Solving Proportions

Learn to find unknown values in proportions using cross multiplication and other strategies.

Intermediate25 minLesson
Printable worksheet

Definition

A proportion is an equation stating that two ratios are equal.
ab=cd\frac{a}{b} = \frac{c}{d}
When one value is unknown, we can solve the proportion to find it.
Cross Multiplication Method: If ab=cd\frac{a}{b} = \frac{c}{d}, then a×d=b×ca \times d = b \times c
This works because multiplying both sides by bdbd gives us:
a×d=b×ca \times d = b \times c
Example: To solve 34=x12\frac{3}{4} = \frac{x}{12}:
  • Cross multiply: 3×12=4×x3 \times 12 = 4 \times x
  • Simplify: 36=4x36 = 4x
  • Divide: x=9x = 9

Try it now

What method is used to solve proportions by multiplying diagonally?

Worked Examples

Solve for xx: 58=x24\frac{5}{8} = \frac{x}{24}

1

Set up cross multiplication

5×24=8×x5 \times 24 = 8 \times x → Cross products are equal

2

Multiply the known values

120=8x120 = 8x

3

Divide both sides by 8

1208=8x8\frac{120}{8} = \frac{8x}{8} → 15=x15 = x

4

Verify by substituting back

58=1524=58\frac{5}{8} = \frac{15}{24} = \frac{5}{8} ✓ → Both ratios equal 58\frac{5}{8}

Common Mistakes

Cross multiplying incorrectly: 34=x12\frac{3}{4} = \frac{x}{12} becoming 3×4=x×123 \times 4 = x \times 12

Why it's wrong: Cross multiplication means diagonal multiplication: numerator of one fraction times denominator of the other.

Correct: Correct: 3×12=4×x3 \times 12 = 4 \times x, so 36=4x36 = 4x, giving x=9x = 9

Setting up the proportion with mismatched units

Why it's wrong: Both ratios must compare the same quantities in the same order (e.g., pencils to cost, not cost to pencils).

Correct: Keep consistent: pencilscost=pencilscost\frac{\text{pencils}}{\text{cost}} = \frac{\text{pencils}}{\text{cost}} or costpencils=costpencils\frac{\text{cost}}{\text{pencils}} = \frac{\text{cost}}{\text{pencils}}

Forgetting to verify the answer

Why it's wrong: Without checking, you might not catch arithmetic errors.

Correct: Always substitute your answer back into the original proportion and verify both ratios are equal.

Interactive Visual

Ratio Tape Diagram

3:4
Part A
14.3
14.3
14.3
= 42.86
Part B
14.3
14.3
14.3
14.3
= 57.14
Total
= 7 parts (100)
Part A:3
Part B:4
If total is:
Ratio:3:4
Fraction form:3/7 and 4/7
Part A value:42.86
Part B value:57.14

Adjust the ratio parts using + and - buttons.

Balance Scale

x + 3=7
x
3
7
Apply to both sides:

Solution: x = 4

Interactive Sandbox

Expression Calculator

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Practice Problems

16 problems
Problem 1 of 16
Easy

What method is used to solve proportions by multiplying diagonally?

Why It Matters

Solving proportions is essential in countless real-world situations:
  • Cooking: If a recipe for 4 people needs 2 cups of flour, how much for 10 people?
  • Maps and Scale Models: A map scale of 1 cm = 50 km lets you calculate real distances
  • Medicine: Doctors calculate medication dosages based on patient weight
  • Photography: Resizing images while maintaining aspect ratio
  • Construction: Scaling blueprints to actual building dimensions
Proportions help us scale quantities up or down while keeping relationships constant!

Real World Applications

Recipe Scaling

Chefs and home cooks use proportions to adjust recipe quantities for different serving sizes.

Example:

A cookie recipe makes 24 cookies using 3 eggs. To make 40 cookies: 324=x40\frac{3}{24} = \frac{x}{40}, so x=5x = 5 eggs.

1Try It Yourself

A pasta recipe for 6 people requires 450 grams of pasta. You're cooking for 10 people.

How many grams of pasta do you need?

Step 1: Write the mathematical expression

Set up the proportion: 4506=x10\frac{450}{6} = \frac{x}{10}

Map Reading

Cartographers use scale ratios to represent real distances on maps.

Example:

If a map scale is 1:100,000, then 1 cm on the map equals 1 km in reality. A 5.5 cm road on the map is actually 5.5 km long.

2Try It Yourself

A hiking map has a scale where 3 cm represents 2 km. You measure a trail that is 12 cm on the map.

How long is the actual trail?

Step 1: Write the mathematical expression

Set up: 32=12x\frac{3}{2} = \frac{12}{x}

Medicine Dosage

Pharmacists calculate proper medication doses based on patient weight using proportions.

Example:

A medication is prescribed at 5 mg per kg of body weight. For a 60 kg patient: 51=x60\frac{5}{1} = \frac{x}{60}, so x=300x = 300 mg.

3Try It Yourself

A children's medicine is dosed at 10 mg per 5 kg of body weight. A child weighs 35 kg.

What dosage should the child receive?

Step 1: Write the mathematical expression

Set up: 105=x35\frac{10}{5} = \frac{x}{35}

Key Takeaways

  • 1A proportion is an equation showing two equal ratios: ab=cd\frac{a}{b} = \frac{c}{d}
  • 2Cross multiplication: if ab=cd\frac{a}{b} = \frac{c}{d}, then a×d=b×ca \times d = b \times c
  • 3The scale factor method multiplies both parts of a ratio by the same number
  • 4Always set up proportions with matching units in the same positions
  • 5Verify your answer by substituting back into the original proportion

Frequently Asked Questions

Cross multiplication is a shortcut for multiplying both sides of the equation by the product of the denominators (bdbd). This clears the fractions and gives us ad=bcad = bc.
Cross multiplication is a shortcut for multiplying both sides of the equation by the product of the denominators (bdbd). This clears the fractions and gives us ad=bcad = bc.
Yes! You can use the scale factor method (find what multiplier transforms one ratio into the other) or multiply both sides by one denominator at a time.
Cross multiplication still works! For 6x=35\frac{6}{x} = \frac{3}{5}, cross multiply: 6×5=x×36 \times 5 = x \times 3, so 30=3x30 = 3x, giving x=10x = 10.

Glossary

Proportion
An equation stating that two ratios are equal, such as ab=cd\frac{a}{b} = \frac{c}{d}
Cross multiplication
A method for solving proportions by multiplying diagonally: if ab=cd\frac{a}{b} = \frac{c}{d}, then ad=bcad = bc
Scale factor
The number you multiply by to transform one ratio into an equivalent ratio
Equivalent ratios
Ratios that express the same relationship, like 23\frac{2}{3} and 46\frac{4}{6}

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