6.2 Percent Increase, Percent Decrease, and Reverse Percentages

6.2 Percent Increase, Percent Decrease, and Reverse Percentages

Quick Summary: Percent change is computed as (new − original) ÷ original, then multiplied by 100. The denominator is always the original value. Reversing a stated change is a different operation: you divide by the multiplier, not apply the same percentage backwards. These two rules account for most of the marks available in this part of the assessment.

Percent change

Requests rose from 240 in one quarter to 300 in the next. What was the percent increase?

Difference: 300 − 240 = 60. Divide by the original: 60 ÷ 240 = 0.25. As a percent: 25%.

The error to guard against is dividing by 300, the new value: 60 ÷ 300 = 0.20, giving 20%. That figure is always among the options, because it is what a candidate produces by reaching for the more recently read number.

A memorable phrasing: change over start.

The decrease case

A caseload fell from 480 to 384. What was the percent decrease?

Difference: 480 − 384 = 96. Divide by the original: 96 ÷ 480 = 0.20 → 20% decrease.

Same rule, same denominator. Report the direction as well as the size; questions frequently offer the right number with the wrong direction.

Plausibility checks

• A decrease can never exceed 100%. A quantity cannot fall by more than all of itself. If you compute a 140% decrease, you have inverted something.
• An increase can exceed 100%. Growth from 200 to 500 is a 150% increase, since 300 ÷ 200 = 1.5.
• Doubling is +100%, not +200%. From 150 to 300, the change is 150 and 150 ÷ 150 = 1.00.
• Halving is −50%.

Reverse percentages

This is the harder of the two ideas and the one most worth practising.

After a 25% increase, a caseload stands at 500. What was it before the increase?

The wrong move is to take 25% off 500: 500 × 0.75 = 375. That is not the original, because the 25% was applied to the smaller original figure, not to 500.

The right move: the new value is 125% of the original, so

original = 500 ÷ 1.25 = 400.

Check it forward: 400 × 1.25 = 500. ✓ And notice that 375 fails this check, since 375 × 1.25 = 468.75.

The general procedure

1. Write the multiplier for the stated change: +25% → 1.25; −30% → 0.70; +8% → 1.08.
2. Divide the known later value by that multiplier.
3. Verify by multiplying forward.

A budget after a 30% cut is $91,000. What was it before?

Multiplier 0.70. Original = 91,000 ÷ 0.70 = $130,000. Check: 130,000 × 0.70 = 91,000. ✓

The wrong route — adding 30% to 91,000 — gives 118,300, which fails the forward check because 118,300 × 0.70 = 82,810.

Spotting a reverse problem

The tell is that the known figure comes after the change. Phrases such as after a 25% increase, following a 30% reduction, the reduced price is, now stands at signal that the value you are given is the end state and the unknown is the start.

Combining change and reversal

A unit's caseload increased by 20% in one year and then decreased by 20% the next, ending at 480. What was the caseload two years earlier?

Combined multiplier: 1.20 × 0.80 = 0.96. Original = 480 ÷ 0.96 = 500. Check: 500 × 1.20 = 600, then 600 × 0.80 = 480. ✓

Note again that the up-then-down pair does not return to the start: the caseload ended below where it began, which is exactly what a combined multiplier of 0.96 means.

A quick reference

Build the habit of verifying every reverse-percentage answer by multiplying forward. It takes one keystroke and it is decisive.