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Ratios, Rates &
Proportions

Master the art of comparing quantities, understanding unit rates, and solving proportional equations for GRE Quantitative success.

GRE Quantitative

What Is a Ratio?

A ratio is a comparison of two quantities by division. If a class has 12 boys and 8 girls, the ratio of boys to girls is \(\frac{12}{8} = \frac{3}{2}\), written as \(3 : 2\).

Boys 12 Girls 8 :
Simplifies to \(3 : 2\)

What Is a Rate?

A rate is a ratio that compares two quantities with different units. A unit rate expresses the amount per one unit of the second quantity.

Rate
\(240\) miles in \(4\) hours: \(\frac{240 \text{ mi}}{4 \text{ hr}}\)
Unit Rate
Divide to get \(\frac{240}{4} = 60\) miles per hour
Why It Matters
Unit rates make comparison easy: is \(60\) mph faster than \(55\) mph?

What Is a Proportion?

A proportion is an equation stating that two ratios are equal. If \(\frac{a}{b} = \frac{c}{d}\), then the four quantities \(a, b, c, d\) are in proportion.

Ratio 1 a b = Ratio 2 c d
Cross multiply: \(a \times d = b \times c\)

The Cross-Multiplication Rule

If \(\frac{a}{b} = \frac{c}{d}\) where \(b \neq 0\) and \(d \neq 0\), then:

\(a \times d = b \times c\)
 a c b d =
\(a \times d\)
\(b \times c\)

Properties of Proportions

Given \(\frac{a}{b} = \frac{c}{d}\), several equivalent forms exist. Knowing these lets you rewrite proportions to match answer choices on the GRE.

1
Cross products: \(a \times d = b \times c\)
2
Reciprocal: \(\frac{b}{a} = \frac{d}{c}\)
3
Alternation: \(\frac{a}{c} = \frac{b}{d}\)
4
Addition: \(\frac{a + b}{b} = \frac{c + d}{d}\)

Ratio vs. Rate vs. Proportion

Ratio Rate Proportion
Definition Comparison of like quantities Comparison of unlike quantities Equation of two ratios
Units Same (e.g., cm to cm) Different (e.g., miles per hour) Matches the ratios involved
Example \(3 : 5\) \(\$4.50\) per lb \(\frac{3}{5} = \frac{6}{10}\)
Key Use Part-to-part or part-to-whole Speed, price, density Solving for unknowns

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

A ratio of \(3:2\) (boys to girls) can be interpreted two ways. Understanding which interpretation the GRE question requires is critical.

Part-to-Part
Boys to girls = \(3:2\). For every 3 boys, there are 2 girls.
Part-to-Whole
Boys to total = \(3:5\). Boys are \(\frac{3}{5}\) of the class.
B B B G G
Total = \(3 + 2 = 5\) parts

How to Solve Proportions

1
Set up the proportion with matching units on top and bottom: \(\frac{\text{same unit}}{\text{same unit}} = \frac{\text{same unit}}{\text{same unit}}\).
2
Cross-multiply to create a linear equation: \(a \times d = b \times c\).
3
Solve for the unknown by isolating the variable.
4
Verify your answer by plugging back in: do both ratios simplify to the same value?

Example 1: Finding the Unknown

If \(\frac{x}{6} = \frac{10}{15}\), find \(x\).

1
Simplify the known ratio: \(\frac{10}{15} = \frac{2}{3}\)
2
Rewrite: \(\frac{x}{6} = \frac{2}{3}\)
3
Cross-multiply: \(x \times 3 = 6 \times 2\), so \(3x = 12\)
4
Solve: \(x = \frac{12}{3} = 4\)

Example 2: Unit Rate Comparison

Store A sells 5 pounds of apples for \(\$8.75\). Store B sells 3 pounds for \(\$4.98\). Which store offers the better deal?

Store A
\(\frac{\$8.75}{5 \text{ lb}} = \$1.75\) per lb
Store B
\(\frac{\$4.98}{3 \text{ lb}} = \$1.66\) per lb

Example 3: Multi-Step Ratio Problem

In a mixture, the ratio of water to concentrate is \(5:2\). If the total mixture is 49 liters, how many liters of concentrate are needed?

1
Total parts: \(5 + 2 = 7\) parts
2
Value of one part: \(\frac{49}{7} = 7\) liters per part
3
Concentrate: \(2 \times 7 = 14\) liters
7L 7L 7L 7L 7L 7L 7L
Water: \(5 \times 7 = 35\) L
Conc: \(2 \times 7 = 14\) L

Example 4: GRE-Style Proportion

A map uses a scale of \(2\) cm \(= 35\) km. If two cities are \(7.5\) cm apart on the map, what is the actual distance?

1
Set up: \(\frac{2 \text{ cm}}{35 \text{ km}} = \frac{7.5 \text{ cm}}{x \text{ km}}\)
2
Cross-multiply: \(2x = 35 \times 7.5 = 262.5\)
3
Solve: \(x = \frac{262.5}{2} = 131.25\) km

Direct vs. Inverse Proportion

Direct Proportion
As \(x\) increases, \(y\) increases at the same rate. Formula: \(y = kx\). The ratio \(\frac{y}{x} = k\) is constant.
Inverse Proportion
As \(x\) increases, \(y\) decreases. Formula: \(y = \frac{k}{x}\). The product \(x \times y = k\) is constant.
x y
Direct: \(y = kx\)
 x y
Inverse: \(y = \frac{k}{x}\)

Common Mistakes to Avoid

✕ Mixing Up Units
Setting up \(\frac{\text{miles}}{\text{hours}} = \frac{\text{hours}}{\text{miles}}\). Keep the same unit in each position: top-left matches top-right.
✕ Part vs. Whole Confusion
A \(3:2\) ratio has 5 total parts, not 3. When finding a fraction, use \(\frac{3}{5}\), not \(\frac{3}{2}\).
✕ Forgetting to Simplify
Working with \(\frac{24}{36}\) instead of simplifying to \(\frac{2}{3}\) first. Larger numbers lead to more arithmetic errors.

Question 1 of 5

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

Question 2 of 5

A car travels \(180\) miles in \(3\) hours. At the same rate, how far will it travel in \(7\) hours?

Question 3 of 5

The ratio of red to blue marbles in a bag is \(4:7\). If there are 56 blue marbles, how many red marbles are there?

Question 4 of 5

If \(y\) is inversely proportional to \(x\), and \(y = 12\) when \(x = 5\), what is \(y\) when \(x = 15\)?

Question 5 of 5

A recipe calls for flour and sugar in a ratio of \(5:3\). If you use \(2.4\) cups of sugar, how many cups of flour do you need?