What a ratio is
A ratio compares two or more quantities of the same kind. If a drink is made with 2 parts juice and 3 parts water, the ratio of juice to water is 2 : 3, read as “2 to 3”. The order matters: the ratio of water to juice would be 3 : 2.
A ratio does not give amounts directly. It gives the relationship between amounts. The drink could be 2 cups of juice with 3 cups of water, or 20 milliliters of juice with 30 milliliters of water. In both cases the ratio is the same.
Ratios can also be written as fractions or in words. In the juice example, juice makes up 2 of the 5 total parts, which is 2/5 of the drink. That is a part-to-whole comparison, as distinct from the part-to-part ratio 2 : 3.
Simplifying ratios
Like a fraction, a ratio can be simplified by dividing every number by a common factor. The ratio 12 : 18 has a common factor of 6, so it simplifies to 2 : 3. Both ratios describe the same relationship.
If the quantities have different units, convert them first. A ratio of 50 cm to 2 m is not 50 : 2. Convert 2 m to 200 cm and the ratio is 50 : 200, which simplifies to 1 : 4. The method for converting units is covered in the article on unit conversion.
What a proportion is
A proportion is a statement that two ratios are equal. For example, 2 : 3 = 4 : 6. Proportions are what you use when you scale a recipe, convert a map distance or compare prices.
Two quantities are in direct proportion if, when one is multiplied by a number, the other is multiplied by the same number. Double the flour and you double the milk. This is also called a constant ratio, and it is why such relationships appear as straight lines through the origin on a graph.
Scaling a recipe
Consider a hypothetical pancake recipe for 4 people: 200 g flour, 300 ml milk and 2 eggs.
Scaling up to 6 people. The scale factor is 6 ÷ 4 = 1.5. Multiply every ingredient by 1.5:
- Flour: 200 × 1.5 = 300 g
- Milk: 300 × 1.5 = 450 ml
- Eggs: 2 × 1.5 = 3
Scaling down to 2 people. The scale factor is 2 ÷ 4 = 0.5:
- Flour: 100 g
- Milk: 150 ml
- Eggs: 1
The key step is finding the single scale factor and applying it to every quantity. If you change only some ingredients, the ratio between them changes and the result will be different. In real cooking, some things do not scale perfectly, such as baking times and pan sizes, so a calculation is a starting point rather than a guarantee.
The unit rate method
A unit rate tells you the amount for one unit. It is often the most reliable way to solve proportion problems.
Suppose 3 notebooks cost 7.50. The unit price is 7.50 ÷ 3 = 2.50 per notebook. For 5 notebooks, multiply: 5 × 2.50 = 12.50.
The same method works for speed. If a train covers 150 km in 2 hours, its speed is 75 km per hour. In 3 hours at the same steady speed, it would cover 75 × 3 = 225 km.
Unit rates are also the best way to compare deals. If one pack has 6 items for 4.20 and another has 10 items for 6.50, the unit prices are 4.20 ÷ 6 = 0.70 and 6.50 ÷ 10 = 0.65, so the second is cheaper per item. Whether it is better value also depends on whether you need that many.
Solving for a missing value
Sometimes one number in a proportion is unknown. Suppose 4 pens cost 6 and you want to know the cost of 10 pens. Write the proportion with a letter for the unknown:
4 / 6 = 10 / c, with pens on top and cost underneath on both sides.
The unit method is often simpler: 6 ÷ 4 = 1.50 per pen, and 10 × 1.50 = 15.
Cross-multiplying gives the same result. From 4/6 = 10/c we get 4c = 60, so c = 15. The step from 4c = 60 to c = 15 is an example of solving a linear equation, which is the topic of that article.
Sharing a total in a ratio
To share 20 in the ratio 3 : 2:
- Add the parts: 3 + 2 = 5.
- Divide the total by the number of parts: 20 ÷ 5 = 4.
- Multiply each share: 3 × 4 = 12 and 2 × 4 = 8.
Check: 12 + 8 = 20.
Ratios and percentages
Percentages are ratios with 100 as the second part. A ratio of 1 : 4 of juice to water means juice is 1 of 5 total parts, or 20%. This link is explored in the article on percentage calculations. Fractions and decimals are connected in the same way, as shown in fraction conversion.
Where this leads
Ratios and proportions underlie a lot of later material: gradients of lines, probability, scale factors in geometry and rates in science. The ideas also appear in spreadsheet basics, where a formula copied down a column applies the same ratio to each row. With practice, the key habit is to ask “what is the one-unit amount?” and then build up from there.