Lesson 5: Solving Proportions and Scale Drawings

Property

A proportion is an equation of the form $\frac{a}{b} = \frac{c}{d}$, where $b \neq 0, d \neq 0$. The proportion states two ratios or rates are equal. For any proportion of this form, its cross products are equal: $a \cdot d = b \cdot c$. Cross products can be used to test whether a proportion is true.

Examples

• The sentence "4 is to 9 as 20 is to 45" is written as the proportion $\frac{4}{9} = \frac{20}{45}$.
• To determine if $\frac{6}{11} = \frac{30}{55}$ is a proportion, we check the cross products. Since $6 \cdot 55 = 330$ and $11 \cdot 30 = 330$, the equation is a proportion.
• To check if $\frac{8}{10} = \frac{30}{40}$ is a proportion, we find the cross products. $8 \cdot 40 = 320$ and $10 \cdot 30 = 300$. Since the products are not equal, it is not a proportion.

Explanation

A proportion is a statement that two ratios are equal, like a balanced scale. The cross-product rule is a quick check: if the products of the numbers on the diagonal are equal, the ratios form a true proportion.

Property

To solve a proportion for a variable, you can use the property of equal cross products. For a proportion like $\frac{a}{b} = \frac{c}{d}$, set the cross products equal, $a \cdot d = b \cdot c$, and then solve the resulting equation for the unknown variable.

Examples

• To solve $\frac{x}{63} = \frac{4}{7}$, set cross products equal: $7 \cdot x = 63 \cdot 4$. This gives $7x = 252$, so $x = 36$.
• To solve $\frac{144}{a} = \frac{9}{4}$, set cross products equal: $144 \cdot 4 = 9 \cdot a$. This gives $576 = 9a$, so $a = 64$.
• To solve $\frac{52}{91} = \frac{-4}{y}$, set cross products equal: $52 \cdot y = 91 \cdot (-4)$. This gives $52y = -364$, so $y = -7$.

Explanation

When a number in a proportion is missing, we use algebra to find it. The cross-product method is the fastest way to turn the proportion into a simple equation that you can solve for the unknown value.