Proportional Relationships in Tables & Graphs

Q: Which numbers do I multiply when I cross multiply?

The diagonals. In $\dfrac{a}{b} = \dfrac{c}{d}$, multiply $a \cdot d$ and $b \cdot c$, then set those two products equal: $ad = bc$. Top-left with bottom-right, bottom-left with top-right.

A proportion is a statement that two ratios are equal: $\dfrac{3}{4} = \dfrac{9}{12}$ . When one of the four numbers is missing — $\dfrac{x}{12} = \dfrac{3}{4}$ — solving the proportion means finding the value that keeps both sides equal. This one skill powers map scales, recipe scaling, similar triangles, and most percent problems.

The standard tool is cross multiplication: multiply diagonally across the equals sign, set the two products equal, and solve the little equation that's left. It works every time, because multiplying both sides of the equation by both denominators clears the fractions in one move.

The three moves

Move 1: cross multiply. In $\dfrac{a}{b} = \dfrac{c}{d}$ , multiply the top-left by the bottom-right and the bottom-left by the top-right. That gives the products $a \cdot d$ and $b \cdot c$ .

Move 2: set the products equal. If the two ratios really are equal, then $a \cdot d = b \cdot c$ . Now the fractions are gone and you have a one-step equation.

Move 3: divide. One side has the variable times a number; divide both sides by that number and you're done. Always finish by plugging your answer back into the original proportion — both sides should simplify to the same fraction.

Why cross multiplication works

It isn't a magic trick. Starting from $\dfrac{x}{12} = \dfrac{3}{4}$ , multiply both sides by $12$ and by $4$ . On the left, the $12$ s cancel and you're left with $4x$ . On the right, the $4$ s cancel and you're left with $3 \cdot 12$ . That's exactly the diagonal products — cross multiplication is just 'multiply both sides by both denominators' done in one step.

Because of that, it only works when each side is a single fraction. If a side looks like $\dfrac{x}{5} + 2$ , you can't cross multiply yet — deal with the $+\,2$ first.

Setting up word problems

The setup matters more than the algebra. Keep the same units in the same position on both sides. If the left ratio is $\dfrac{\text{miles}}{\text{inches}}$ , the right ratio must also be $\dfrac{\text{miles}}{\text{inches}}$ — never flipped. A quick sanity check: read each ratio out loud with its units. If the two sides don't say the same kind of sentence, the setup is wrong.

Worked examples

Example 1: variable on top

Solve $\dfrac{x}{12} = \dfrac{3}{4}$ .

Cross multiply

4 \cdot x = 3 \cdot 12

Simplify each product

4x = 36

Divide both sides by

4

x = 9

Check:

\dfrac{9}{12} = \dfrac{3}{4}

✓

Answer: $x = 9$

Example 2: variable on the bottom

Solve $\dfrac{5}{8} = \dfrac{30}{n}$ .

Cross multiply

5 \cdot n = 8 \cdot 30

Simplify each product

5n = 240

Divide both sides by

5

n = 48

Check:

\dfrac{30}{48} = \dfrac{5}{8}

✓

Answer: $n = 48$

Example 3: a word problem

Three identical buses carry $132$ students in total. At that rate, how many students can $7$ buses carry?

Set up matching ratios of students to buses

\dfrac{132}{3} = \dfrac{s}{7}

Cross multiply

3 \cdot s = 132 \cdot 7

Simplify the right side

3s = 924

Divide both sides by

3

s = 308

Answer: $308$ students. (Sanity check: $132 \div 3 = 44$ per bus, and $44 \cdot 7 = 308$ .)

Example 4: a decimal answer

On a map, $2$ inches represents $15$ miles. How many miles does $7$ inches represent?

Set up matching ratios of inches to miles

\dfrac{2}{15} = \dfrac{7}{m}

Cross multiply

2 \cdot m = 15 \cdot 7

Simplify the right side

2m = 105

Divide both sides by

2

m = 52.5

Answer: $52.5$ miles — a decimal is a perfectly good answer.

Try one yourself

Common questions

Which numbers do I multiply when I cross multiply?

The diagonals. In $\dfrac{a}{b} = \dfrac{c}{d}$ , multiply $a \cdot d$ and $b \cdot c$ , then set those two products equal: $ad = bc$ . Top-left with bottom-right, bottom-left with top-right.

Does it matter which side the variable is on, or whether it's on top or bottom?

No. Cross multiplication handles all four positions the same way — after the diagonal products, you always have a one-step equation. If the variable ends up multiplied by a number, divide by that number.

Can I solve a proportion without cross multiplying?

Often, yes. If one denominator is a clean multiple of the other — like $\dfrac{x}{12} = \dfrac{3}{4}$ , where $4 \cdot 3 = 12$ — just multiply the top by the same factor: $3 \cdot 3 = 9$ . Use the shortcut when the numbers cooperate and cross multiplication when they don't.

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Proportional Relationships in Tables & Graphs

The three moves

Why cross multiplication works

Setting up word problems

Worked examples

Example 1: variable on top

Example 2: variable on the bottom

Example 3: a word problem

Example 4: a decimal answer

Try one yourself

Common questions

Keep learning

Unit Rate as Slope

Percent of a Number

AA Similarity