How to Solve Proportions Using Cross Multiplication

Grade: 6-7 | Topic: Arithmetic

What You Will Learn

After this lesson you will know how to determine whether two ratios form a proportion, solve for a missing value in a proportion using cross multiplication, and apply proportions to scale problems, recipes, and maps. You will also understand why cross multiplication works mathematically.

Theory

What is a proportion?

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

$\frac{a}{b} = \frac{c}{d}$

For example, $\frac{3}{4} = \frac{6}{8}$ is a proportion because both fractions simplify to the same value.

Testing whether two ratios form a proportion

To check if $\frac{a}{b}$ and $\frac{c}{d}$ are a proportion, compute the cross products:

$a \times d \stackrel{?}{=} b \times c$

If the cross products are equal, the ratios form a proportion. If not, they do not.

Example: Is $\frac{4}{6} = \frac{10}{15}$ a proportion?

$4 \times 15 = 60 \qquad 6 \times 10 = 60$

Since $60 = 60$, yes — it is a proportion.

Cross multiplication to solve for an unknown

When one value in a proportion is unknown, cross multiplication converts the proportion into a simple equation:

$\frac{a}{b} = \frac{c}{x} \implies a \times x = b \times c \implies x = \frac{b \times c}{a}$

Why it works: Multiplying both sides of $\frac{a}{b} = \frac{c}{x}$ by $bx$ gives:

$bx \cdot \frac{a}{b} = bx \cdot \frac{c}{x}$

$ax = bc$

This is exactly the cross-multiplication result. It is simply a shortcut for clearing both denominators at once.

Setting up a proportion correctly

The key to using proportions is making sure corresponding quantities are in the same positions. Keep the same type of measurement in each row (or column):

$\frac{\text{item A quantity 1}}{\text{item A quantity 2}} = \frac{\text{item B quantity 1}}{\text{item B quantity 2}}$

For example, if 3 apples cost $2.25 and you want to find the cost of 7 apples:

$\frac{3 \text{ apples}}{\$2.25} = \frac{7 \text{ apples}}{\$x}$

Both numerators are apples; both denominators are dollars.

Worked Examples

Example 1: Basic proportion (easy)

Problem: Solve for $x$: $\dfrac{5}{8} = \dfrac{x}{24}$

Step 1: Cross-multiply.

$5 \times 24 = 8 \times x$

$120 = 8x$

Step 2: Divide both sides by 8.

$x = \frac{120}{8} = 15$

Answer: $x = 15$

Check: $\frac{5}{8} = \frac{15}{24}$. Simplify: $\frac{15}{24} = \frac{5}{8}$ ✓

Example 2: Unknown in the denominator (easy)

Problem: Solve for $n$: $\dfrac{9}{n} = \dfrac{3}{7}$

Step 1: Cross-multiply.

$9 \times 7 = n \times 3$

$63 = 3n$

Step 2: Divide both sides by 3.

$n = 21$

Answer: $n = 21$

Check: $\frac{9}{21} = \frac{3}{7}$ ✓

Example 3: Recipe scaling (medium)

Problem: A cookie recipe uses 2 cups of flour for every 3 cups of sugar. If you use 5 cups of flour, how many cups of sugar do you need?

Step 1: Set up the proportion with matching positions.

$\frac{2 \text{ flour}}{3 \text{ sugar}} = \frac{5 \text{ flour}}{x \text{ sugar}}$

Step 2: Cross-multiply.

$2x = 3 \times 5 = 15$

Step 3: Divide both sides by 2.

$x = \frac{15}{2} = 7.5$

Answer: You need 7.5 cups of sugar.

Example 4: Map scale problem (medium)

Problem: On a map, 4 cm represents 30 km. If two cities are 11 cm apart on the map, what is the actual distance?

Step 1: Set up the proportion.

$\frac{4 \text{ cm}}{30 \text{ km}} = \frac{11 \text{ cm}}{d \text{ km}}$

Step 2: Cross-multiply.

$4d = 30 \times 11 = 330$

Step 3: Divide by 4.

$d = \frac{330}{4} = 82.5$

Answer: The actual distance is 82.5 km.

Example 5: Multi-step proportion with decimals (challenging)

Problem: A car uses 7.5 liters of fuel to travel 100 km. How much fuel is needed to travel 340 km?

Step 1: Set up the proportion.

$\frac{7.5 \text{ L}}{100 \text{ km}} = \frac{f \text{ L}}{340 \text{ km}}$

Step 2: Cross-multiply.

$7.5 \times 340 = 100 \times f$

$2550 = 100f$

Step 3: Divide by 100.

$f = 25.5$

Answer: The car needs 25.5 liters of fuel.

Check: $\frac{7.5}{100} = 0.075$ L/km. $0.075 \times 340 = 25.5$ ✓

Common Mistakes

Mistake 1: Mixing up the positions in the proportion

❌ $\frac{3 \text{ apples}}{5 \text{ oranges}} = \frac{x \text{ oranges}}{12 \text{ apples}}$ (apples and oranges are swapped between fractions)

✅ $\frac{3 \text{ apples}}{5 \text{ oranges}} = \frac{x \text{ apples}}{12 \text{ oranges}}$ (same units in the same positions)

Why this matters: Both numerators must represent the same type of quantity, and both denominators must represent the same type. Mixing them up produces a wrong answer.

Mistake 2: Cross-multiplying incorrectly

❌ $\frac{4}{x} = \frac{6}{9} \implies 4 \times 6 = x \times 9$ (multiplied numerator by numerator)

✅ $\frac{4}{x} = \frac{6}{9} \implies 4 \times 9 = x \times 6 \implies 36 = 6x \implies x = 6$

Why this matters: Cross multiplication means diagonal multiplication — numerator of one fraction times denominator of the other. Multiplying straight across is ordinary fraction multiplication, which is a different operation entirely.

Mistake 3: Forgetting to verify the answer

❌ Solving and moving on without checking.

✅ Substitute your answer back: $\frac{4}{6} = \frac{6}{9} \rightarrow \frac{2}{3} = \frac{2}{3}$ ✓

Why this matters: A quick check catches arithmetic errors before they become wrong answers on a test.

Practice Problems

Try these on your own before checking the answers:

1. Solve for $x$: $\dfrac{7}{3} = \dfrac{x}{9}$
2. Solve for $n$: $\dfrac{12}{n} = \dfrac{4}{5}$
3. If 5 notebooks cost $8.50, how much do 12 notebooks cost?
4. A model car is built at a scale of 1:24. If the model is 18 cm long, how long is the real car in cm?
5. A printer prints 45 pages in 3 minutes. How many pages does it print in 8 minutes?

Click to see answers

1. Cross-multiply: $7 \times 9 = 3x \implies 63 = 3x \implies x = 21$
2. Cross-multiply: $12 \times 5 = 4n \implies 60 = 4n \implies n = 15$
3. $\frac{5}{8.50} = \frac{12}{x}$. Cross-multiply: $5x = 8.50 \times 12 = 102 \implies x = 20.40$. Cost is $20.40.
4. $\frac{1}{24} = \frac{18}{x}$. Cross-multiply: $x = 24 \times 18 = 432$. The real car is 432 cm (4.32 m).
5. $\frac{45}{3} = \frac{p}{8}$. Cross-multiply: $3p = 360 \implies p = 120$. It prints 120 pages.

Summary

• A proportion states that two ratios are equal: $\frac{a}{b} = \frac{c}{d}$.
• Cross multiplication converts a proportion into a simple equation: $a \times d = b \times c$.
• Always set up proportions so that matching quantities are in the same positions (numerator-to-numerator, denominator-to-denominator).
• Cross-multiply diagonally — not straight across.
• Verify your answer by substituting it back into the original proportion.

Related Topics

• Ratios and Proportions — Complete Guide with Examples
• How to Simplify Ratios — Methods and Examples
• Ratio Word Problems with Step-by-Step Solutions

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