MISSION · Number sense
Percent Increase and Decrease
Discounts, tips, markups, and why they don't cancel
LEVEL 1 · EXPLORE
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LEVEL 2 · LEARN
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The idea in one sentence
A percent change is always measured against what you started with, so the same percent applied twice measures against two different amounts.
Two ways to do it
A £40 coat is reduced by \(25\%\).
Find the change, then apply it. \(25\%\) of 40 is 10, so the coat costs \(40 - 10 = 30\).
Or find what’s left, in one step. Taking off \(25\%\) leaves \(75\%\), and \(0.75 \times 40 = 30\).
The second method is worth learning properly. One multiplication, no subtraction, and it chains — which matters the moment two changes stack up.
\[\text{new} = \text{original} \times (1 + \text{change})\]with the change written as a decimal, negative for a decrease. A \(20\%\) rise is \(\times 1.2\); a \(20\%\) fall is \(\times 0.8\); a \(5\%\) tip is \(\times 1.05\).
They don’t cancel
Take £100. Add \(10\%\) to get £110. Now take \(10\%\) off: that \(10\%\) is of £110, which is £11, leaving £99.
You do not end up where you started, and the order doesn’t rescue you — going down first and up after also lands on £99. In multiplier form the reason is plain:
\[1.1 \times 0.9 = 0.99\]Two changes of the same size in opposite directions always lose you a little, and the bigger the percentage the worse it gets. A \(50\%\) rise followed by a \(50\%\) fall leaves you at 75% of where you began.
Finding the change
Going the other way, the formula is:
\[\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\]The denominator is always the original. A price going from 40 to 50 is a \(25\%\) increase, because the £10 rise is measured against the 40 you started at. Coming back down from 50 to 40 is a \(20\%\) decrease, because the same £10 is now measured against
- Same money, different percentages, and both are correct.
Reversing one
If a sale price of £30 is after \(25\%\) off, the original was not £37.50. You multiply to go forward, so you divide to come back:
\[30 \div 0.75 = 40\]Adding \(25\%\) back to the sale price is the classic wrong move, and it under-shoots every time.
Common mix-ups
- Measuring against the new amount instead of the original.
- Assuming an increase and a decrease of the same percent cancel. They never do.
- Adding percentages of different totals. Two \(10\%\) discounts on different items are not \(20\%\) off anything.
- Reversing a change by applying it backwards rather than dividing.
Check yourself
A £60 jacket rises by \(20\%\), then goes into a \(20\%\)-off sale. What does it cost, and what is the overall percent change?
\(60 \times 1.2 = 72\), then \(72 \times 0.8 = 57.60\). Overall \(\frac{57.60 - 60}{60} = -0.04\), a \(4\%\) decrease.
LEVEL 3 · REFLECT
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