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What Is Proportion? Definition, Formula & Examples

TL;DR

A proportion is an equation that says two ratios are equal, written ( \frac{a}{b} = \frac{c}{d} ) or ( a:b::c:d ). This article covers the definition, the cross-multiplication rule, direct and inverse proportion, continued proportion, six worked examples, and the slips that cost marks.

What Is a Proportion?

A proportion is a statement that two ratios are equal. If the ratio ( a:b ) equals the ratio ( c:d ), then the four quantities are in proportion, written:

( \frac{a}{b} = \frac{c}{d} \quad \text{or} \quad a : b :: c : d ).

Read ( a:b::c:d ) as "aaa is to bbb as ccc is to ddd." The outer terms aaa and ddd are the extremes; the inner terms bbb and ccc are the means. A ratio compares two quantities; a proportion sets two ratios equal. That one-word difference — compares versus sets equal — is the line between the two ideas.

So is a ratio the same as a proportion? No. A ratio is a single comparison, like 3:5. A proportion needs two ratios and an equals sign between them, like ( 3:5=6:10 ). You can't have a proportion with only one ratio, the way you can't have an equation with only one side.

The Cross-Multiplication Rule

The most useful fact about any proportion is this: the product of the means equals the product of the extremes.

If ( \frac{a}{b} = \frac{c}{d} ), then ( a \cdot d = b \cdot c ). This is cross-multiplication, and it is the tool that solves almost every proportion problem.

How Do You Solve a Proportion?

You solve a proportion by cross-multiplying, then dividing to isolate the unknown.

Say a recipe uses 2 cups of flour for every 3 cups of milk, and you pour in 9 cups of milk. How much flour?

1. Write the proportion: ( \frac{2}{3} = \frac{x}{9} ).
2. Cross-multiply: ( 3x = 2 \times 9 = 18 ).
3. Divide: ( x = 6 ).

So 6 cups of flour keeps the mixture in proportion. The structure never changes — name the four positions, cross-multiply, solve.

Types of Proportion

Two quantities can be in proportion in two different ways:

• Direct proportion. As one quantity grows, the other grows by the same factor. Written ( y = kx ), where k is the constant of proportionality.
• Inverse proportion. As one quantity grows, the other shrinks so the product stays fixed. Written ( y = \frac{k}{x} ) or ( xy = k ).
• Continued proportion. Three or more quantities form a chain of equal ratios, as in ( a:b = b:c ).

Properties of a Proportion

Every proportion behaves in a few predictable ways:

• The cross products are equal. ( a \times d = b \times c ).
• You can flip both sides. If ( \frac{a}{b} = \frac{c}{d} ), then ( \frac{b}{a} = \frac{d}{c} ).
• You can swap the means or the extremes. ( \frac{a}{c} = \frac{b}{d} ) also holds.

Examples of Proportion

Example 1

Are the ratios 4:6 and 6:9 in proportion?

Cross-multiply: 4×9=36 and 6×6=36. The products match, so ( \frac{4}{6} = \frac{6}{9} ).

Final answer: yes, they are in proportion.

Example 2

A car travels 150 km on 10 litres of fuel. How far on 4 litres, at the same rate?

Set the two rates equal and cross-multiply: ( \frac{150}{10} = \frac{x}{4} ) ( \Rightarrow; 10x = 600 ) ( \Rightarrow; x = 60. )

Final answer: 60 km.

Example 3

Find the value of x in ( \frac{x}{8} = \frac{15}{20} ).

Cross-multiply: ( 20x = 8 \times 15 = 120), so ( x=6. )

Final answer: x = 6.

Example 4

666 workers build a wall in 12 days. How long for 888 workers?

Set the products equal: ( 6 \times 12 = 8 \times t ) ( \Rightarrow; 72 = 8t ) ( \Rightarrow; t = 9. )

Final answer: 9 days.

Example 5

Find the mean proportional between 4 and 25.

In a continued proportion ( 4:b=b:25 ), the mean proportional satisfies ( b^2 = 4 \times 25 = 100 ), so ( b=10. )

Final answer: 10.

Example 6

A photo 15 cm wide and 10 cm tall is enlarged to a width of 24 cm. What is the new height?

Set width-to-height ratios equal: ( \frac{15}{10} = \frac{24}{h} ) ( \Rightarrow; 15h = 240 ) ( \Rightarrow; h = 16. )

Final answer: 16 cm.

Why Proportion Shows Up Everywhere

"The first scientific theory of proportion was written to compare quantities of any kind."

Today that idea quietly runs:

• Maps and scale models. A scale of 1:50,000 is a proportion.
• Recipes and chemistry. Doubling a recipe is solving a proportion.
• Currency and unit conversion. Exchange rates are direct proportions.
• Engineering and medicine. Drug doses scale in proportion to body weight.

Where Proportion Trips Students Up

Mistake 1: Setting up the ratios in mismatched order

The correct way: Keep the same quantity on top in both ratios.

Mistake 2: Using direct-proportion setup on an inverse-proportion problem

The correct way: For inverse proportion, set the products equal.

Mistake 3: Cross-multiplying without understanding why

The correct way: Cross-multiplication is only valid when two ratios are joined by an equals sign.

Key Takeaways

• A proportion is an equation stating that two ratios are equal.
• The product of the means equals the product of the extremes.
• Direct proportion grows together; inverse proportion trades one quantity for the other.
• The most common mistake is mismatching the order of the ratios.

Practice These Three Before Moving On

1. Is 5:8 and 8:24 in proportion? Use cross-multiplication to check.
2. If 9 pens cost $12, how much do 15 pens cost at the same rate?
3. 4 taps fill a tank in 90 minutes. How long would 6 taps take? (Decide first whether this is direct or inverse.)

Frequently Asked Questions

What does a proportion mean in simple terms?

Two ratios are equal — two comparisons that match.

What is the symbol for proportion?

The double colon ( :: ).

What is the constant of proportionality?

In a direct proportion ( y = kx ), the fixed number k is the constant of proportionality.

How is a proportion different from a quadratic equation?

A proportion is two equal ratios; a quadratic involves a squared variable.

Can a proportion have more than two ratios?

Yes — a continued proportion chains three or more equal ratios.