Division is the operation most people dread, mostly because long division on paper has so many steps. Mentally, it is easier to treat division as a question about multiplication: how many lots of this number fit into that one? Once you think of it that way, a handful of tricks makes most divisions quick. This guide covers them, shows how to handle remainders, and ends with a practice set.
Every multiplication fact gives you a division fact for free. If 7 × 8 = 56, then 56 ÷ 8 = 7 and 56 ÷ 7 = 8. This is the idea of fact families, and it is why quick times tables make division quick too.
So the first question to ask is: is this a times-table fact I already know? For 63 ÷ 9, ask what times 9 makes 63. The answer is 7.
For bigger numbers, take away a large, easy multiple first, usually 10 lots, then deal with what is left.
For 156 ÷ 12: 10 × 12 = 120, which leaves 156 − 120 = 36. Then 36 is 3 × 12. So 10 + 3 lots of 12 make 156, and 156 ÷ 12 = 13.
For 1,008 ÷ 12: 80 × 12 = 960, which leaves 1,008 − 960 = 48, and 48 is 4 × 12. So 1,008 ÷ 12 = 84.
If both numbers share a factor, divide both by it. The answer doesn't change, just as 6 ÷ 3 and 60 ÷ 30 are both 2.
For 840 ÷ 24, divide both by 4 to get 210 ÷ 6. That still looks awkward, so divide both by 3 to get 70 ÷ 2 = 35. So 840 ÷ 24 = 35.
Prime factors take this idea as far as it goes. For 924 ÷ 28, break both numbers into primes: 924 = 2² × 3 × 7 × 11 and 28 = 2² × 7. Cancel the factors they share (2² × 7), which leaves 3 × 11. The result is 33. Check: 33 × 28 = 924.
These rules tell you whether a division will come out exactly, without doing it.
Take 567 ÷ 9. The digits add up to 5 + 6 + 7 = 18, a multiple of 9, so the division is exact. Then 9 × 60 = 540 and 567 − 540 = 27, which is 3 lots of 9. So 567 ÷ 9 equals 63.
For more on these rules, including the less familiar ones for 7 and 11, see divisibility in Vedic Mathematics.
Estimation tells you roughly where the answer is, so you can home in on it. For instance, when dividing 789 by 17, round 789 to 800 and 17 to 20. That gives 800 ÷ 20 = 40, a quick ballpark.
Rounding 17 up to 20 means dividing by a bigger number, so the real answer is larger than 40. Try 17 × 45 = 765, then 17 × 46 = 782, which is as close as you can get without going over. And 789 − 782 = 7. So 789 ÷ 17 = 46 remainder 7, or about 46.4.
When a division isn't exact, the answer can be written three ways. Take 100 ÷ 7: 7 × 14 = 98, so 2 are left over.
Which one is right depends on the question. If 100 sweets are shared between 7 children, each gets 14 and 2 are left over. If 100 metres of rope is cut into 7 equal pieces, each piece is about 14.29 metres. Percentages work the same way; our guide to quick percentage calculations turns divisions like 18 ÷ 24 into a percentage.
Multiply your answer by the number you divided by, then add any remainder. You should get the number you started with.
Answers:
Since every division is a multiplication fact in reverse, the best practice is the facts themselves. Play the multiplication game, or choose division in practice mode to work on it directly. For another set of division shortcuts, read Vedic Maths division tricks.
Take away a large, easy multiple first, such as 10 or 100 lots of the divisor, then divide what is left. You can also divide both numbers by a common factor to make them smaller before you start.
Use the divisibility rules. For example, a number divides exactly by 3 or 9 when its digits add up to a multiple of 3 or 9, and by 4 when its last two digits make a multiple of 4.
It depends on what you are dividing. Things that can't be split, like people or sweets, need a remainder. Things that can, like money or length, usually need a decimal.