Ratios and Proportions With Everyday Examples

What a ratio says

A ratio compares two or more quantities by how many times one contains the other. "3 parts water to 2 parts juice" is written 3:2 and read "three to two". A ratio does not tell you the actual amounts, only their relationship: 3 cups to 2 cups and 300 ml to 200 ml are both 3:2.

Ratios behave like fractions. Multiplying or dividing every part by the same number does not change the ratio, which is how you simplify one.

12:8: both numbers divide by 4, so it simplifies to 3:2.
45:60:75: all three divide by 15, giving 3:4:5.
1.5:2.5: multiply both by 2 to clear the decimals, giving 3:5.

The method for finding the greatest common factor is the same as for fractions; see the fractions guide.

Part-to-part versus part-to-whole

This is the single most important idea in ratio problems. A ratio of 3:2 compares the two parts with each other. The whole has 3 + 2 = 5 parts.

The first part is 3 out of 5, which is 3/5 = 0.6 = 60% of the whole.
The second part is 2/5 = 40%.
The first part is 3/2 = 1.5 times the second, or 150% of it.

A drink mixed 1:4 (one part concentrate, four parts water) is therefore 20% concentrate, not 25%, because the whole has 5 parts. Mixing up "1:4" with "1/4" is the most common ratio mistake there is. The percentage calculator turns any part and whole into a percentage.

Splitting an amount in a ratio

To share an amount in a given ratio, add the parts, find the value of one part, and multiply.

Example: split 250 in the ratio 3:2.

Total parts: 3 + 2 = 5.
One part: 250 ÷ 5 = 50.
Shares: 3 × 50 = 150 and 2 × 50 = 100.

Check: 150 + 100 = 250, and 150:100 simplifies to 3:2.

Three-way example: 1,200 grams of mixed nuts split 2:3:5. Total parts 10, one part is 120 g, so the shares are 240 g, 360 g and 600 g.

When the amount does not divide evenly, decide what to do with the remainder before you start. Splitting 100 coins 2:1 gives 66.67 and 33.33, which cannot be paid in whole coins; you might give 67 and 33 and note who got the extra one.

Proportions and cross-multiplication

A proportion says two ratios are equal, such as 3/4 = x/20. To find the unknown, cross-multiply: multiply diagonally and set the products equal.

3 × 20 = 4 × x, so 60 = 4x and x = 15. Check: 15/20 simplifies to 3/4.

Example: 5 notebooks cost 7.50. How much do 8 cost at the same price each? Set up 5/7.50 = 8/x. Cross-multiplying gives 5x = 8 × 7.50 = 60, so x = 12.00. The "unit rate" route gives the same result: 7.50 ÷ 5 = 1.50 each, and 8 × 1.50 = 12.00.

Always keep the same units in the same positions. If notebooks are on top on the left, notebooks must be on top on the right too.

Scaling a recipe

Scaling a recipe is a proportion problem with a scale factor: new servings ÷ original servings. Multiply every ingredient by that factor.

Ingredient	Serves 4	Serves 6 (× 1.5)	Serves 3 (× 0.75)
Flour	300 g	450 g	225 g
Milk	250 ml	375 ml	187.5 ml
Eggs	2	3	1.5
Salt	1.5 tsp	2.25 tsp	about 1 tsp

Practical cooks round sensibly: 187.5 ml of milk can be 190 ml, and 1.5 eggs can be one large egg plus a little extra milk. Two things do not scale in a simple way. Cooking time depends on the thickness of the food, not on the total amount. And tin or pan size scales with area: a round tin 24 cm across holds (24 ÷ 20)^2 = 1.44 times as much as a 20 cm tin, not 1.2 times. So a recipe for a 20 cm tin needs about 1.44 times the ingredients for a 24 cm tin of the same depth.

Map scales

A map scale such as 1:25,000 means 1 unit on the map equals 25,000 of the same units on the ground. Use centimetres and convert at the end: 100 cm = 1 m and 100,000 cm = 1 km.

4 cm on a 1:25,000 map is 4 × 25,000 = 100,000 cm = 1 km.
7.3 cm on a 1:50,000 map is 365,000 cm = 3.65 km.
Going the other way, 2.5 km on a 1:25,000 map is 250,000 cm ÷ 25,000 = 10 cm.

Areas scale by the square of the ratio. On a 1:25,000 map, 1 cm stands for 250 m, so 1 square centimetre stands for 250 m × 250 m = 62,500 square metres. The same "square" rule applies to the recipe tins above and to any scale drawing or model.

Screens, photos and other fixed shapes

Aspect ratios keep shapes from stretching. A 16:9 image that is 1,920 pixels wide is 1,920 × 9 ÷ 16 = 1,080 pixels tall. Resized to 1,280 wide, it must be 720 tall. If you change only one dimension, the picture distorts.

Inverse proportion

Sometimes one quantity goes down when the other goes up. If 4 identical taps fill a tank in 6 hours, the job takes 4 × 6 = 24 "tap-hours". With 3 taps it takes 24 ÷ 3 = 8 hours, not 4.5. Ask yourself whether more of one thing should mean more or less of the other before you set up the equation.

Comparing ratios

Which drink is stronger: concentrate to water 2:3, or 3:5? Convert each to the part-to-whole fraction. 2:3 is 2/5 = 40% concentrate, and 3:5 is 3/8 = 37.5%. The 2:3 mix is stronger. Converting to decimals or percentages is usually the fastest way to compare; the fractions and decimals guide shows how.

Common mistakes

Treating 1:4 as one quarter. 1:4 is 1 part in 5, which is 20%.
Adding instead of multiplying. Scaling 4 servings to 6 is × 1.5, not + 2 of everything.
Mixing units. Convert map distances to the same unit before using the scale.
Scaling areas linearly. Doubling width and length quadruples the area.
Assuming direct proportion everywhere. More taps mean less time, not more.

FAQ

Is a ratio the same as a fraction?
Closely related. A part-to-whole ratio is a fraction; a part-to-part ratio such as 3:2 becomes the fractions 3/5 and 2/5 of the whole.

Does the order in a ratio matter?
Yes. 3:2 water to juice is different from 3:2 juice to water. Always say what each number refers to.

How do I write a ratio as "1 to something"?
Divide both parts by the first. 4:10 becomes 1:2.5, which is handy for comparing several ratios at once.

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