Percentages and Ratios: Choose the Correct Base

Start with the denominator

A percentage is a part divided by a chosen nonzero base, multiplied by 100. Before calculating, write “percentage of what?” A change from 40 to 50 is 10 units, but a 25% increase because the original base is 40.

Percentage change = (new value − original value) / original value × 100. The original value must be nonzero; the increase and decrease examples here use positive original values. If x% of a quantity equals y, the whole is 100y/x, provided x is nonzero.

Successive changes

Use multipliers. A 20% increase followed by a 20% decrease gives 1.20 × 0.80 = 0.96, a 4% net decrease. The second percentage acts on a different base. To return from 80 to 100 requires 20/80 × 100 = 25%, not 20%.

Ratios and shares

If A:B = 3:5 and their total is 640, there are 8 equal parts; one part is 80, so A = 240 and B = 400. The ratio 3:5 means A is 3/5 of B, but A is 3/8 of the total. These are different denominators.

Combining ratios

A:B = 2:3 and B:C = 4:5. Make B equal: A:B = 8:12 and B:C = 12:15. Therefore A:B:C = 8:12:15. Multiplying unrelated ratio terms directly does not preserve the shared quantity.

Worked expenditure problem

A household spends 60% of income and saves the rest. Income rises 20%, while expenditure rises 10%. Choose initial income 100: expenditure 60 and saving 40. New income 120, expenditure 66, saving 54. Saving rises by 14/40 = 35%. Choosing a convenient base avoids unnecessary algebra.

Practice

1. 18% of a number is 72. Answer: 400.
2. A price rises 25% and then falls 20%. Answer: no net change, since 1.25 × 0.80 = 1.
3. A:B = 4:7 and B − A = 90. Answer: A = 120, B = 210.
4. A score rises from 30% to 42%. Answer: 12 percentage points, or a 40% relative increase.
5. A quantity falls 30%. Required increase to recover? Answer: 30/70 × 100 = 42 6/7%.

Check your work

Estimate whether the result should be above or below the original. Distinguish percentage points from percentage change. Use fractions such as 1/8 = 12.5% when they simplify arithmetic, but preserve the correct base.