Four steps
Long division works through the number being divided from left to right, one digit at a time. Every round has the same four steps: Divide, Multiply, Subtract, Bring down.
For 528 ÷ 12, look at the first digit, 5. 12 does not go into 5, because 5 is less than 12. So use the first two digits instead: 52.
528 ÷ 12 set out for long division: 12 outside, and 528 under the bar.
Divide and multiply
Divide: how many 12s are in 52? 12 × 4 = 48 and 12 × 5 = 60, so four 12s fit, and five would be too many. Write 4 above the 2 of 528.
Multiply: 4 × 12 = 48. Write 48 under the 52.
4 is written above the 2, and 4 × 12 = 48 is written under the 52.
Subtract and bring down
Subtract: 52 − 48 = 4. This 4 is what is left of the 52 after four 12s are taken out, so it must be less than 12. If it is 12 or more, another 12 fits, and the digit above needs to be bigger.
Bring down: bring the next digit of 528, the 8, down beside the 4. Together they make 48.
52 − 48 = 4, and the 8 comes down beside it to make 48.
Go round again
Repeat the four steps with 48. Divide: 48 ÷ 12 = 4, so write 4 above the 8. Multiply: 4 × 12 = 48. Subtract: 48 − 48 = 0. There are no more digits to bring down, so the division is finished: 528 ÷ 12 = 44.
Multiply back to check: 44 × 12 = 440 + 88 = 528.
The second round leaves 0, and the answer above the bar is 44.
What each round does
Each round deals with one place value. The first 4 is written above the tens digit of 528, so it stands for 4 tens, which is 40. The first round took forty 12s out of 528: 40 × 12 = 480. The 48 that was left is four more 12s.
So long division is dividing by splitting, done one place at a time. 528 = 480 + 48, and 480 ÷ 12 + 48 ÷ 12 = 40 + 4 = 44.
The first jump is forty 12s, which is 480. The second jump is four more 12s, which is 48, and it lands on 528.
A one-digit divisor
Dividing by a one-digit number takes the same four steps. For 952 ÷ 7: 7 goes into 9 once, and 9 − 7 = 2 is left. Bring down the 5 to make 25. 7 goes into 25 three times, 3 × 7 = 21, and 4 is left. Bring down the 2 to make 42. 7 goes into 42 six times exactly. So 952 ÷ 7 = 136.
952 ÷ 7 = 136: three rounds, one for each digit of 952.
When the divisor does not fit
Every digit that is brought down gets a digit above it, even when that digit is 0. For 630 ÷ 6: 6 goes into 6 once. Bring down the 3. 6 does not go into 3, so write 0 above the 3; the 3 is still left. Bring down the 0 to make 30, and 6 goes into 30 five times. So 630 ÷ 6 = 105.
Leaving out the 0 gives 15, and 15 × 6 = 90, nowhere near 630.
The 0 above the 3 says that no 6 fits in 3. The answer is 105.
Worked example: School Fair Money Shared Equally Among the Classes
Question At a school fair, the parents' association raises $3024. The money is for class outings and is shared equally among the school's 18 classes. (a) How much does each class receive? (b) A local business then offers to pay for two classes' outings, so the whole $3024 is shared equally among the other 16 classes instead. How much more does each of those 16 classes receive than before?
1.Draw the $3024 as one bar cut into 18 equal shares, one for each class. To find one share, divide 3024 by 18 by long division.
The $3024 is one bar cut into 18 equal shares, one for each class. 2.Since 3 is less than 18, start with 30: 30 ÷ 18 = 1 remainder 12. Bring down the 2 to make 122.
3 is less than 18, so start with 30: 30 ÷ 18 = 1 remainder 12. Bring down the 2 to make 122. 3.18 × 6 = 108, so 122 ÷ 18 = 6 remainder 122 − 108 = 14. Bring down the 4 to make 144.
18 × 6 = 108 and 122 − 108 = 14. Bring down the 4 to make 144. 4.(a) 18 × 8 = 144, so 144 ÷ 18 = 8 with nothing left over. Each class receives $168.
(a) 18 × 8 = 144, nothing left over. Each class receives $168. 5.The two shares no longer needed are 2 × $168 = $336, and they are shared among the other 16 classes.
The two shares no longer needed are 2 × $168 = $336. 6.(b) Long division gives 336 ÷ 16 = 21, so each of the 16 classes receives $21 more. Check: 16 × (168 + 21) = 16 × 189 = 3024.
(b) 336 ÷ 16 = 21: each of the 16 classes receives $21 more.
Answer: (a) $168; (b) $21 more
Common mistakes
- Answering $189 for part (b). That is the new share; the question asks how much more it is than the old share of $168.
- Stopping the long division after two digits and answering $16. Every digit of 3024 must be brought down, so there is a third round, 144 ÷ 18 = 8, and the share is $168.