Percentage Bases and Successive Changes: Avoid the Common Traps

In this note, ask: What is the 100% base? The denominator is the reference quantity, not necessarily the larger value.

Assumptions: Assume a positive original, matching units, and no extra fixed adjustments. Successive changes use the immediately preceding value. Recovery requires a nonzero combined multiplier. All calculations are exact.

Choose the method

Question type	Calculation	Base or safeguard
A is what percent of B?	A ÷ B × 100	B is the reference.
Percentage change	(new − old) ÷ old × 100	Use the starting value.
Rate: p% to q%	q − p percentage points	Relative change: (q − p) ÷ p × 100, for p > 0.
Two successive changes	Multiply their factors	The second base is updated.
Recover the original	final ÷ combined multiplier	Divide; do not “subtract back”.

An r% increase uses 1 + r/100; a decrease uses 1 − r/100. The new value is the original plus or minus the stated fraction of it. For combined multiplier M, net percentage change is (M − 1) × 100; a negative result means a decrease.

Four worked contrasting examples

Example 1 — Same difference, different denominators

Two strips measure 120 cm and 150 cm. Difference = 150 − 120 = 30 cm.

The longer is 30 ÷ 120 × 100 = 25% longer than the shorter. The shorter is 30 ÷ 150 × 100 = 20% shorter than the longer. Meanwhile, 120 ÷ 150 × 100 = 80% describes the shorter length as a share, not a decrease. “Compared with” or “than” identifies the reference.

Example 2 — Percentage points versus percentages

A task-completion rate rises from 40% to 50%.

Difference = 50 − 40 = 10 percentage points. Relative increase = 10 ÷ 40 × 100 = 25%, not 10%. Subtraction measures the gap between rates; division measures growth relative to the starting rate.

Example 3 — Successive changes do not simply add

From 100 units, successive rises of 20% and 10% give 100 × 1.20 = 120, then 120 × 1.10 = 132. Net increase = (132 − 100) ÷ 100 × 100 = 32%, not 30%.

Contrast a 20% rise followed by a 20% fall: 100 × 1.20 = 120; 120 × 0.80 = 96. Net decrease = (100 − 96) ÷ 100 × 100 = 4%. The fall removes 24 units; the rise added only 20. For an r% rise and r% fall, with 0 < r < 100, the multiplier is (1 + r/100)(1 − r/100) = 1 − (r/100)²; net loss = (r²/100)%.

Example 4 — Recover the original by division

After a 25% rise and a 12% fall, a quantity becomes 220 units.

Multiplier = 1.25 × 0.88 = 1.10, a net 10% increase. Original = 220 ÷ 1.10 = 200 units. Check: 200 × 1.25 = 250; 250 × 0.88 = 220. A 10% reduction instead gives 220 × 0.90 = 198: it wrongly uses 220 as the base.

Practice questions

1. How much longer is 80 cm than 50 cm, and how much shorter is 50 cm than 80 cm, in percentages?
2. A completion rate rises from 60% to 72%. Find the percentage-point increase and relative percentage increase.
3. A quantity of 400 units rises by 25%, then falls by 25%. Find the final quantity and net percentage change.
4. A quantity falls by 20%, then rises by 10%, becoming 264 units. Find its original value.

Worked answers

1. Difference = 80 − 50 = 30 cm. Longer: 30 ÷ 50 × 100 = 60%. Shorter: 30 ÷ 80 × 100 = 37.5%. The reference length changes.
2. Increase = 72 − 60 = 12 percentage points. Relative increase = 12 ÷ 60 × 100 = 20%, using the starting rate.
3. First: 400 × 1.25 = 500. Then: 500 × 0.75 = 375 units. Change = (375 − 400) ÷ 400 × 100 = −6.25%, a 6.25% decrease.
4. Multiplier = 0.80 × 1.10 = 0.88. Original = 264 ÷ 0.88 = 300 units. Check: 300 × 0.80 = 240; 240 × 1.10 = 264.

Revision checklist

Identify the reference; distinguish shares, changes and percentage points; multiply successive factors; divide to recover; check by applying the changes forward.