Ratio and Proportion

A ratio compares two quantities of the same kind, and a proportion says two ratios are equal. These ideas show up in age problems, mixtures, partnerships and sharing money, so a solid grip on ratio makes several other topics easier.

What a ratio is

The ratio of a to b, written a : b, is the fraction a / b. Both quantities must be in the same unit. The ratio of 40 cm to 2 m is 40 : 200 = 1 : 5, not 40 : 2.

A ratio does not change if you multiply or divide both terms by the same non-zero number, so 6 : 9 = 2 : 3. Always reduce a ratio to its simplest form.

Comparing ratios

To decide which of two ratios is bigger, compare them as fractions. Is 3 : 4 larger than 5 : 7? Cross-multiply: 3 × 7 = 21 and 5 × 4 = 20. Since 21 > 20, 3 : 4 is the larger ratio.

Proportion

If a : b = c : d, the four numbers are in proportion, and a × d = b × c (product of extremes = product of means). This lets you find a missing term. If 5 : 8 = x : 40, then 8x = 200 and x = 25.

• Third proportional to a and b: the number c such that a : b = b : c, so c = b² / a.
• Fourth proportional to a, b, c: the number d such that a : b = c : d, so d = (b × c) / a.
• Mean proportional between a and b: the number m such that a : m = m : b, so m = √(a × b).

Dividing a quantity in a ratio

To divide N in the ratio a : b, first add the parts: a + b. Then the first share is N × a / (a + b) and the second is N × b / (a + b). Example: divide 2,100 in the ratio 4 : 3. The total parts are 7, so one part is 300. The shares are 1,200 and 900.

Age problems

Give each age as a multiple of a common unit. If two ages are in the ratio 5 : 7 and their sum is 72, one part is 72 / 12 = 6, so the ages are 30 and 42. When the question talks about "after n years" or "n years ago", add or subtract n from each actual age, not from the ratio parts.

Mixtures

If a mixture has two ingredients in the ratio a : b, the share of the first is a / (a + b) of the total. In 60 litres of milk and water mixed 2 : 1, milk is 40 litres and water is 20 litres. To make the ratio 1 : 1, add enough water to bring it to 40 litres, which is 20 more litres. Notice that adding a substance changes only that one part. The other part stays fixed, and that fixed part is what you build the new ratio around.

Worked examples

Example 1. The salaries of A and B are in the ratio 7 : 5, and A earns 6,000 more than B. The difference is 2 parts = 6,000, so one part is 3,000. A earns 21,000 and B earns 15,000.

Example 2. A bag holds 1, 2 and 5 rupee coins in the ratio 4 : 3 : 2, and there are 90 coins in all. The parts add up to 9, so one part is 10 coins. That gives 40 one-rupee coins, 30 two-rupee coins and 20 five-rupee coins, worth 40 + 60 + 100 = 200 rupees in total.

Example 3. A and B invest 30,000 and 20,000 for the same time and earn 10,000. Profit is shared in the ratio of investments, 3 : 2, so A gets 6,000 and B gets 4,000.

Common mistakes

• Not converting units before forming a ratio.
• Adding a number to both terms of a ratio, which changes it. Only multiplying or dividing keeps it the same.
• Giving the share in "parts" instead of the actual amount.
• In age problems, applying "after 5 years" to the ratio instead of to the actual ages.

Exam tips

1. When the total or a difference is given, find the value of one part first.
2. Check that your shares add up to the total.
3. For mixtures, track the quantity that does not change, and build the new ratio around it.

Try the quiz and read each explanation, especially for the questions that involve incomes, expenses and mixtures.