Ratio and proportion compare quantities and describe how those quantities change together. They occur frequently in partnership, ages, mixtures, maps, work, and direct or inverse variation problems.

What is a ratio?

A ratio compares two quantities in the same units. The ratio of a to b is written as a:b or a/b. Simplify a ratio by dividing both terms by their highest common factor.

• 12:18 = 2:3 after dividing both terms by 6.
• Convert units before forming a ratio, such as metres to metres.
• Equivalent ratios are made by multiplying or dividing both terms by the same non-zero number.

Dividing a quantity in a ratio

To divide a total T in the ratio a:b, first find the total number of parts, a + b. The two shares are T x a/(a + b) and T x b/(a + b).

For example, Rs. 900 divided in the ratio 2:3 gives Rs. 360 and Rs. 540, because the total has 5 parts.

What is a proportion?

Two ratios are in proportion when they are equal: a:b = c:d. This is written as a:b::c:d, and it gives the useful rule a x d = b x c.

• In a:b::c:d, a and d are the extremes.
• b and c are the means.
• The fourth proportional to a, b, c is (b x c)/a.
• The mean proportional between a and b is the square root of a x b.

Direct and inverse proportion

• Direct proportion: both quantities increase or decrease together. If y is directly proportional to x, y/x is constant.
• Inverse proportion: one quantity increases as the other decreases. If y is inversely proportional to x, x x y is constant.
• Distance and time are directly proportional at constant speed.
• Workers and days are inversely proportional when work rate is constant.

Common applications

• Partnership: distribute profit in the ratio of capital x time.
• Mixtures: use the quantities of ingredients to form the required ratio.
• Ages: compare present or future ages, then use their difference or total.
• Scale: keep map and actual distances in the same unit before applying the scale ratio.

How to solve questions

1. Identify the quantities being compared and make their units consistent.
2. Simplify the ratio or write the proportional relationship.
3. For a total, add ratio parts before calculating individual shares.
4. Use cross multiplication for proportions.
5. Decide whether the relationship is direct or inverse before calculating.