A price rises from $80 to $100.
The increase is $20.
What is the percentage increase?
The answer is not 20% merely because the numerical increase is 20.
The change must be compared with the original base.
20/80 × 100% = 25%.
Percentage change is always relative to a reference quantity. The numerator is the change; the denominator tells us what that change is being measured against.
The core formulas
Percentage increase:
(new−original)/original × 100%.
Percentage decrease:
(original−new)/original × 100%.
In both cases, the original value is the reference base.
Worked example: percentage increase
A quantity rises from 240 to 300.
Change:
300−240=60.
Relative change:
60/240 = 0.25.
Percentage increase:
25%.
Worked example: percentage decrease
A price falls from $150 to $120.
Decrease:
150−120=30.
Percentage decrease:
30/150×100% = 20%.
The same absolute change can represent different percentages when the original base changes.
A $20 increase is not always a 20% increase
From $80 to $100:
$20 is 25% of $80.
From $200 to $220:
$20 is 10% of $200.
Absolute change and relative change answer different questions.
Use multipliers for efficient increase and decrease
An increase of 15% means the new value is:
100%+15%=115% of original.
Multiplier:
1.15.
A decrease of 15% means:
85% of original.
Multiplier:
0.85.
Worked example: percentage multiplier
A salary of $3,200 rises by 6%.
New salary:
3,200×1.06 = $3,392.
The increase itself is:
3,200×0.06=$192.
Both routes describe the same change.
Reverse percentage: recover the original
After a 20% discount, a jacket costs $96.
The sale price is 80% of original.
So:
0.8×original=96.
original=96÷0.8=$120.
Adding 20% to $96 would be wrong because 20% was measured from the original $120, not from the discounted price.
Percentage increase and decrease are not symmetric operations because the reference base changes after the first move.
A 20% increase followed by a 20% decrease does not return to the start
Start with 100.
Increase 20%:
100×1.2=120.
Then decrease 20%:
120×0.8=96.
Final value=96.
Overall change:
4% decrease from the original.
The second 20% acts on 120, not 100.
Repeated percentage change is multiplicative
A quantity increases by 10% and then by another 10%.
Overall multiplier:
1.1×1.1=1.21.
Overall increase:
21%.
It is not 20% because the second increase applies to the already increased value.
Worked example: increase then decrease
A population rises by 8%, then falls by 5%.
Overall multiplier:
1.08×0.95=1.026.
Overall change:
2.6% increase.
The two percentage changes cannot simply be combined as +8−5=+3% because their bases differ.
Percentage points are different
A pass rate rises from 40% to 50%.
Difference in percentage points:
50%−40%=10 percentage points.
Relative percentage increase:
(50−40)/40×100%=25%.
Percentage points compare percentage values directly. Percentage change compares the difference with the original percentage as a base.
Another percentage-point example
An interest rate rises from 3% to 4%.
Increase:
1 percentage point.
Relative increase in the rate itself:
(4−3)/3 = 1/3 ≈ 33.3%.
Saying “the rate increased by 1%” can therefore be ambiguous or misleading.
Percentage change needs a non-zero reference base
If a quantity changes from 0 to 10, the ordinary percentage-increase formula would require division by zero.
The percentage increase is therefore undefined under the usual formula.
The absolute increase is 10, but no finite relative percentage increase from zero can be computed in the usual way.
Percentage decrease can reach 100% but not exceed it for a non-negative quantity that cannot go below zero
If a positive quantity falls to zero:
decrease/original = 1.
Percentage decrease=100%.
But percentage increase can exceed 100%.
For example, 40 rises to 100:
increase=60.
60/40=1.5.
Percentage increase=150%.
“Increased to 120%” versus “increased by 120%”
If a value is increased to 120% of original:
new=1.2×original.
This is a 20% increase.
If a value is increased by 120%:
new=2.2×original.
The language changes the multiplier completely.
Discount and markup use different bases
An item costs a shop $80 and is marked up by 25% on cost.
Markup:
25% of 80 = 20.
Selling price=$100.
Now a 10% discount is applied to selling price:
10% of 100=10.
Final price=$90.
The markup percentage used cost as its base.
The discount used marked selling price as its base.
Same percentage language, different reference quantities.
Worked example: compare two changes fairly
Store A sales rise from 100 to 130.
Increase=30%.
Store B sales rise from 400 to 460.
Increase=60/400=15%.
Store B gains more units in absolute terms.
Store A grows faster in relative terms.
Both statements can be true.
Percentage change is dimensionless
Change and original carry the same unit.
When divided, the units cancel.
That is why percentage change can compare growth across quantities measured on different absolute scales, provided the comparison itself is meaningful.
Graph interpretation: base still matters
If a graph rises from 20 to 30, the absolute increase is 10.
Relative increase=10/20=50%.
If another series rises from 100 to 120, the absolute increase is 20 but relative increase is only 20%.
Visual height alone does not determine percentage growth.
Reverse percentage with increase
After a 12% increase, a value becomes 448.
448 represents 112% of original.
Original:
448÷1.12=400.
Do not subtract 12% of 448 because that uses the new value as the base.
Reverse repeated change
A value rises 5% per year for two years and becomes 220.50.
Two-year multiplier:
1.05²=1.1025.
Original:
220.50÷1.1025=200.
Repeated percentages are naturally handled through multiplier products.
Common misconception 1: divide by the new value
Ordinary percentage change uses the original value as the base.
Common misconception 2: percentage points and percent are interchangeable
A change from 40% to 50% is 10 percentage points but a 25% relative increase.
Common misconception 3: equal increase and decrease cancel
They act on different bases unless the value somehow returns to the same reference before the second change.
Common misconception 4: successive changes can always be added
Use multipliers when percentage changes act sequentially.
Common misconception 5: an increase from zero has an ordinary finite percentage increase
The usual formula is undefined because the reference denominator is zero.
A diagnostic ladder
- Can the learner identify original and new values?
- Can the learner calculate absolute change?
- Can the learner choose the original base?
- Can the learner calculate percentage increase?
- Can the learner calculate percentage decrease?
- Can the learner use multipliers?
- Can the learner solve reverse-percentage problems?
- Can the learner combine repeated changes multiplicatively?
- Can the learner distinguish percentage points from relative percentage change?
- Can the learner explain why equal percentage rise/fall do not cancel?
- Can the learner identify cases where the ordinary percentage-change formula is undefined?
How this fits Secondary Mathematics
Percentage increase and decrease connect proportional reasoning, finance, statistics, graphs, compound change and later exponential models. The crucial habit is preserving the reference base through each change.
Exact subject-level scope should be checked against the current MOE/SEAB Mathematics syllabus for the learner’s cohort.
The deeper lesson: percentages are relationships, not labels
The number 20 can be a $20 change, a 20-point difference or a 20% relative change depending on what it is compared with.
Percentage reasoning becomes reliable when every percentage carries an invisible question beside it: “percentage of what?”