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Ratio &
Proportion

Master proportional reasoning to solve ratio problems with confidence: from setting up equations to cross-multiplying your way to the answer.

SSAT Upper Level: Quantitative

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
\(\frac{a}{b}\)
Colon Form
\(a : b\)
Word Form
"a to b"

Visualizing Ratios

A bar model shows how parts relate to the whole. Below, a \(3 : 2\) ratio splits 5 equal parts: 3 blue and 2 gold.

1 2 3 4 5 3 parts 2 parts

What Is a Proportion?

A proportion is an equation stating that two ratios are equal:

If you know three of the four values, you can find the missing one using cross-multiplication:

The Cross-Multiplication Rule

a b = c d
\(\color{#818cf8}a \color{white}\times \color{#f87171}d \color{white}= \color{#4ade80}b \color{white}\times \color{#fbbf24}c\)
Cross products are equal in every true proportion.

Ratio vs. Proportion

Ratio Proportion
Definition Comparison of two quantities Equation of two equal ratios
Form \(a : b\) or \(\frac{a}{b}\) \(\frac{a}{b} = \frac{c}{d}\)
Purpose Describe a relationship Solve for an unknown
Example \(3 : 5\) \(\frac{3}{5} = \frac{x}{20}\)

Part-to-Part vs. Part-to-Whole

A basket has 6 apples and 4 oranges (10 fruits total).

Part-to-Part
Apples to oranges: \(6 : 4 = 3 : 2\)
Part-to-Whole
Apples to total: \(6 : 10 = 3 : 5\)

How to Solve a Proportion

1
Set up the proportion. Keep units consistent. Either put matching units across from each other (\(\frac{\text{unit A}}{\text{unit B}} = \frac{\text{unit A}}{\text{unit B}}\)) or in the same fraction (\(\frac{\text{unit A}}{\text{unit A}} = \frac{\text{unit B}}{\text{unit B}}\)).
2
Cross-multiply. Multiply diagonally to get an equation: \(a \times d = b \times c\).
3
Solve for the unknown. Divide both sides by the coefficient of the variable.
4
Check your answer. Plug back in and verify both ratios simplify to the same value.

Example 1: Finding a Missing Value

If 3 notebooks cost $7.50, how much do 8 notebooks cost?

1
Set up: \(\dfrac{3}{7.50} = \dfrac{8}{x}\)
2
Cross-multiply: \(3 \times x = 7.50 \times 8\)
3
Simplify: \(3x = 60\)
4
Solve: \(x = \dfrac{60}{3} = 20\)

Example 2: Splitting a Total

Ali and Ben share $180 in the ratio \(2 : 3\). How much does each receive?

$36 $36 $36 $36 $36 Ali: $72 Ben: $108
1
Total parts: \(2 + 3 = 5\)
2
Value per part: \(\dfrac{180}{5} = 36\)
3
Ali: \(2 \times 36 =\) $72   Ben: \(3 \times 36 =\) $108

Example 3: Multi-Step Ratio

A recipe uses flour, sugar, and butter in the ratio \(5 : 3 : 2\). If you need 750 g of flour, how much sugar and butter do you need?

1
Find the multiplier: Flour is 5 parts, so \(\dfrac{750}{5} = 150\) g per part.
2
Sugar (3 parts): \(3 \times 150 = \) 450 g
3
Butter (2 parts): \(2 \times 150 = \) 300 g

Equivalent Ratios & Scaling

Two ratios are equivalent if one can be obtained by multiplying (or dividing) both terms of the other by the same non-zero number.

\(2 : 5\)
× 2
\(4 : 10\)
× 3
\(12 : 30\)
simplest equivalent equivalent
\(18 : 24\)
÷ 6
\(3 : 4\)
simplest form

Direct vs. Inverse Proportion

Direct Proportion
As one quantity increases, the other increases at the same rate. \(y = kx\). Example: more hours worked, more pay earned.
Inverse Proportion
As one quantity increases, the other decreases so the product stays constant. \(xy = k\). Example: more workers, fewer hours to finish.

Graphing Proportional Relationships

x y Direct: y = 2x 1 2 3 4 2 4 6 8 x y Inverse: xy = 12 1 2 3 4 2 4 6 8

Common Mistakes

✕ Swapped Terms
Writing \(\frac{3}{5}\) when the problem says "5 to 3". Always match the order stated in the question.
✕ Adding Instead of Multiplying
Cross-multiplication means \(a \times d\), not \(a + d\). The operation is multiplication, not addition.
✕ Forgetting Units
Mixing units (e.g., inches with feet) in the same proportion yields a wrong answer. Convert first.

Quick Reference Summary

Concept Formula / Rule Example
Ratio \(a : b\) \(3 : 7\)
Proportion \(\frac{a}{b} = \frac{c}{d}\) \(\frac{3}{7} = \frac{9}{21}\)
Cross-Multiply \(ad = bc\) \(3 \times 21 = 7 \times 9 = 63\)
Splitting a Total Part \(= \frac{\text{share}}{\text{total parts}} \times \text{total}\) \(\frac{3}{5} \times 100 = 60\)
Equivalent Ratios Multiply/divide both terms by same number \(4 : 6 = 2 : 3\)

Question 1 of 5

If \(\frac{x}{12} = \frac{5}{4}\), what is the value of \(x\)?

Question 2 of 5

A map uses a scale of \(1 : 50{,}000\). Two cities are 8 cm apart on the map. What is the actual distance in kilometers?

Question 3 of 5

Red and blue beads are in the ratio \(3 : 5\). If there are 40 beads in total, how many are red?

Question 4 of 5

If 4 workers can paint a fence in 6 hours, how many hours would it take 12 workers (at the same rate)?

Question 5 of 5

The ratio of cats to dogs at a shelter is \(7 : 4\). If there are 28 cats, which proportion correctly finds the number of dogs, \(d\)?