Lessons 39: Proportions

New Concept

A proportion is a statement that two ratios are equal. The cross products of equal ratios are equal.

What’s next

This lesson introduces the core idea of proportions. Next, you'll work through examples showing how to use cross products to find a missing value in a proportion.

Property

A proportion is a statement that two ratios are equal. Read $\frac{16}{20} = \frac{4}{5}$ as "sixteen is to twenty as four is to five."

Examples

• Cost check: $\frac{2 \text{ apples}}{1 \text{ dollar}} = \frac{10 \text{ apples}}{5 \text{ dollars}}$.
• A false proportion: $\frac{1}{2} = \frac{3}{5}$.
• A true proportion: $\frac{3}{4} = \frac{9}{12}$.

Explanation

Proportions show balanced relationships. If 2 apples cost 1 dollar, a proportion confirms 10 apples cost 5 dollars. Things stay fair as they scale up or down!

Property

Multiply the upper term of one ratio by the lower term of the other. The cross products of equal ratios are equal.

Examples

• Check $\frac{4}{5} = \frac{8}{10}$: $4 \times 10 = 40$ and $5 \times 8 = 40$. Equal!
• Check $\frac{2}{3} = \frac{5}{8}$: $2 \times 8 = 16$ and $3 \times 5 = 15$. Not equal!

Explanation

Think of the cross product as a super-fast trick to check if two ratios are truly equal partners. Multiply diagonally! If the answers match, it’s a valid proportion. It's a trusty lie detector for ratios.

Property

Step 1: Find the cross products, which are equal. Step 2: Divide the known product by the known factor to find the missing term.

Examples

• Solve $\frac{6}{x} = \frac{3}{5}$: $3x = 30$, so $x=10$.
• Solve $\frac{n}{12} = \frac{2}{3}$: $3n = 24$, so $n=8$.
• Solve $\frac{5}{8} = \frac{10}{d}$: $5d = 80$, so $d=16$.

Explanation

Finding a missing number is a treasure hunt. Cross-multiplication gives you a map ($20n=360$), and division leads you straight to the treasure ($n=18$). X marks the spot!