Chapter 5 of 8

All 8 chapters
  1. 1Multiplying and dividing in your head, readunderstood
  2. 2Short multiplication, readunderstood
  3. 3Long multiplication, readunderstood
  4. 4Short division, readunderstood
  5. 5Long division, readunderstood
  6. 6Order of operations, readunderstood
  7. 7Estimate, then check, readunderstood
  8. 8Choosing what to do, readunderstood

M3.2.5 Chapter 5

Long division

How do I divide 7 870 by 23 on paper?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Divide a 4 digit number by a 2 digit number
  • Make a list of multiples to help with long division

Mei has 7 870 seeds. She plants them in rows of 23. How many full rows does she make? How many seeds are left over?

Short division is hard when the divisor is big, because the calculations in your head grow. Long division writes the working down. First, make a list of multiples of 23. A multiple is a number in a times table.

A list of the first ten multiples of 23, in two tables with the columns times and answer. 1 times is 23, 2 times is 46, 3 times is 69, 4 times is 92, 5 times is 115, 6 times is 138, 7 times is 161, 8 times is 184, 9 times is 207, 10 times is 230.timesanswer1232463694925115timesanswer613871618184920710230
The first ten multiples of 23. Each one is 23 more than the one before.

You can make the list by adding 23 each time. Check the last line. 10 × 23 is 23 with a 0 after it, which is 230.

Draw it first

Long division, round by round

  1. The calculation 7 870 divided by 23 written as long division. The divisor 23 is outside the bracket and 7 870 is inside. The answer 342 builds above the bracket, with r 4 at the end. Under the dividend are the rows 69, 97, 92, 50, 46 and 4.237870

    1 Write 23 outside the bracket and 7 870 inside it. 7 is less than 23, so start with 78.

  2. The calculation 7 870 divided by 23 written as long division. The divisor 23 is outside the bracket and 7 870 is inside. The answer 342 builds above the bracket, with r 4 at the end. Under the dividend are the rows 69, 97, 92, 50, 46 and 4.237870396−97

    2 3 × 23 = 69, and 4 × 23 is too big. Write 3 above the 8. Take away: 78 − 69 = 9. Bring down the 7 to make 97.

  3. The calculation 7 870 divided by 23 written as long division. The divisor 23 is outside the bracket and 7 870 is inside. The answer 342 builds above the bracket, with r 4 at the end. Under the dividend are the rows 69, 97, 92, 50, 46 and 4.237870396−97429−50

    3 4 × 23 = 92. Write 4 above the 7. Take away: 97 − 92 = 5. Bring down the 0 to make 50.

  4. The calculation 7 870 divided by 23 written as long division. The divisor 23 is outside the bracket and 7 870 is inside. The answer 342 builds above the bracket, with r 4 at the end. Under the dividend are the rows 69, 97, 92, 50, 46 and 4.237870396−97429−50264−4r 4

    4 2 × 23 = 46. Write 2 above the 0. Take away: 50 − 46 = 4. Nothing is left to bring down, so the answer is 342 remainder 4.

Every round has the same four moves. Divide: how many 23s fit? Multiply: write the product. Take away: find the leftover. Bring down: add the next digit.

Worked example

Work out 5 376 ÷ 21.

  1. Start with 53, because 5 is less than 21. Then 2 × 21 = 42, and 53 − 42 = 11.
  2. Bring down the 7 to make 117. Then 5 × 21 = 105, and 117 − 105 = 12.
  3. Bring down the 6 to make 126. Then 6 × 21 = 126, and 126 − 126 = 0.
  4. Nothing is left over. The answer is 256.

Answer: 256

Worked example

Work out 9 126 ÷ 34.

  1. Start with 91. Then 2 × 34 = 68, and 91 − 68 = 23.
  2. Bring down the 2 to make 232. Then 6 × 34 = 204, and 232 − 204 = 28.
  3. Bring down the 6 to make 286. Then 8 × 34 = 272, and 286 − 272 = 14.
  4. The answer is 268 remainder 14.

Answer: 268 remainder 14

Worked example

Work out 6 150 ÷ 15.

  1. Start with 61. Then 4 × 15 = 60, and 61 − 60 = 1.
  2. Bring down the 5 to make 15. Then 1 × 15 = 15, and 15 − 15 = 0.
  3. Bring down the 0. Nothing was left over, so the number is 0. 15 goes into 0 no times, so write 0 above it.
  4. The answer is 410.

Answer: 410

See it three ways

With things

Take 78 stones. Make piles of 23 stones. Keep making piles until you cannot make another full pile.

As a picture

A bar model of 78 stones. The bar has four parts: 23, 23, 23 and 9. Over the bar, the total is 78.232323978

In symbols

78 ÷ 23 = 3 remainder 9, because 3 × 23 = 69 and 78 − 69 = 9.

Try it

Make your own list of multiples

You need: paper and a pencil.

  1. Write 1 × 17 = 17 at the top of the page.
  2. Add 17 to get the next line, 2 × 17 = 34. Keep adding 17 down to 10 × 17.
  3. Check your last line. It should be 17 with a 0 after it.

What you will see: Each line is the line above plus 17. The fifth line, 85, is half of the last line, 170. A list like this takes one minute and saves many mistakes.

If you cannot try it, this is what happens.
A list of the first ten multiples of 17, in two tables with the columns times and answer. 1 times is 17, 2 times is 34, 3 times is 51, 4 times is 68, 5 times is 85, 6 times is 102, 7 times is 119, 8 times is 136, 9 times is 153, 10 times is 170.timesanswer117234351468585timesanswer610271198136915310170

A wrong idea many people have

To divide by 23, divide by 2 and then by 3.

Why it feels right: Times 6 splits into times 2 and times 3. The digits 2 and 3 sit inside 23.

The test that breaks it: Try 7 870 ÷ 2 = 3 935. Then 3 935 ÷ 3 = 1311 remainder 2. But 7 870 ÷ 23 = 342 remainder 4.

What is true: Dividing by 2 and then by 3 is the same as dividing by 6, because 2 × 3 = 6. The digits of 23 are not two separate divisors.

Key words

long division
dividing on paper by writing each round of working down.
multiples
the numbers in a times table, such as 23, 46 and 69.

Back to your guess

The answer is C: Make a list of the 23 times table. Then find how many 23s fit into each part, and write the working down. The answer is 342 with 4 left over.

Long division writes down each round of working, with a list of multiples to help. 7 870 ÷ 23 = 342 with 4 left over. Dividing by 2 and then by 3 is the same as dividing by 6, not by 23.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. In 7 870 ÷ 23, which number do you start with?

    The long division bracket with 23 outside it and 7 870 inside it. Nothing is worked out yet.237870
    Tap your answer.

    1. Yes, that is right. 7 is smaller than 23, so 23 cannot go into it. The first two digits make 78, and 23 goes into 78 three times.

    2. Not this one. 23 does not go into 7, because 7 is smaller. Take one more digit, so the number is 78. Go to M3.2.5. Try another.

    3. Not this one. That is one digit too many. 23 already goes into 78, so start there. Go to M3.2.5. Try another.

    4. Not this one. 23 is the divisor, the number you divide by. You start with digits from the number inside the bracket. Go to M3.2.5. Try another.

    See the answer

    78

    7 is smaller than 23, so 23 cannot go into it. The first two digits make 78, and 23 goes into 78 three times.

    • If you chose B: 23 does not go into 7, because 7 is smaller. Take one more digit, so the number is 78. Go to M3.2.5.
    • If you chose C: That is one digit too many. 23 already goes into 78, so start there. Go to M3.2.5.
    • If you chose D: 23 is the divisor, the number you divide by. You start with digits from the number inside the bracket. Go to M3.2.5.

  2. The list of multiples of 23 is 23, 46, 69, 92 and 115. How many 23s fit into 97?

    A list of the first ten multiples of 23, in two tables with the columns times and answer. 1 times is 23, 2 times is 46, 3 times is 69, 4 times is 92, 5 times is 115, 6 times is 138, 7 times is 161, 8 times is 184, 9 times is 207, 10 times is 230.timesanswer1232463694925115timesanswer613871618184920710230
    Tap your answer.

    1. Not this one. 5 × 23 = 115, which is bigger than 97. So five 23s do not fit. Go to M3.2.5. Try another.

    2. Yes, that is right. 4 × 23 = 92 is not more than 97, and 5 × 23 = 115 is. So four 23s fit.

    3. Not this one. 3 × 23 = 69 fits, but 97 − 69 = 28 is still more than 23. One more 23 fits. Go to M3.2.5. Try another.

    4. Not this one. 92 is the product, 4 × 23. The question asks how many 23s fit, so the answer is 4. Go to M3.2.5. Try another.

    See the answer

    4

    4 × 23 = 92 is not more than 97, and 5 × 23 = 115 is. So four 23s fit.

    • If you chose A: 5 × 23 = 115, which is bigger than 97. So five 23s do not fit. Go to M3.2.5.
    • If you chose C: 3 × 23 = 69 fits, but 97 − 69 = 28 is still more than 23. One more 23 fits. Go to M3.2.5.
    • If you chose D: 92 is the product, 4 × 23. The question asks how many 23s fit, so the answer is 4. Go to M3.2.5.

  3. You have worked out 78 − 69 = 9 in 7 870 ÷ 23. What do you do next?

    Tap your answer.

    1. Not this one. The digits come down in order, from the left. After 78, the next digit is the 7. The 0 comes after it. Go to M3.2.5. Try another.

    2. Not this one. The 9 is a leftover, not an answer digit. The answer digit was the 3 you wrote above the 8. Go to M3.2.5. Try another.

    3. Not this one. There are still digits in 7 870 to bring down. You stop only when none are left. Go to M3.2.5. Try another.

    4. Yes, that is right. The leftover 9 and the next digit, 7, make 97. Then you ask how many 23s fit into 97.

    See the answer

    Bring down the 7 next to the 9 to make 97.

    The leftover 9 and the next digit, 7, make 97. Then you ask how many 23s fit into 97.

    • If you chose A: The digits come down in order, from the left. After 78, the next digit is the 7. The 0 comes after it. Go to M3.2.5.
    • If you chose B: The 9 is a leftover, not an answer digit. The answer digit was the 3 you wrote above the 8. Go to M3.2.5.
    • If you chose C: There are still digits in 7 870 to bring down. You stop only when none are left. Go to M3.2.5.

  4. Read the working. What is 7 870 ÷ 23?

    The long division of 7 870 by 23, finished. The answer above the bracket is 342 r 4. The rows under it are 69, 97, 92, 50, 46 and 4.237870396−97429−50264−4r 4
    Tap your answer.

    1. Not this one. The 46 is the last product, 2 × 23. The remainder is the leftover after taking it away, which is 4. Go to M3.2.5. Try another.

    2. Not this one. This leaves out the remainder. 23 × 342 = 7 866, and 7 870 is 4 more. Go to M3.2.5. Try another.

    3. Yes, that is right. The answer above the bracket is 342. The last leftover is 4. So 7 870 ÷ 23 = 342 remainder 4.

    4. Not this one. You may have left out the last answer digit. Each round writes one digit above the bracket. Check by multiplying back: 23 × 34 = 782, far from 7 870. Go to M3.2.5. Try another.

    See the answer

    342 remainder 4

    The answer above the bracket is 342. The last leftover is 4. So 7 870 ÷ 23 = 342 remainder 4.

    • If you chose A: The 46 is the last product, 2 × 23. The remainder is the leftover after taking it away, which is 4. Go to M3.2.5.
    • If you chose B: This leaves out the remainder. 23 × 342 = 7 866, and 7 870 is 4 more. Go to M3.2.5.
    • If you chose D: You may have left out the last answer digit. Each round writes one digit above the bracket. Check by multiplying back: 23 × 34 = 782, far from 7 870. Go to M3.2.5.

  5. Which list shows the first five multiples of 17?

    Tap your answer.

    1. Yes, that is right. 17, then 17 + 17 = 34, then 34 + 17 = 51, then 51 + 17 = 68, then 68 + 17 = 85.

    2. Not this one. You added 10 each time. For multiples of 17, add 17 each time. Go to M3.2.5. Try another.

    3. Not this one. You doubled each time. That is 1, 2, 4, 8 and 16 times 17. Multiples go up one time at a time. Go to M3.2.5. Try another.

    4. Not this one. This starts at 2 × 17. The list starts with 1 × 17, which is 17 itself. Go to M3.2.5. Try another.

    See the answer

    17, 34, 51, 68, 85

    17, then 17 + 17 = 34, then 34 + 17 = 51, then 51 + 17 = 68, then 68 + 17 = 85.

    • If you chose B: You added 10 each time. For multiples of 17, add 17 each time. Go to M3.2.5.
    • If you chose C: You doubled each time. That is 1, 2, 4, 8 and 16 times 17. Multiples go up one time at a time. Go to M3.2.5.
    • If you chose D: This starts at 2 × 17. The list starts with 1 × 17, which is 17 itself. Go to M3.2.5.

  6. Work out 4 836 ÷ 12.

    See the answer

    403

    Start with 48. 4 × 12 = 48, so write 4 and nothing is left. Bring down the 3. 12 goes into 3 no times, so write 0. Bring down the 6. 12 goes into 36 three times. The answer is 403.

  7. Work out 9 126 ÷ 34. Give the answer and the remainder.

    See the answer

    268 remainder 14

    34 × 268 = 9 112, and 9 126 − 9 112 = 14.

  8. Write the first five multiples of 17.

    See the answer

    17, 34, 51, 68 and 85

    Start at 17 and add 17 each time: 17, 34, 51, 68, 85.

  9. From an earlier chapter

    Work out 6 207 ÷ 9. Give the answer and the remainder.

    See the answer

    689 remainder 6

    9 goes into 62 six times, leaving 8. 9 goes into 80 eight times, leaving 8. 9 goes into 87 nine times, leaving 6. So the answer is 689 remainder 6.

Keep it

Copy this into your exercise book.

Make a list of multiples of the divisor. Then repeat four moves: divide, multiply, take away, bring down. The last leftover is the remainder.

Draw: Draw the long division of 7 870 by 23. Write the list of multiples of 23 beside it.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

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