Ratio, Proportion and Average: Formulas and Worked Examples

Ratio

A ratio a:b compares two quantities of the same kind and behaves like the fraction a/b. Multiplying or dividing both terms by the same number does not change it.

• To divide a quantity in the ratio a:b, the shares are a/(a + b) and b/(a + b) of the total.
• The compound ratio of a:b, c:d and e:f is (a × c × e):(b × d × f).

Proportion

Four numbers are in proportion when a:b = c:d, which means a × d = b × c (product of extremes = product of means).

• Fourth proportional to a, b, c is x where a:b = c:x, so x = b × c ÷ a.
• Mean proportional between a and b is the square root of a × b.
• Direct proportion: both quantities rise together and x1/y1 = x2/y2 (more notebooks, more cost).
• Inverse proportion: one rises as the other falls and x1 × y1 = x2 × y2 (more workers, fewer days).

Partnership

Profit is shared in the ratio of capital × time.

• Same period: share in the ratio of capitals.
• Different periods: multiply each capital by the number of months it stayed in the business, then take the ratio.

Mixtures and alligation

When a cheaper ingredient (price c) is mixed with a dearer one (price d) to give a mean price m:

Quantity of cheaper : Quantity of dearer = (d - m) : (m - c)

For questions that add water or milk to a mixture, the quantity of the ingredient that is not added stays fixed; use it to find the new total.

Average

• Average = Sum of observations ÷ Number of observations, so Sum = Average × Number.
• Weighted (combined) average = (n1 × a1 + n2 × a2) ÷ (n1 + n2). It lies nearer the average of the larger group.
• Correcting an error: new sum = old sum - wrong value + right value.
• Overlapping groups: add the group sums and subtract the overall sum to get the item counted twice.

Worked examples

1. Partnership. A starts with Rs 40,000; B joins after 6 months with Rs 60,000; the year's profit is Rs 35,000. A: 40,000 × 12 = 480,000. B: 60,000 × 6 = 360,000. Ratio 4:3. A gets 4/7 × 35,000 = Rs 20,000 and B gets Rs 15,000.
2. Alligation. Rice at Rs 90 per kg and Rs 120 per kg is mixed to cost Rs 100 per kg. Cheaper : dearer = (120 - 100):(100 - 90) = 20:10 = 2:1. Check: (2 × 90 + 1 × 120) ÷ 3 = 100.
3. Changing a mixture. 60 litres of milk and water in the ratio 2:1 holds 40 litres of milk and 20 litres of water. For a ratio of 1:2 with milk unchanged, water must be 80 litres, so 60 litres of water is added.
4. Combined average. 40 students average 65 and 60 students average 75. Combined = (40 × 65 + 60 × 75) ÷ 100 = (2,600 + 4,500) ÷ 100 = 71.
5. Overlap. Eleven numbers average 50 (sum 550). The first six average 49 (sum 294) and the last six average 52 (sum 312). The sixth number is counted twice: 294 + 312 - 550 = 56.

Exam pointers

• Check the order in which the ratio is asked; 2:1 and 1:2 will both be among the options.
• In partnership never ignore time: a partner who joins late has capital multiplied by fewer months.
• For inverse proportion multiply, do not cross-divide: workers × days stays constant.
• The combined average is not the simple mean of two averages unless the groups are equal in size.
• In alligation write the mean price in the middle and subtract diagonally; the result is the ratio of quantities.
• When something is added to a mixture, anchor on the ingredient that did not change.