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Short and long division
Remainders as whole numbers, decimals and fractions, and what a remainder means in context.
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Division shares a number into equal groups. In this lesson you will divide big numbers without a calculator, and decide what to do with anything left over.
In this lesson you will learn:
- short division, set out as a bus stop
- three ways to write a remainder: as a whole number, a fraction and a decimal
- long division, for dividing by a 2-digit number
- what a remainder means in a real situation: round up, round down, or keep it exact
1. Short division: the bus stop
Key idea
Work from the left. Ask how many times the divisor goes into each digit. Write the answer on top and
carry the remainder to the next digit.
Example: 4,728 ÷ 6
078864475248
- 4: 6 does not go into 4. Carry the 4 to the next digit, making 47.
- 47: 6 × 7 = 42, so 6 goes 7 times, remainder 5. Write 7, carry 5, making 52.
- 52: 6 × 8 = 48, so 8 times, remainder 4. Write 8, carry 4, making 48.
- 48: 6 × 8 = 48 exactly. Write 8.
Answer788
Check with the inverse: 788 × 6 = 4,728.
Watch out
If the divisor does not go into a digit, write 0 on top and carry. 824 ÷ 4 = 206, not 26.
20648224
The division workshop: short division
How it works
- Tap how many times the divisor goes in. The remainder carries itself across.
- Two right with no slips moves you up a level. Level 5 divides by 11 and 12.
2. Remainders three ways
Key idea
A remainder is what is left over. You can write it as a whole number, a
fraction or a decimal.
4,729 ÷ 6 is 788, with 1 left over. We write the answer as 788 r 1:
0788r 164475249
| 17 ÷ 4 written as… | Answer | How |
|---|---|---|
| a remainder | 4 r 1 | 4 fours make 16, with 1 left over. |
| a fraction | 4 14 | The 1 left over is shared between 4: 14 each. |
| a decimal | 4.25 | Keep dividing past the decimal point, with zeros. |
The decimal: 17 ÷ 4, keeping going
4.25417.1020
- Write 17 as 17.00. Put the decimal point on top, straight above the one below.
- 4 into 17 is 4, remainder 1. Carry the 1 past the point: 10 tenths.
- 4 into 10 is 2, remainder 2. Carry 2: 20 hundredths. 4 into 20 is 5.
Answer4.25
Watch out
A remainder is not the decimal part. 17 ÷ 4 = 4 r 1, but it is 4.25, not 4.1.
Share it out: pick the numbers and watch them shared
Share
into groups of
Things to try
- Share 17 into 4 groups, then 18, 19, 20. When does the remainder disappear?
- Try dividing by 3. Why does the decimal never end?
3. Long division
Key idea
For a 2-digit divisor, write the working underneath. Each step is: divide, multiply,
subtract, bring down the next digit.
Top tip
Write out the first few multiples of the divisor before you start. For 24: 24, 48, 72, 96, 120…
Example: 7,488 ÷ 24
312247488−7228−2448−480
- Divide: 24 does not go into 7, so use 74. 24 × 3 = 72, so write 3 on top.
- Multiply and subtract: 74 − 72 = 2.
- Bring down the next digit, 8: 28. 24 goes once. 28 − 24 = 4.
- Bring down the last 8: 48. 24 × 2 = 48 exactly.
Answer312
Check: 312 × 24 = 7,488.
With a remainder: 9,175 ÷ 23
398r 21239175−69227−207205−18421
- 23 into 91: 23 × 3 = 69. 91 − 69 = 22. Bring down 7: 227.
- 23 into 227: 23 × 9 = 207. 227 − 207 = 20. Bring down 5: 205.
- 23 into 205: 23 × 8 = 184. 205 − 184 = 21. Nothing left to bring down.
Answer398 r 21
Check: 398 × 23 = 9,154, and 9,154 + 21 = 9,175.
The division workshop: long division
How it works
- Tap how many times the divisor goes in. The multiply, subtract and bring down then happen for you.
- Use the multiples strip at the top. Level 4 brings in remainders; level 5 has zeros in the answer.
4. What does the remainder mean?
Key idea
In a real situation, decide what the remainder means. You may need to round up,
round down, or keep it exact.
| Situation | 75 ÷ 16 = 4 r 11, so… |
|---|---|
| Round up when everyone or everything must fit. 75 children, minibuses that hold 16. |
5 minibuses. 4 would leave 11 children behind. |
| Round down when only full groups count. 75 eggs, boxes of 16. How many full boxes? |
4 full boxes, with 11 eggs over. |
| Keep it exact when it can be split. 75 pizzas shared between 16 children. |
4 1116 pizzas each. The 11 left over are cut up and shared too. |
Watch out
Never give “4 r 11 minibuses” as an answer. Read the question again and decide.
Round up, round down, or keep it exact?
5. Practice: pick your level
Start with Foundation if you are in Year 4 or just starting. Use rough paper and set each one out as a bus stop. Move up when you are getting most of them right.
The mistakes that cost marks
- Leaving out a 0 in the answer. 824 ÷ 4 written as 26. if the divisor does not go into a digit, write 0 on top: 824 ÷ 4 = 206.
- Forgetting to carry the remainder. 4,728 ÷ 6 worked out one digit at a time, as if each digit stood alone. carry each remainder to the next digit: 47, then 52, then 48. The answer is 788.
- A remainder bigger than the divisor. 31 ÷ 6 written as 4 r 7. if the remainder is as big as the divisor, it goes once more: 31 ÷ 6 = 5 r 1.
- Writing the remainder as the decimal. 17 ÷ 4 = 4 r 1 written as 4.1. keep dividing with zeros: 17.00 ÷ 4 = 4.25. Or use a fraction: 4 14.
- Rounding the wrong way. 75 children, minibuses of 16: answering 4. everyone must fit, so round up: 5 minibuses.
- Not checking. Most slips are easy to catch. multiply back: answer × divisor + remainder should give the number you started with.
Quick checklist
Tick each one when you can do it without help.
All ticked. Brilliant work! You have finished Lesson 2.3.
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