What is the difference between a ratio and a proportion?

A ratio compares two quantities (like 3:4), while a proportion says two ratios are equal (like \frac{3}{4} = \frac{6}{8}). Proportions are useful when one ratio is missing a number and you need to find it.

When should I use cross-multiplication instead of just solving the equation?

Cross-multiplication is a shortcut specifically for proportions—it turns \frac{a}{b} = \frac{c}{d} into the simpler equation ad = bc without fractions. Use it whenever you see two fractions set equal to each other.

Can both a and b be zero?

No. If b = 0, the left side \frac{a}{b} is undefined, so the proportion \frac{a}{b} = \frac{c}{d} doesn't exist. Similarly, d cannot be zero. But a or c can be zero.

Do I need to simplify fractions before setting up a proportion?

No—you can cross-multiply right away. But simplifying first often makes the arithmetic easier, especially in word problems. Either way, you'll get the same answer.

Proportion

Use Proportion when two ratios are equal to find a missing value by cross-multiplying to solve equations with fractions.

\frac{a}{b} = \frac{c}{d} \iff ad = bc

Solve a problem with Proportion

Type the problem. The solver will use Proportion where Proportion is the right tool, and tell you when it is not.

What each symbol means

What Proportion takes

Symbol	Meaning
$a$	The first numerator (top value of the left-hand fraction); if you confuse it with the denominator, the proportion flips and gives a wrong answer.
$b$	The first denominator (bottom value of the left-hand fraction); if $b = 0$ , the fraction $\frac{a}{b}$ is undefined and makes the whole proportion invalid.
$c$	The second numerator (top value of the right-hand fraction); must match the type of quantity as $a$ (e.g., both distances or both costs) for the proportion to make sense.
$d$	The second denominator (bottom value of the right-hand fraction); if $d = 0$ , the fraction $\frac{c}{d}$ is undefined and the proportion fails.

When to use it

When you see two equal fractions with one unknown value and need to find it.

Level

Usually taught in: Pre-Algebra · Appears on: SAT, ACT

Worked examples

1. Find the missing value in a simple proportion

Problem

Solve

\frac{3}{4} = \frac{x}{8}

for

x

$3 \cdot 8 = 4 \cdot x$
Cross-multiply the proportion by pairing diagonals: the numerator of each fraction multiplies with the denominator of the other.
$24 = 4x$
Compute $3 \cdot 8 = 24$ on the left side.
$x = 6$
Divide both sides by 4 to isolate $x$ .

Answer: $x = 6$

This is a straightforward proportion—you know three values and cross-multiply to find the fourth. The proportion $\frac{3}{4} = \frac{6}{8}$ is true because both simplify to $\frac{3}{4}$ .

2. Solve a proportion with a negative denominator

Problem

Solve

\frac{x}{-5} = \frac{12}{20}

for

x

$x \cdot 20 = (-5) \cdot 12$
Cross-multiply: the left numerator $x$ times the right denominator 20, equals the left denominator $-5$ times the right numerator 12.
$20x = -60$
Carefully multiply $(-5) \cdot 12 = -60$ , keeping the negative sign.
$x = -3$
Divide both sides by 20. You can verify: $\frac{-3}{-5} = \frac{3}{5}$ and $\frac{12}{20} = \frac{3}{5}$ ✓.

Answer: $x = -3$

Negative numbers are easy to mishandle. The key is to track the sign through the cross product: $(-5) \cdot 12 = -60$ , giving $20x = -60$ , so $x = -3$ . Always check your answer by substituting back into the original proportion.

3. Use proportion to split work between two crews

Problem

Two construction crews split a job proportionally to their available hours. Crew A works 40 hours and completes 15 units of work. Crew B works 32 hours at the same productivity rate. How many units of work does Crew B complete?

$\frac{15}{40} = \frac{w}{32}$
Set up the proportion so both sides have the same units: $\frac{\text{units}}{\text{hours}}$ for each crew. Let $w$ be the units Crew B completes.
$15 \cdot 32 = 40 \cdot w$
Cross-multiply the proportion: the numerators and denominators pair diagonally.
$480 = 40w$
Compute $15 \cdot 32 = 480$ on the left side.
$w = 12$
Divide both sides by 40 to solve for $w$ .

Answer: $w = 12 \text{ units}$

Proportions let you scale rates across different contexts. Since both crews work at the same rate (units per hour), the proportion captures this without needing to compute the rate explicitly. Crew B completes 12 units in 32 hours, maintaining the same ratio as Crew A's 15 units in 40 hours.

Common mistakes

Where Proportion usually goes wrong

The mistake	Why it is wrong	The fix
Given $\frac{4}{6} = \frac{x}{9}$ , a student cross-multiplies as $4 \cdot x = 6 \cdot 9$ instead of $4 \cdot 9 = 6 \cdot x$	Cross-multiplication must pair diagonal terms: outer times outer and inner times inner, not adjacent pairs.	$4 \cdot 9 = 36$ and $6 \cdot x$ , so $36 = 6x$ and $x = 6$ . Check: $\frac{4}{6} = \frac{2}{3}$ and $\frac{6}{9} = \frac{2}{3}$ ✓.
When solving $\frac{10}{x+3} = \frac{2}{5}$ , a student gets $x = 22$ and doesn't verify that the denominator is defined	The original proportion is undefined if $x + 3 = 0$ , so you must check that your answer doesn't violate this.	Verify: $x + 3 = 22 + 3 = 25 \neq 0$ , so the answer is valid. If solving had given $x = -3$ , it would be rejected.
In a scale problem, 'the ratio of boys to girls is 3 to 5,' a student writes $\frac{5}{3}$ instead of $\frac{3}{5}$	The phrasing 'A to B' specifies the order: A is first, so it belongs in the numerator.	Boys to girls is $\frac{\text{boys}}{\text{girls}} = \frac{3}{5}$ . If you reversed it, you'd be comparing girls to boys ( $\frac{5}{3}$ ), which is a different proportion.

Tips and when to use something else

Use Proportion to find a missing value when two ratios are equal—cross-multiply to solve for the unknown term.
For cleaner arithmetic, simplify both fractions before cross-multiplying; $\frac{6}{9} = \frac{8}{12}$ becomes $\frac{2}{3} = \frac{2}{3}$ , an identity.
In word problems, 'for every' and 'per' signal a proportion setup—e.g., 'for every 3 red marbles there are 5 blue' becomes $\frac{3}{5}$ .
If you need to find only one ratio's value without a comparison or equation, use Unit Rate instead—Proportion is for when you have two ratios and one term is missing.

Related formulas

RatioA ratio compares two quantities to show their relationship, and can be written as a fraction or with a colon to help you scale or find missing values.Unit RateA unit rate is a ratio that compares a quantity to exactly one unit of measure, helping you compare values or find the cost per item.Equivalent FractionsMultiply or divide both parts of a fraction by the same non-zero number to create an equivalent fraction with the same value.Multiplying FractionsMultiply fractions by multiplying numerators and denominators separately, letting you combine fractional amounts and solve fraction problems efficiently.

Frequently asked questions

What is the difference between a ratio and a proportion?: A ratio compares two quantities (like $3:4$ ), while a proportion says two ratios are equal (like $\frac{3}{4} = \frac{6}{8}$ ). Proportions are useful when one ratio is missing a number and you need to find it.
When should I use cross-multiplication instead of just solving the equation?: Cross-multiplication is a shortcut specifically for proportions—it turns $\frac{a}{b} = \frac{c}{d}$ into the simpler equation $ad = bc$ without fractions. Use it whenever you see two fractions set equal to each other.
Can both $a$ and $b$ be zero?: No. If $b = 0$ , the left side $\frac{a}{b}$ is undefined, so the proportion $\frac{a}{b} = \frac{c}{d}$ doesn't exist. Similarly, $d$ cannot be zero. But $a$ or $c$ can be zero.
Do I need to simplify fractions before setting up a proportion?: No—you can cross-multiply right away. But simplifying first often makes the arithmetic easier, especially in word problems. Either way, you'll get the same answer.

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Reviewed 2026-09-18