Division and fractions in your head
Divide by 4 and 8 with nothing but halving, turn ÷5 into a doubling, and learn the handful of fraction-to-percent pairs that make a quarter of daily arithmetic instant.
HALVING CHAINS
÷4 is two halves, ÷8 is three
4 = 2 × 2 and 8 = 2 × 2 × 2, so dividing by them is just halving repeatedly. The point isn't that it's clever, it's that you only ever hold one number at a time and each step is small enough not to drop.
296 ÷ 8: half is 148, half again is 74, half again is 37. 96 ÷ 4: 48, then 24.
296 ÷ 8 (a bill split eight ways)
halve: 296 → 148
halve: 148 → 74
halve: 74 → 3737 eachCount the halvings on your fingers, because losing track of how many you've done is the only way this goes wrong. Three fingers for 8, two for 4, four for 16.
Check yourself
456 chairs go into 8 equal rows. How many chairs in a row?
- 57
- 28.5
- 114
Show the answer
57
Yes. 456 → 228 → 114 → 57. Three halvings, because 8 is three doublings of 1.
DIVIDING BY 5
÷5 is double it, then ÷10
5 is 10 ÷ 2, so dividing by 5 is dividing by 10 and multiplying by 2. Doubling and moving a decimal point are both free; long division by 5 is not. The same idea gives you ÷50 as double, then ÷100.
340 ÷ 5: double to 680, then 68. 1,250 ÷ 5: double to 2,500, then 250. 900 ÷ 50: double to 1,800, then 18.
Dividing by a number with no shortcut
- A 1,440,000 bill between 6 people
6 isn't a power of 2, so pure halving won't finish the job. But 6 = 2 × 3, and you can do those one after the other.
- Halve it: 720,000
The easy half of the work, done in a second.
- Now divide by 3: 240,000
72 ÷ 3 = 24, so 720,000 ÷ 3 = 240,000. Each person pays 240,000.
- The general move
Break the divisor into factors and divide in stages: ÷12 is ÷4 then ÷3, ÷15 is ÷5 then ÷3, ÷20 is ÷2 then ÷10.
Check yourself
A bill of 1,350,000 split between 5 people comes to 270,000 each.
Show the answer
True
True. Double 1,350,000 to get 2,700,000, then move the point one place: 270,000. Check it the other way if you like: five lots of 270,000 is 1,350,000.
| fraction | percent | quick way |
|---|---|---|
| 1/2 | 50% | halve |
| 1/4 | 25% | halve twice |
| 1/5 | 20% | double, ÷10 |
| 1/8 | 12.5% | halve three times |
| 1/3 | about 33% | a third, never exact |
Check yourself
Match each fraction to the percentage you'd use instead
Show the answer
- 1/8 → 12.5%
- 1/5 → 20%
- 2/5 → 40%
- 3/4 → 75%
Divide, or take a fraction?
Divide in stages
Use it when the divisor breaks into small factors and the number is untidy: 1,440,000 ÷ 6 as halve, then thirds. You work with the real number the whole way.
Take a known fraction
Use it when the fraction is one you know as a percentage: 3/8 of 2,400,000 is 12.5% three times, and 12.5% of 2,400,000 is 300,000. Fewer steps, and it reads straight off the bill.
Check yourself
What's the fastest route for each?
- 672 ÷ 8
- 430 ÷ 5
- 75% of 200
- 96 ÷ 4
- 1,250 ÷ 5
- 2/3 of 90
Show the answer
Halve, halve, halve: 672 ÷ 8, 96 ÷ 4
Double, then ÷10: 430 ÷ 5, 1,250 ÷ 5
Use a fraction you know: 75% of 200, 2/3 of 90
Make it a habit
- Before dividing, factor the divisor: 8 is 2·2·2, 12 is 4·3, 15 is 5·3, 20 is 2·10.
- ÷5 is double then ÷10. ÷50 is double then ÷100. Never divide by 5 the long way.
- Learn 1/8 = 12.5% by heart. It turns eighths of a bill into a percentage you can build.
- 1/3 is not 33%. It is a bit more. For money, divide properly rather than using 33%.
Check yourself
The hard one. A project costs 2,400,000 and Reza's share is 3/8 of it. What does he owe?
- 300,000
- 800,000
- 960,000
- 900,000
Show the answer
900,000
Exactly. An eighth is 2,400,000 halved three times: 1,200,000, 600,000, 300,000. Three of those is 900,000.
Lesson recap
- ÷4 is halving twice and ÷8 is halving three times; count the halvings so you don't stop early.
- ÷5 is double then ÷10, because 5 is 10 ÷ 2. Same trick gives ÷50.
- For any other divisor, break it into factors and divide in stages: ÷6 is ÷2 then ÷3.
- Know 1/2, 1/4, 1/5 and 1/8 as percentages, and build the rest from them: 3/8 is three 12.5%s.