The percent swap, and percent change
Two things that save you daily: x% of y is the same as y% of x, and a price that goes up then down by the same percent does not come back. Stacked discounts too.
THE SWAP
x% of y equals y% of x
x% of y means x × y ÷ 100. Swap the two numbers and you get y × x ÷ 100, which is the same multiplication in a different order. So you are always allowed to put the awkward number in the percent slot and the easy one in front.
4% of 75 = 75% of 4 = 3. 8% of 125 = 125% of 8 = 10. 18% of 50 = 50% of 18 = 9.
4% of 75
hard way: 75 ÷ 100 = 0.75, then × 4
swap: 75% of 4
half of 4 = 2
quarter of 4 = 1
2 + 1 = 33The swap is worth doing whenever one of the two numbers is 4, 5, 8, 20, 25, 50 or 100. Those make lovely percentages, and awful things to take a percentage of.
Check yourself
A bank charges 8% on a transfer of 125. Swap it round. What's the fee?
- 10
- 12.5
- 100
Show the answer
10
Yes. 8% of 125 is 125% of 8, which is 8 plus a quarter of 8, so 10.
CHANGE
Up then down does not bring you home
A rise of 20% multiplies the price by 1.2. A fall of 20% multiplies it by 0.8. Do both and you have multiplied by 1.2 × 0.8 = 0.96, so you end 4% below where you started. The second percent is taken from a different, bigger number, which is why the two never cancel.
500,000 up 20% is 600,000. Then 20% off 600,000 is 120,000, not 100,000, so you land at 480,000.
a 500,000 price, up 20%, then down 20%
up: 10% of 500,000 = 50,000
20% = 100,000
500,000 + 100,000 = 600,000
down: 10% of 600,000 = 60,000
20% = 120,000
600,000 − 120,000 = 480,000480,000 — not 500,000Notice the two corrections are different sizes, 100,000 and 120,000, because they are slices of different totals. That gap is the whole phenomenon.
Check yourself
A shop raises a 200 item by 10% for the season, then cuts it by 10% afterwards. The price is back at 200.
Show the answer
False
False. 200 goes up to 220, and 10% of 220 is 22, so it comes down to 198. You always end slightly below, because the cut is a slice of the bigger number.
Two discounts, one after the other
- A jacket at 1,000,000 with 30% off
30% of 1,000,000 is 300,000, so the shelf price is 700,000. Or think of it as keeping 70%.
- Then a further 20% off at the till
That 20% comes off the 700,000, not the original. 20% of 700,000 is 140,000.
- The final price is 560,000
700,000 − 140,000 = 560,000. You kept 70% and then 80% of that, so you kept 56%.
- So the real discount is 44%, not 50%
You paid 560,000 out of 1,000,000. Two discounts never simply add, because the second one is smaller than it looks.
Check yourself
A coat is 800,000. The sale takes 25% off, and a voucher takes a further 10% at the till. What does Negar pay?
- 520,000
- 560,000
- 540,000
- 600,000
Show the answer
540,000
Right. 25% off leaves 600,000, and 10% of that is 60,000, so she pays 540,000.
Check yourself
Apply each change and match it to the new number
Show the answer
- 200, up 15% → 230
- 200, down 15% → 170
- 80, up 25% → 100
- 80, down 25% → 60
Make it a habit
- Before taking a percentage, ask whether the swap makes it easier. 16% of 25 is 25% of 16, which is 4.
- Think in what you keep, not what you lose: 20% off means × 0.8, and two discounts multiply.
- Any percent change asks 'percent of what?'. The answer is always the number you had just before.
- A rise and a fall of the same percent always leave you slightly below the start, never exactly back.
Check yourself
The hard one. A 4,000,000 salary is cut by 10% in a bad year, then raised by 10% the next year. What is it now?
- 4,000,000
- 3,600,000
- 3,960,000
Show the answer
3,960,000
Exactly. The cut takes it to 3,600,000, and 10% of that is only 360,000, so the rise brings it to 3,960,000. Still 1% short.
Lesson recap
- x% of y is always the same as y% of x, so put the awkward number in the percent slot.
- A percent change is a multiplication: up 20% is × 1.2, down 20% is × 0.8.
- Up then down by the same percent lands below the start, because each percent is taken from a different total.
- Stacked discounts multiply what you keep: 30% off then 20% off leaves 56%, a 44% saving.