TL;DR:
Change is measured against the starting value:
(new − old) / old.A rise from 40 to 50 is
+25%, but the fall from 50 back to 40 is−20%, because the base is different.Successive changes multiply. They do not add.
A jacket costs $40 and goes up to $50. How much did the price rise, in percent? Then the shop puts it back to $40. Is that a 25% drop? It is not, and seeing why clears up most of the confusion around percent change.
Change over the original
A percent change answers "how big was the change compared with where we started?" The recipe is one line: percent change = (new − old) / old × 100. The starting value, the "old", is always the denominator. That is the part people forget.
Worked example: a price rising from 40 to 50 dollars
Find the change:
50 − 40 = 10.Divide by the old value:
10 / 40 = 0.25.Convert to a percentage by multiplying by 100:
25%.
The price rose by 25%. Check using a multiplier: 40 × 1.25 = 50. It works, so 1.25 is the growth factor.
Now go back. The change is 40 − 50 = −10, and the old value is now 50, so −10 / 50 = −0.2. The fall is 20%, not 25%. Check: 50 × 0.8 = 40. The same ten dollars is a smaller share of the bigger price.
The multiplier shortcut
Every percent change is a multiplication. A rise of r% multiplies by 1 + r/100, and a fall multiplies by 1 − r/100. A 15% increase is a multiplier of 1.15. A 15% discount is a multiplier of 0.85. Using multipliers makes chains of changes easy and removes the need to add or subtract percent figures.
Why successive changes do not add
Say a $100 item goes up by 20% and then goes down by 20%. Many people expect $100 again. Compute it: 100 × 1.20 = 120, then 120 × 0.80 = 96. The total change is 96 / 100 = 0.96, a fall of 4%. The second 20% applies to $120, a bigger base, so it removes more dollars (24) than the first added (20).
Two stacked discounts fall into the same trap. Take an $80 item, 20% off, then 10% off the lowered price. That is 80 × 0.8 = 64, then 64 × 0.9 = 57.60, so you save 80 − 57.60 = 22.40. As a share of the original, 22.40 / 80 = 0.28, which is 28% and not 30%. Multiply the factors to confirm: 0.8 × 0.9 = 0.72, and 1 − 0.72 = 0.28.
Getting back to where you started
After a 20% discount, what rise brings the price back? If the price was $80 and is now $64, you need 80 / 64 = 1.25, a 25% rise. In general, undoing a fall by r means dividing by 1 − r/100, not adding r back.
Percent points are not percent
If a rate goes from 40% to 50%, it rose by 10 percentage points. In relative terms it rose by 10 / 40 = 0.25, a 25% rise. News stories often blur the two, so ask which one is meant. A rise "to" a value and a rise "by" an amount are also different sentences.
Try one yourself: 250 down to 200
A club had 250 members and now has 200. The change is 200 − 250 = −50. Divide by the old value: −50 / 250 = −0.2, a fall of 20%. Check: 250 × 0.8 = 200.
To grow back to 250, the change needed is +50 on a base of 200: 50 / 200 = 0.25, a rise of 25%. Check: 200 × 1.25 = 250. The pair 20% down and 25% up is the same pair you saw with 40 and 50, and it comes from the same reason: the base differs by the direction.
When you read "up by 50%", ask three things: up from what, over what time, and is it a rate or a count? A rise from 2 to 3 cases is 1 / 2 = 50%, which sounds dramatic, but it is one extra case. Always turn the percent back into the actual amounts. The multiplier makes this easy: 1.5 × 2 = 3.
How to check your answer
Rebuild the new value. Multiply the old value by your multiplier and see whether you land on the new value.
Check the sign. A rise is positive, a fall is negative.
Check the base. Did you divide by the starting value and not the ending one?
Sanity bound. A fall cannot exceed
100%, since the value would go below zero, but a rise can.
Common mistakes
Dividing by the new value. For 40 to 50 this gives
10/50 = 20%, which is the reverse change.Adding percentages from successive changes, such as taking
20%then10%as30%.Treating a
50%rise followed by a50%fall as returning to the start. It gives1.5 × 0.5 = 0.75, a25%fall overall.Mixing up percentage points and percent.
Rounding the multiplier early, for instance using 1.3 for
1.28.
Where you will use it next
The same multiplier idea drives interest that compounds each year, as in compound interest step by step. It also underlies scale factors in ratios and proportions with maps and recipes, where a recipe is scaled up or down by one steady factor.
Frequently asked questions
How do I find the original price after a discount?
Divide the sale price by the multiplier. If a jacket costs $64 after 20% off, the multiplier is 0.8, and 64 / 0.8 = 80. Check: 80 × 0.8 = 64.
Is a `100%` rise the same as doubling?
Yes. The multiplier is 1 + 100/100 = 2.
Get started
