represent the missing value

Looking at the denominators, we must multiply 9 by 7 to get 63. To create an equivalent ratio, we need to multiply both the numerator and denominator by 7:

\displaystyle \dfrac{1 \cdot 7}{9 \cdot 7}

\displaystyle =

\displaystyle \dfrac{7}{63}

Creating an equivalent ratio

Therefore, the missing value in the table is 7, and Jenny can run 7 miles in 63 minutes.

A more efficient algebraic method is the Means Extremes Property. For the proportion a:b=c:d, the extremes are a and d while the means are b and c.

Means Extremes Property

A proportion can be solved by determining the product of the means and the product of the extremes. These products will be equal.

a:b=c:d \implies a \cdot d = b \cdot c

Example:

5:12=10:24

5 \cdot 24 = 12 \cdot 10

120=120

We can use the Means Extremes Property to express proportions in multiple ways:

Diagram illustrating various methods to express proportions: equivalent fractions where a over b is equal to c over d; equal ratios where a is to b as c is to d with a and d as the extremes and b and c as the means; and the cross multiplication rule a times d equals b times c indicating that the product of the extremes equals the product of the means.

We can solve the proportion in our previous example using the Means Extremes property as well.

\displaystyle \dfrac{1}{9}	\displaystyle =	\displaystyle \dfrac{d}{63}	Equivalent ratios
\displaystyle 1 \cdot 63	\displaystyle =	\displaystyle 9 \cdot d	Applying the Means Extremes Property
\displaystyle 63	\displaystyle =	\displaystyle 9d	Simplifying the multiplication
\displaystyle 7	\displaystyle =	\displaystyle d	Divide each side by 9

This confirms that Jenny can run 7 miles in 63 minutes.

Examples

Example 1

Write proportions that could be used to solve each problem.

Joey wants to buy 8 watermelon. He knows 3 watermelon cost \$5. Write a proportion that we could solve to find Joey's total price.

Worked Solution

Create a strategy

We can use the equation \dfrac{a}{b} = \dfrac{c}{d}, where numerators a,\,c represent the cost, and the denominators b,\,d represent the number of watermelons.

Apply the idea

Let x represent the cost that we do not know.

\displaystyle \dfrac{a}{b}	\displaystyle =	\displaystyle \dfrac{c}{d}	We can start with this equation
\displaystyle \dfrac{5}{3}	\displaystyle =	\displaystyle \dfrac{x}{8}	Substitute a=5,\,b=3,\,c=x, and d=8

Martin wants the ratio of black tiles to white tiles in their bathroom to be 5:7. They need 1200 total tiles. Write a proportion to figure out how many black tiles they need.

Worked Solution

Create a strategy

We can use the equation \dfrac{a}{b} = \dfrac{c}{d}, where the numerators a,\,c represent the black tiles, and the denominators b,\,d represent the total tiles.

Apply the idea

The ratio of "black tiles is to white tiles" is 5:7. So, the ratio of "black tiles is to total tiles" must be 5:12.

Let x represent the total number of black tiles that we do not know.

\displaystyle \dfrac{a}{b}	\displaystyle =	\displaystyle \dfrac{c}{d}	We can start with this equation
\displaystyle \dfrac{5}{12}	\displaystyle =	\displaystyle \dfrac{x}{1200}	Substitute a=5,\,b=12,\,c=x, and d=1200

Example 2

Consider the proportion:

\dfrac{7}{70} = \dfrac{a}{5}

Create a ratio table with at least 4 entries to find the missing value in the proportion.

Worked Solution

Create a strategy

Start by adding the ratios in the given proportion to the ratio table. Recall that every ratio in a ratio table is equivalent.

Apply the idea

x	\text{ }	\text{ }	5	7
y	\text{ }	\text{ }	a	70

First, find the unit rate:

\displaystyle \dfrac{70}{7}

\displaystyle =

\displaystyle \dfrac{x}{1}

Set up the equivalent ratios

Looking at the denominators, we have to divide 7 by 7 to get 1. To create an equivalent fraction, we must also divide the numerator by 7.

\displaystyle \dfrac{70 \div 7}{7 \div 7}

\displaystyle =

\displaystyle \dfrac{10}{1}

Divide by 7

x	1	3	5	7
y	10	\text{ }	a	70

Noitce we have found the unit rate which means every 1 unit increase in x will result in a 10 unit increase in y. We can multiply each x-value by 10 to complete our missing y-values.

x	1	3	5	7
y	10	30	50	70

We have shown that a = 50.

Determine what x-value in this proportional relationship would have a y-value of 85.

Worked Solution

Create a strategy

We will use the ratio table from part (a) to find the missing value.

Apply the idea

Let a represent the missing value.

\displaystyle \dfrac{10}{1}

\displaystyle =

\displaystyle \dfrac{85}{a}

Set up the proportion

Looking at the numerators, we have to multiply 10 by 8.5 to get 85 since 85 \div 10 = 8.5. To create an equivalent fraction, we must also multiply the denominator by 8.5.

A table displaying values for x: 1, 2, 5, 7 and a variable a, along with corresponding y values 10, 30, 50, 70, 85. The table demonstrates that multiplying 10 by 8.5 results in 85. Therefore, to determine the value of the variable a from 1, the same multiplier of 8.5 should be applied.

\displaystyle \dfrac{10 \cdot 8.5}{1 \cdot 8.5}	\displaystyle =	\displaystyle \dfrac{85}{a}	Multiply by 8.5
\displaystyle a	\displaystyle =	\displaystyle 8.5	Solve for a

x	1	3	5	7	8.5
y	10	30	50	70	85

The x-value is 8.5 when the y-value is 85.

Reflect and check

We could also solve for this value using the Means Extremes Property.

\displaystyle \dfrac{10}{1}

\displaystyle =

\displaystyle \dfrac{85}{a}

Set up the proportion

\displaystyle 10 \cdot a	\displaystyle =	\displaystyle 1 \cdot 85	Use the Means Extremes Property
\displaystyle 10 \cdot a	\displaystyle =	\displaystyle 85	Evaluate the multiplication
\displaystyle a	\displaystyle =	\displaystyle 8.5	Dividing both sides by 10

Example 3

Frank serves 2 cups of coffee every 5 minutes. Write and solve a proportion for the number of cups of coffee he serves in 1 hour.

Worked Solution

Create a strategy

Set up the proportion and then use the Means Extremes property.

Apply the idea

It is important for our units across our proportion to match. We will write 1 hour as 60 minutes for this problem.

Let x represent the missing value.

\displaystyle \dfrac{2}{5}

\displaystyle =

\displaystyle \dfrac{x}{60}

Set up the proportion

\displaystyle 2 \cdot 60	\displaystyle =	\displaystyle 5 \cdot x	Use the Means Extremes Property
\displaystyle 120	\displaystyle =	\displaystyle 5 \cdot x	Evaluate the multiplication
\displaystyle 24	\displaystyle =	\displaystyle x	Dividing both sides by 5

Frank can serve 24 cups of coffee in 1 hour.

Idea summary

Proportions are equivalent ratios and can generally be written in the form:\dfrac{a}{b} = \dfrac{c}{d} We can use the Means Extremes Property to represent and solve proportions:

Outcomes

7.CE.2

The student will solve problems, including those in context, involving proportional relationships.

7.CE.2b

Write and solve a proportion that represents a proportional relationship between two quantities to find a missing value, including problems in context.

7.CE.2c

Apply proportional reasoning to solve problems in context, including converting units of measurement, when given the conversion factor.

2.02 Write and solve proportions

Ideas

Write and solve proportions

Examples

Example 1

Example 2

Example 3

Outcomes

7.CE.2

7.CE.2b

7.CE.2c

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