12.2 Ratios, rates and relative change

Most questions ask for a comparison, and a comparison is almost always a ratio. When a quantity changes from an initial value \(x_1\) to a final value \(x_2\), the absolute change is \(x_2 - x_1\) and the relative (or fractional) change is

\[\boxed{r = \frac{x_2 - x_1}{x_1}},\]

which is multiplied by 100 to give a percentage change. The base is the initial value. A rise from 50 to 200 is a relative change of \(150/50 = 3\), that is 300 per cent, whereas the larger absolute rise from 200 to 300 is only 50 per cent. A fall can never exceed 100 per cent, while a rise is unbounded. When the question speaks of the largest relative change without specifying its sign, the magnitude \(\lvert r\rvert\) is compared.

Many quantities of interest are not shown directly but are ratios of two displayed quantities: production per unit of water, seats won per seat contested, gross domestic product per head of population, population per unit area, mass per unit volume. If a chart shows \(A\) and another shows \(B\) for the same items, the derived quantity \(A/B\) must be formed item by item; the item with the largest \(A\) need not have the largest \(A/B\). When items are only to be ranked, any common unit factor may be ignored, and two fractions \(a/b\) and \(c/d\) with positive denominators compare as \(ad\) and \(bc\).

A rate printed per unit of one quantity must be inverted when the question asks about the other. If a region produces \(k\) kg of rice per lakh litres of water, it uses \(1/k\) lakh litres per kg; the region with the highest yield is the one that uses the least water per kg.

Prices quoted per unit of size are a frequent trap. If a diamond of \(w\) carats is priced at \(R\) per carat, its price is \(wR\). When the rate itself rises with the size, the price of a single stone grows faster than its weight. To buy the greatest number of items with a fixed budget, one buys the cheapest items first and moves to dearer ones only when the cheaper stock is exhausted.