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\(3\)
\(6\)
\(4\)
\(8\)
\(=\)

Determining
Proportions

Learn to decide whether two ratios are equivalent and form a true proportion: using simplification, cross products, and logical reasoning.

Pre-Algebra

What Is a Ratio?

A ratio compares two quantities by division. If a class has 12 boys and 8 girls, the ratio of boys to girls is \(\frac{12}{8}\), which simplifies to \(\frac{3}{2}\).

Fraction Form
\(\frac{a}{b}\) where \(b \neq 0\)
Colon Form
\(a : b\) read as "a to b"
Word Form
"a to b" in plain language

What Is a Proportion?

A proportion is an equation that states two ratios are equal. If the ratio \(\frac{a}{b}\) equals the ratio \(\frac{c}{d}\), we write:

Ratio 1
\(\frac{1}{2}\)
 Ratio 2
\(\frac{3}{6}\)
 =
Both simplify to \(\frac{1}{2}\) ✓

Why Do Proportions Matter?

Proportions appear everywhere: scaling recipes, reading maps, converting units, and comparing rates. Recognizing when two ratios are truly equivalent keeps calculations accurate.

Cooking
Doubling a recipe keeps ingredient ratios the same.
Maps
Scale factors relate map distance to real distance.
Unit Rates
Comparing prices: "Is \(\frac{\$6}{3\text{ lb}}\) the same deal as \(\frac{\$10}{5\text{ lb}}\)?"

The Cross-Product Rule

\(a\)
\(b\)
 =
\(c\)
\(d\)
 Cross products:
\(a\) × \(d\) = \(b\) × \(c\)

Simplifying to Check

Another way to verify a proportion is to simplify both ratios to lowest terms. If they reduce to the same fraction, they form a proportion.

\(\frac{6}{9}\)
 ÷ 3 each
\(\frac{2}{3}\)
\(\frac{10}{15}\)
 ÷ 5 each Same simplified form ✓ Proportion

Ratio vs. Proportion

Ratio Proportion
What is it? A comparison of two quantities An equation of two equal ratios
Form \(\frac{a}{b}\) \(\frac{a}{b} = \frac{c}{d}\)
Example \(\frac{3}{4}\) \(\frac{3}{4} = \frac{6}{8}\)
How many ratios? One Two (and they must be equal)

How to Determine a Proportion

1
Write both ratios as fractions. Ensure numerator and denominator match the same type of quantity.
2
Choose a method: simplify to lowest terms or compute cross products.
3
Simplify: Divide each numerator and denominator by their GCF. Compare the results.
4
Cross products: Compute \(a \times d\) and \(b \times c\). If they are equal, it is a proportion.
5
State your conclusion with justification: "These ratios do / do not form a proportion because..."

Example 1: Simple Check

Do \(\frac{4}{6}\) and \(\frac{10}{15}\) form a proportion?

Cross-Product Method
\(4\) × \(15\) = \(60\)
\(6\) × \(10\) = \(60\)
\(60 = 60\)
 ✓ True Proportion

Example 2: Not a Proportion

Do \(\frac{3}{5}\) and \(\frac{7}{10}\) form a proportion?

Cross-Product Method
\(3\) × \(10\) = \(30\)
\(5\) × \(7\) = \(35\)
\(30 \neq 35\)
 ✕ Not a Proportion

Example 3: Larger Numbers

Do \(\frac{12}{18}\) and \(\frac{20}{30}\) form a proportion?

1
Simplify the first ratio: GCF of 12 and 18 is 6. \(\frac{12}{18} = \frac{12 \div 6}{18 \div 6} = \frac{2}{3}\)
2
Simplify the second ratio: GCF of 20 and 30 is 10. \(\frac{20}{30} = \frac{20 \div 10}{30 \div 10} = \frac{2}{3}\)
3
Compare: Both simplify to \(\frac{2}{3}\), so they are proportional.

Example 4: Real-World Context

A store sells 5 apples for \(\$3\). Another sells 8 apples for \(\$5\). Are the price-per-apple ratios proportional?

1
Write ratios: Store A: \(\frac{3}{5}\), Store B: \(\frac{5}{8}\) (dollars per apple)
2
Cross products: \(3 \times 8 = 24\) and \(5 \times 5 = 25\)
3
Conclusion: \(24 \neq 25\), so the ratios are not proportional. The stores charge different rates per apple.

Common Mistakes to Avoid

✕ Swapping the Order
Writing \(\frac{3}{4}\) when the ratio should be \(\frac{4}{3}\). Order matters: boys-to-girls is not girls-to-boys.
✕ Cross-Multiplying Wrong
For \(\frac{a}{b} = \frac{c}{d}\), multiplying \(a \times b\) instead of \(a \times d\). Always go diagonally.
✕ Not Simplifying Fully
Reducing \(\frac{8}{12}\) to \(\frac{4}{6}\) instead of \(\frac{2}{3}\). Always divide by the greatest common factor.

Two Methods, One Answer

Simplify Cross Products
What you do Divide each ratio by its GCF Multiply diagonally and compare
Best when Numbers share obvious common factors Numbers are large or have no obvious GCF
Result if proportional Same simplified fraction Equal cross products
Example \(\frac{6}{9} = \frac{2}{3}\) \(6 \times 15 = 9 \times 10\)

Quick Reference Summary

Proportion
\(\frac{a}{b} = \frac{c}{d}\) when the two ratios are equivalent.
Cross Products
If \(a \times d = b \times c\), then the ratios form a proportion.
Simplify
Divide numerator and denominator by the GCF. Compare results.
Not Proportional
Cross products are unequal, or simplified forms differ.

Question 1 of 5

Do \(\frac{5}{8}\) and \(\frac{15}{24}\) form a proportion?

Question 2 of 5

Which pair of ratios forms a proportion?

Question 3 of 5

A recipe uses 3 cups of flour for every 2 cups of sugar. If you use 9 cups of flour, how many cups of sugar keep the proportion?

Question 4 of 5

What are the cross products of \(\frac{7}{12}\) and \(\frac{5}{9}\)?

Question 5 of 5

A student says \(\frac{8}{14}\) and \(\frac{12}{21}\) form a proportion because both numerators are even. Is the student correct?