Percentages

1. Meaning, base and useful conversions

A percentage is a ratio with denominator 100. Thus, 24% means 24/100, or 0.24. Although a percentage has no unit, the quantity calculated from it retains the original unit: 24% of ₹500 is ₹120, while 24% of 500 kilometres is 120 kilometres. Before calculating, identify the part, the whole and the relationship being asked.

The three basic forms are: part = percentage × whole/100; percentage = part/whole × 100; and whole = part × 100/percentage. If 72 candidates constitute 30% of a group, the group contains 72 × 100/30 = 240 candidates. Reverse questions are common traps because candidates sometimes apply the stated percentage to the part rather than recovering the whole.

Frequently used conversions make mental calculation faster: 1/2 = 50%, 1/3 = 33⅓%, 1/4 = 25%, 1/5 = 20%, 1/6 = 16⅔%, 1/8 = 12.5%, 1/10 = 10%, 1/12 = 8⅓%, 1/16 = 6.25% and 3/8 = 37.5%. Learn these as exact relationships, not merely rounded decimals.

• Use symmetry when helpful: a% of b = b% of a. Therefore, 16% of 25 = 25% of 16 = 4.
• Break awkward percentages into familiar parts: 17.5% of 640 = 10% + 5% + 2.5% of 640 = 64 + 32 + 16 = 112.
• For percentage-only comparisons, assume a convenient base such as 100 or a suitable multiple; do not do this when an actual value is fixed by the question.

2. Percentage change and reverse comparisons

Percentage change always uses the original value as the denominator. If attendance increases from 240 to 300, the increase is 60/240 × 100 = 25%. If it subsequently falls from 300 to 240, the decrease is 60/300 × 100 = 20%. The absolute difference is identical, but the comparison bases are not.

If A is p% more than B, then A = B(100 + p)/100. Consequently, B is 100p/(100 + p)% less than A. If A is p% less than B, then B is 100p/(100 − p)% more than A, provided p is less than 100. Thus, a value 25% above another corresponds to the second being 20% below the first.

Reverse calculations require division by the multiplier. A price that becomes ₹920 after a 15% increase was originally 920/1.15 = ₹800. A price that becomes ₹720 after a 10% reduction was originally 720/0.90 = ₹800. Subtracting 15% from ₹920 would not undo the original increase.

Distinguish percentage change from percentage-point change. If a success rate rises from 40% to 50%, it increases by 10 percentage points, but the relative increase is 10/40 × 100 = 25%. Likewise, a rise from 6% to 6.5% is 0.5 percentage point, or 50 basis points; one basis point equals 0.01 percentage point.

• Translate wording carefully: ‘increased by 120%’ means the final value is 220% of the original; ‘increased to 120%’ means a 20% rise.
• For a positive starting value, a decrease exceeding 100% is generally inconsistent with ordinary non-negative quantities such as population or stock.

3. Successive changes and multiplicative relationships

Successive percentage changes multiply; they are not ordinarily added. For changes a% and b%, use signed values, with decreases negative. The net change is a + b + ab/100 percent. An increase of 20% followed by a decrease of 10% gives 20 − 10 − 2 = 8% growth. Equivalently, 1.20 × 0.90 = 1.08.

An increase and decrease of the same p% produce a net decrease of p²/100 percent. For example, a 20% rise followed by a 20% fall leaves 96% of the original value. To recover from a fall of p%, the required increase is 100p/(100 − p)%. A 20% loss therefore requires a 25% gain for full recovery.

For repeated growth at r% per period over n periods, final value = initial value × (1 + r/100)^n. For repeated decline, use (1 − r/100)^n. This is the structure behind compound interest, depreciation and constant-rate population projections. Use each period’s actual rate if rates vary; an arithmetic average generally does not reproduce compound growth.

The same multiplier method applies to products. Since expenditure = price × quantity, a 25% price rise requires quantity to fall to 1/1.25 = 80% of its earlier level to keep expenditure unchanged. The required reduction is 20%. Similarly, if a rectangle’s length increases by 10% and breadth by 20%, its area increases by 1.10 × 1.20 − 1 = 32%.

• Two successive discounts of 20% and 10% give an effective discount of 28%, not 30%.
• Pure percentage multipliers can be rearranged without changing their product; this shortcut does not automatically apply when fixed additions, charges or rounding intervene.

4. Combined groups, changing denominators and examination problems

The overall percentage for multiple groups is a weighted average, with group sizes as weights. If 60% of 100 candidates and 80% of 300 candidates qualify, total qualifiers are 60 + 240 = 300 out of 400. The combined qualification rate is 75%, not the simple average of 70%. A simple average is valid when the relevant group sizes are equal.

Nested percentages require multiplication. If 40% of a school’s students are girls and 25% of those girls use the library daily, these users constitute 0.40 × 0.25 = 10% of all students. Always ask whether the second percentage refers to the entire population or only to a subgroup.

Election and examination questions often change denominators midway. Suppose 80% of 1,000 registered voters vote and 10% of votes cast are invalid. Valid votes number 1,000 × 0.80 × 0.90 = 720. A candidate receiving 60% of valid votes gets 432 votes, equivalent to 43.2% of registered voters.

Read marks-based conditions equally carefully. If a candidate scores 144 and falls short of the pass requirement by 16 marks, the pass mark is 160. If passing requires 40% of maximum marks, the maximum is 160/0.40 = 400. The shortfall is an absolute number, not a percentage.

• Do not add overlapping categories unless their common membership is accounted for.
• A constant subgroup count can acquire a larger percentage share when the total shrinks; a rising share does not necessarily mean a rising count.

5. A reliable CSAT solving strategy

Begin by naming the base and converting verbal statements into equations or multipliers. Keep fractions exact until the final step, particularly for values such as 16⅔% and 33⅓%. Estimate the likely range before calculating: after a 20% discount, the original price must exceed the selling price, and reversing that discount requires division by 0.8.

Use options as a checking tool. In a reverse-percentage problem, substitute a plausible option and apply the stated change. For successive changes, start with 100 when only the net percentage is required. In data interpretation, examine headings, units and whether percentages describe totals, subgroups or changes over time before using the figures.

Practise accuracy before speed. Maintain an error log organised by wrong base, additive treatment of successive changes, incorrect averaging and percentage-point confusion. These recurring conceptual mistakes matter more than memorising many specialised shortcuts.

• Final check: denominator correct, units consistent, direction sensible and answer compatible with the options.

Previous year questions

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Practice questions