Year 6 Arithmetic (SATs) · Multiplication and division · Lesson 2.4
Year 6 Arithmetic (SATs): Long division by a two-digit number
Part of the Year 6 Arithmetic (SATs) multiplication and division chapter. Chapter overview · whole chapter on one page
2.4
Long division by a two-digit number
Story. A school has 1,046 pupils. Each class has room for 25 pupils. The office needs to know how many classes are needed, and the remainder decides whether one more class is needed.
Rule
In long division, divide by the two-digit number one digit of the answer at a time. Multiply the divisor by each digit, subtract, and bring down the next digit of the starting number.
The remainder can be written in three ways: as a whole number (98 r 0), as a fraction (41 2/3), or rounded in context (a class of 25 needs 42 classes if 21 pupils are left over).
Key ideas
- Each step: divide, multiply, subtract, bring down.
- The remainder must be smaller than the two-digit divisor.
- The question tells you which form of the remainder to give.
- A fraction remainder is the remainder over the divisor, simplified.
Worked example 2.4.1
Work out 1,568 ÷ 16.
Method. Divide one digit of the answer at a time.
- Bring down the thousands digit to make 1. 1 is less than 16, so this digit of the answer is 0.1 < 16, so 0 with remainder 1
- Bring down the hundreds digit to make 15. 15 is less than 16, so this digit of the answer is 0.15 < 16, so 0 with remainder 15
- Bring down the tens digit to make 156. 16 goes into 156 9 times, and the remainder is what is left.16 × 9 = 144, 156 - 144 = 12
- Bring down the ones digit to make 128. 16 goes into 128 8 times, and the remainder is what is left.16 × 8 = 128, 128 - 128 = 0
- Write the answer. 98
Check. 16 × 98 = 1,568. ✓
Watch out. The remainder here is 0, so the division is exact.
Worked example 2.4.2
Work out 1,000 ÷ 24. Give the remainder as a fraction.
Method. Divide as usual. The remainder becomes a fraction over 24.
- Bring down the thousands digit to make 1. 1 is less than 24, so this digit of the answer is 0.1 < 24, so 0 with remainder 1
- Bring down the hundreds digit to make 10. 10 is less than 24, so this digit of the answer is 0.10 < 24, so 0 with remainder 10
- Bring down the tens digit to make 100. 24 goes into 100 4 times, and the remainder is what is left.24 × 4 = 96, 100 - 96 = 4
- Bring down the ones digit to make 40. 24 goes into 40 1 times, and the remainder is what is left.24 × 1 = 24, 40 - 24 = 16
- Write the remainder as a fraction. The remainder is 16 out of 24. Simplify it.16/24 = 2/3
- Write the answer. 41 2/3
Check. 41 + 2/3 = 41 2/3, and 24 × 41 + 16 = 1,000. ✓
Worked example 2.4.3
A school has 1,046 pupils and each class holds 25 pupils. How many classes are needed?
Method. Divide to find the full classes, then decide what the remainder means.
- Bring down the thousands digit to make 1. 1 is less than 25, so this digit of the answer is 0.1 < 25, so 0 with remainder 1
- Bring down the hundreds digit to make 10. 10 is less than 25, so this digit of the answer is 0.10 < 25, so 0 with remainder 10
- Bring down the tens digit to make 104. 25 goes into 104 4 times, and the remainder is what is left.25 × 4 = 100, 104 - 100 = 4
- Bring down the ones digit to make 46. 25 goes into 46 1 times, and the remainder is what is left.25 × 1 = 25, 46 - 25 = 21
- Write the answer. 42 classes
Check. 41 full classes hold 1025 pupils, and 21 pupils need one more class. ✓
Watch out. A remainder of 21 pupils still needs a class, so round up, not down.
Exercise 2D Practice: long division and remainders
Non-calculator1
FWork out 2,736 ÷ 36. Give the remainder as a whole number.
76
Method. Divide one digit at a time with long division.
- Bring down the thousands digit to make 2. 2 is less than 36, so this digit of the answer is 0.2 < 36, so 0 with remainder 2
- Bring down the hundreds digit to make 27. 27 is less than 36, so this digit of the answer is 0.27 < 36, so 0 with remainder 27
- Bring down the tens digit to make 273. 36 goes into 273 7 times, and the remainder is what is left.36 × 7 = 252, 273 - 252 = 21
- Bring down the ones digit to make 216. 36 goes into 216 6 times, and the remainder is what is left.36 × 6 = 216, 216 - 216 = 0
- Write the answer. 76
Check. 36 × 76 + 0 = 2,736. ✓
2
FWork out 3,822 ÷ 27. Give the remainder as a whole number.
141 remainder 15
Method. Divide one digit at a time with long division.
- Bring down the thousands digit to make 3. 3 is less than 27, so this digit of the answer is 0.3 < 27, so 0 with remainder 3
- Bring down the hundreds digit to make 38. 27 goes into 38 1 times, and the remainder is what is left.27 × 1 = 27, 38 - 27 = 11
- Bring down the tens digit to make 112. 27 goes into 112 4 times, and the remainder is what is left.27 × 4 = 108, 112 - 108 = 4
- Bring down the ones digit to make 42. 27 goes into 42 1 times, and the remainder is what is left.27 × 1 = 27, 42 - 27 = 15
- Write the answer. 141 remainder 15
Check. 27 × 141 + 15 = 3,822. ✓
3
FWork out 4,760 ÷ 35. Give the remainder as a whole number.
136
Method. Divide one digit at a time with long division.
- Bring down the thousands digit to make 4. 4 is less than 35, so this digit of the answer is 0.4 < 35, so 0 with remainder 4
- Bring down the hundreds digit to make 47. 35 goes into 47 1 times, and the remainder is what is left.35 × 1 = 35, 47 - 35 = 12
- Bring down the tens digit to make 126. 35 goes into 126 3 times, and the remainder is what is left.35 × 3 = 105, 126 - 105 = 21
- Bring down the ones digit to make 210. 35 goes into 210 6 times, and the remainder is what is left.35 × 6 = 210, 210 - 210 = 0
- Write the answer. 136
Check. 35 × 136 + 0 = 4,760. ✓
4
FWork out 1,584 ÷ 22. Give the remainder as a whole number.
72
Method. Divide one digit at a time with long division.
- Bring down the thousands digit to make 1. 1 is less than 22, so this digit of the answer is 0.1 < 22, so 0 with remainder 1
- Bring down the hundreds digit to make 15. 15 is less than 22, so this digit of the answer is 0.15 < 22, so 0 with remainder 15
- Bring down the tens digit to make 158. 22 goes into 158 7 times, and the remainder is what is left.22 × 7 = 154, 158 - 154 = 4
- Bring down the ones digit to make 44. 22 goes into 44 2 times, and the remainder is what is left.22 × 2 = 44, 44 - 44 = 0
- Write the answer. 72
Check. 22 × 72 + 0 = 1,584. ✓
5
FWork out 5,376 ÷ 48. Give the remainder as a whole number.
112
Method. Divide one digit at a time with long division.
- Bring down the thousands digit to make 5. 5 is less than 48, so this digit of the answer is 0.5 < 48, so 0 with remainder 5
- Bring down the hundreds digit to make 53. 48 goes into 53 1 times, and the remainder is what is left.48 × 1 = 48, 53 - 48 = 5
- Bring down the tens digit to make 57. 48 goes into 57 1 times, and the remainder is what is left.48 × 1 = 48, 57 - 48 = 9
- Bring down the ones digit to make 96. 48 goes into 96 2 times, and the remainder is what is left.48 × 2 = 96, 96 - 96 = 0
- Write the answer. 112
Check. 48 × 112 + 0 = 5,376. ✓
6
FWork out 6,009 ÷ 45. Give the remainder as a whole number.
133 remainder 24
Method. Divide one digit at a time with long division.
- Bring down the thousands digit to make 6. 6 is less than 45, so this digit of the answer is 0.6 < 45, so 0 with remainder 6
- Bring down the hundreds digit to make 60. 45 goes into 60 1 times, and the remainder is what is left.45 × 1 = 45, 60 - 45 = 15
- Bring down the tens digit to make 150. 45 goes into 150 3 times, and the remainder is what is left.45 × 3 = 135, 150 - 135 = 15
- Bring down the ones digit to make 159. 45 goes into 159 3 times, and the remainder is what is left.45 × 3 = 135, 159 - 135 = 24
- Write the answer. 133 remainder 24
Check. 45 × 133 + 24 = 6,009. ✓
7
FWork out 2,115 ÷ 25. Give the remainder as a whole number.
84 remainder 15
Method. Divide one digit at a time with long division.
- Bring down the thousands digit to make 2. 2 is less than 25, so this digit of the answer is 0.2 < 25, so 0 with remainder 2
- Bring down the hundreds digit to make 21. 21 is less than 25, so this digit of the answer is 0.21 < 25, so 0 with remainder 21
- Bring down the tens digit to make 211. 25 goes into 211 8 times, and the remainder is what is left.25 × 8 = 200, 211 - 200 = 11
- Bring down the ones digit to make 115. 25 goes into 115 4 times, and the remainder is what is left.25 × 4 = 100, 115 - 100 = 15
- Write the answer. 84 remainder 15
Check. 25 × 84 + 15 = 2,115. ✓
8
FWork out 3,920 ÷ 40. Give the remainder as a whole number.
98
Method. Divide one digit at a time with long division.
- Bring down the thousands digit to make 3. 3 is less than 40, so this digit of the answer is 0.3 < 40, so 0 with remainder 3
- Bring down the hundreds digit to make 39. 39 is less than 40, so this digit of the answer is 0.39 < 40, so 0 with remainder 39
- Bring down the tens digit to make 392. 40 goes into 392 9 times, and the remainder is what is left.40 × 9 = 360, 392 - 360 = 32
- Bring down the ones digit to make 320. 40 goes into 320 8 times, and the remainder is what is left.40 × 8 = 320, 320 - 320 = 0
- Write the answer. 98
Check. 40 × 98 + 0 = 3,920. ✓
9
PSA shop packs 1,300 cans into full cases of 24. How many full cases can it make?
54 full cases
Method. Find the full cases, ignoring the remainder.
- Bring down the thousands digit to make 1. 1 is less than 24, so this digit of the answer is 0.1 < 24, so 0 with remainder 1
- Bring down the hundreds digit to make 13. 13 is less than 24, so this digit of the answer is 0.13 < 24, so 0 with remainder 13
- Bring down the tens digit to make 130. 24 goes into 130 5 times, and the remainder is what is left.24 × 5 = 120, 130 - 120 = 10
- Bring down the ones digit to make 100. 24 goes into 100 4 times, and the remainder is what is left.24 × 4 = 96, 100 - 96 = 4
- Write the answer. 54 full cases
Check. 54 × 24 = 1,296, and 4 cans are left. ✓
10
PS96 pupils go on a trip. Each coach has 40 seats. How many coaches are needed?
3 coaches
Method. Divide, then round up because the 16 pupils left over need a seat.
- Bring down the tens digit to make 9. 9 is less than 40, so this digit of the answer is 0.9 < 40, so 0 with remainder 9
- Bring down the ones digit to make 96. 40 goes into 96 2 times, and the remainder is what is left.40 × 2 = 80, 96 - 80 = 16
- Write the answer. 3 coaches
Check. 2 coaches carry 80 pupils, so the other 16 need a third coach. ✓
Watch out. Round up when every person needs a place.
11
PSWrite 85 ÷ 12 as a mixed number.
7 1/12
Method. Divide, then write the remainder as a fraction over 12.
- Bring down the tens digit to make 8. 8 is less than 12, so this digit of the answer is 0.8 < 12, so 0 with remainder 8
- Bring down the ones digit to make 85. 12 goes into 85 7 times, and the remainder is what is left.12 × 7 = 84, 85 - 84 = 1
- Write the fraction. The remainder 1 goes over 12.1/12 = 1/12
- Write the answer. 7 1/12
Check. 7 × 12 + 1 = 85, so 85 ÷ 12 = 7 1/12. ✓
12
PSWrite 1,000 ÷ 24 as a mixed number.
41 2/3
Method. The remainder over 24 simplifies to 2/3.
- Bring down the thousands digit to make 1. 1 is less than 24, so this digit of the answer is 0.1 < 24, so 0 with remainder 1
- Bring down the hundreds digit to make 10. 10 is less than 24, so this digit of the answer is 0.10 < 24, so 0 with remainder 10
- Bring down the tens digit to make 100. 24 goes into 100 4 times, and the remainder is what is left.24 × 4 = 96, 100 - 96 = 4
- Bring down the ones digit to make 40. 24 goes into 40 1 times, and the remainder is what is left.24 × 1 = 24, 40 - 24 = 16
- Simplify the fraction. Divide the top and bottom of 16/24 by 8.16/24 = 2/3
- Write the answer. 41 2/3
Check. 41 2/3 is 41 and two thirds. ✓
13
RAli says 1,000 ÷ 24 is 41 remainder 16, and he rounds it to 41 to the nearest whole number. Is he right? Write 41 or 42.
42
Method. The remainder is 16 out of 24, which is more than half, so the rounded answer is 42.
- Bring down the thousands digit to make 1. 1 is less than 24, so this digit of the answer is 0.1 < 24, so 0 with remainder 1
- Bring down the hundreds digit to make 10. 10 is less than 24, so this digit of the answer is 0.10 < 24, so 0 with remainder 10
- Bring down the tens digit to make 100. 24 goes into 100 4 times, and the remainder is what is left.24 × 4 = 96, 100 - 96 = 4
- Bring down the ones digit to make 40. 24 goes into 40 1 times, and the remainder is what is left.24 × 1 = 24, 40 - 24 = 16
- Check the size of the remainder. Half of 24 is 12, and 16 is more than 12.16 > 12
- Write the answer. 42
Check. 16/24 = 2/3, which is more than half, so it rounds up to 42. ✓
Watch out. Ali is wrong: a remainder over half the divisor rounds up.
14
RWork out 2,000 ÷ 36 and give the remainder as a fraction.
55 5/9
Method. Divide, then write the remainder over 36.
- Bring down the thousands digit to make 2. 2 is less than 36, so this digit of the answer is 0.2 < 36, so 0 with remainder 2
- Bring down the hundreds digit to make 20. 20 is less than 36, so this digit of the answer is 0.20 < 36, so 0 with remainder 20
- Bring down the tens digit to make 200. 36 goes into 200 5 times, and the remainder is what is left.36 × 5 = 180, 200 - 180 = 20
- Bring down the ones digit to make 200. 36 goes into 200 5 times, and the remainder is what is left.36 × 5 = 180, 200 - 180 = 20
- Write the fraction. 20/36 simplifies to 5/9.20/36 = 5/9
- Write the answer. 55 5/9
Check. 36 × 55 + 20 = 2,000. ✓