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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](/content/math/terms/ratio/index.html) 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.