Problem 1: Finding X% of Y - The Multiplier Method
"What is 35% of 240?" You see this type of question everywhere: figuring out a discount at checkout, calculating a tip, working out how much tax gets added to something. The cleanest way to handle it is the multiplier method. Convert the percentage to a decimal and multiply. Done.
What is 35% of 240?
= 0.35 × 240
= 84
What is 8.5% of 600?
= 0.085 × 600
= 51
To get the decimal multiplier, move the decimal point two places to the left: 35% becomes 0.35, 8.5% becomes 0.085, 125% becomes 1.25. And yes, if the percentage is over 100%, the result will be bigger than your starting number. That comes up with things like compound growth.
| Percentage | Multiplier | Applied to 200 |
|---|---|---|
| 10% | 0.10 | 20 |
| 17.5% | 0.175 | 35 |
| 50% | 0.50 | 100 |
| 110% | 1.10 | 220 |
| 5.25% | 0.0525 | 10.5 |
Problem 2: What Percentage Is X of Y?
"A student scores 54 out of 72. What percentage is that?" You've got both numbers, and you need to know the relationship between them as a percentage. Simple: divide the part by the whole, then multiply by 100.
54 out of 72:
= (54 ÷ 72) × 100
= 0.75 × 100
= 75%
17 out of 40:
= (17 ÷ 40) × 100
= 0.425 × 100
= 42.5%
One thing to get right every time: divide part by whole first, then multiply by 100. Not the other way around. Dividing 100 by the whole before you multiply gives a completely different, wrong number.
You can use this same formula whenever you're comparing two numbers in the same units. Say you've spent £3,200 out of an £8,000 budget: (3200 ÷ 8000) × 100 = 40%. Same exact method.
Problem 3: Percentage Increase and Decrease
Two things fall under percentage change: finding out how much something changed as a percentage, and applying a known percentage change to get the new value.
Finding the Percentage Change
Price rises from £45 to £54:
= ((54 − 45) ÷ 45) × 100
= (9 ÷ 45) × 100
= 0.2 × 100
= 20% increase
Attendance falls from 800 to 680:
= ((680 − 800) ÷ 800) × 100
= (−120 ÷ 800) × 100
= −15% (a 15% decrease)
Applying a Percentage Change
Instead of working out the change amount separately and then adding or subtracting it, just use one multiplier. For an increase of p%, multiply by (1 + p/100). For a decrease, multiply by (1 − p/100).
Decrease £80 by 15%: 80 × 0.85 = £68
It's faster and you're less likely to make a mistake, especially when you're stacking multiple percentage changes on top of each other.
Problem 4: Reverse Percentage - Finding the Original Value
This is the one that trips people up most often. A jacket costs £63 after a 30% discount. What was the original price? The instinct is to add 30% back to £63. But that's wrong. 30% of £63 isn't the same as 30% of the original price, so you can't just reverse it that simply.
Here's the right move: the sale price is 70% of the original, because 30% was taken off. So you divide the price you know by the decimal version of that percentage.
Sale price £63 is 70% of the original:
Original = 63 ÷ 0.70 = £90
Verify: £90 × 0.70 = £63 ✓
Price after 20% increase is £96:
This is 120% of the original.
Original = 96 ÷ 1.20 = £80
Figure out what percentage the current value is of the original, convert it to a decimal, then divide. Same method every time, whether you're working backwards from a tax-inclusive price, a post-increase figure, or a discounted amount.
Common Mistakes to Avoid
Even when you know the formulas, these mistakes still catch people out:
Adding Percentages of Different Wholes
A 10% pay rise followed by a 10% pay cut doesn't bring you back to where you started. The rise is calculated on the original salary, but the cut is calculated on the higher salary. You end up with slightly less than you began with.
+10%: £100 × 1.10 = £110
−10%: £110 × 0.90 = £99 (not £100)
Using the Wrong Base for Percentage Change
Percentage change is always measured against the original value, not the new one. If you accidentally put the new value in the denominator, you get a different number that doesn't mean much.
Confusing Percentage Points with Percent Change
If a pass rate goes from 60% to 75%, that's a 15 percentage point increase. But the percentage change in the pass rate is (15 ÷ 60) × 100 = 25%. Both of those statements are true. They just measure different things. Mixing them up is one of the most common ways statistics get twisted.