Concept · Pre-Algebra
What Is a Proportion? Two Equal Ratios
A ratio compares two quantities. A proportion claims two ratios are the same — and that claim is what you can solve.
Ratio first, proportion second
The answer to what is a proportion begins with the difference between the two words, which are used interchangeably in speech and mean different things in mathematics.
| Example | What it is | |
|---|---|---|
| Ratio | 3 : 4 | a comparison of two quantities |
| Proportion | 3/4 = 9/12 | a claim that two ratios are equal |
A ratio on its own is a description. A proportion is a statement that can be true or false, and that is what makes it solvable — exactly the same distinction as between an expression and an equation.
What makes two ratios equivalent
Multiplying both parts of a ratio by the same number does not change the relationship it describes. 3 : 4 and 9 : 12 are equivalent ratios, because the second is the first scaled by 3.
This is the same machinery as equivalent fractions, and it is why proportions can be solved at all. Writing 3/4 = 9/x asserts that the two ratios are equivalent, and solving finds the value that makes the assertion true.
- a⁄b = c⁄dthe proportion
- a⁄b · bd = c⁄d · bdmultiply both sides by bd
- ad = bcevery denominator cancels
Part to part, part to whole
A ratio can compare two parts of a group, or one part to the whole group, and confusing them is a common source of wrong answers.
- Part to part. 3 red balls to 2 blue is the ratio 3 : 2. Writing it as the fraction 3/2 describes something different from what most questions mean.
- Part to whole. 3 red out of 5 balls is the fraction 3/5, and this one genuinely is a fraction of the group.
The question decides which is wanted. 'What fraction are red' asks for 3/5; 'what is the ratio of red to blue' asks for 3 : 2.
Direct and inverse proportion
Two quantities are in direct proportion when doubling one doubles the other. That is the case a simple proportion handles.
| Relationship | As one doubles | Example |
|---|---|---|
| Direct | the other doubles | distance at constant speed |
| Inverse | the other halves | time taken as speed increases |
| Neither | no fixed pattern | area of a circle against radius |
Inverse proportion needs a different setup: the product stays constant rather than the ratio. Four workers taking six hours means 24 worker-hours, so eight workers take three.
Reading a ratio in its simplest form
Ratios are usually reduced, in exactly the way fractions are, by dividing every part by the greatest common factor.
- 12 : 18both parts share a factor of 6
- 2 : 3the same relationship, smaller numbers
A ratio can also have more than two parts — 2 : 3 : 5 for a mixture — and the same rule applies. What it cannot do is be compared with another ratio unless both are in the same order of quantities.
Where proportions turn up
Once you recognise the shape, a large share of everyday mathematics is one proportion.
| Situation | The proportion |
|---|---|
| Scale model at 1:50 | 1/50 = model / real |
| Percentage of a total | part / whole = rate / 100 |
| Unit rate or price per kilo | cost / amount = cost / 1 |
| Similar triangles | side / matching side = side / matching side |
The percentage row is the one worth memorising. Nearly every percentage question is that single proportion with a different value missing.
Questions about what is a proportion
What is the difference between a ratio and a proportion?
A ratio compares two quantities, like 3 : 4. A proportion states that two ratios are equal, like 3/4 = 9/12, and that statement is what you solve.
Is a ratio the same as a fraction?
A part-to-whole ratio is — 3 out of 5 is 3/5. A part-to-part ratio, 3 red to 2 blue, is not; writing it as 3/2 means something different.
What is inverse proportion?
When one quantity doubles as the other halves, so their product stays constant. More workers means less time, and workers × hours does not change.
How do I know if two things are proportional?
Check whether doubling one doubles the other. Area against radius fails that test, so a proportion cannot be used there.