Chapter 4 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

M3.2.4 Chapter 4

Short division

How do I divide 5 217 by 7 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 1 digit number
  • Write a remainder as a whole number or as a fraction

Leila shares 5 217 beads equally between 7 bags. How many beads go in each bag? How many are left over?

This is short division. It is also called the bus stop method, because of the shape of the bracket. The number you divide by is the divisor. Work from the left. When a number does not divide exactly, carry what is left over to the next digit.

Draw it first

Short division, one digit at a time

  1. The calculation 5 217 divided by 7 written as a bus stop. The divisor 7 is outside the bracket and 5 217 is inside. The answer 745 builds above the bracket, with r 2 at the end. Small carried digits 5, 3 and 3 sit in front of the digits 2, 1 and 7.75217

    1 Write the divisor 7 outside the bracket. Write 5 217 inside it, one digit in each place.

  2. The calculation 5 217 divided by 7 written as a bus stop. The divisor 7 is outside the bracket and 5 217 is inside. The answer 745 builds above the bracket, with r 2 at the end. Small carried digits 5, 3 and 3 sit in front of the digits 2, 1 and 7.752175

    2 7 goes into 5 no times. Write nothing above. Carry the 5, so the next number is 52.

  3. The calculation 5 217 divided by 7 written as a bus stop. The divisor 7 is outside the bracket and 5 217 is inside. The answer 745 builds above the bracket, with r 2 at the end. Small carried digits 5, 3 and 3 sit in front of the digits 2, 1 and 7.75217573

    3 7 goes into 52 seven times, because 7 × 7 = 49. Write 7 above the 2. Carry the 3, so the next number is 31.

  4. The calculation 5 217 divided by 7 written as a bus stop. The divisor 7 is outside the bracket and 5 217 is inside. The answer 745 builds above the bracket, with r 2 at the end. Small carried digits 5, 3 and 3 sit in front of the digits 2, 1 and 7.7521757343

    4 7 goes into 31 four times, because 7 × 4 = 28. Write 4 above the 1. Carry the 3, so the next number is 37.

  5. The calculation 5 217 divided by 7 written as a bus stop. The divisor 7 is outside the bracket and 5 217 is inside. The answer 745 builds above the bracket, with r 2 at the end. Small carried digits 5, 3 and 3 sit in front of the digits 2, 1 and 7.75217573435r 2

    5 7 goes into 37 five times, because 7 × 5 = 35. Write 5 above the 7. The last leftover is 2.

The 2 left at the end is the remainder. Each bag gets 745 beads, and 2 beads are left over. You write it as 745 r 2.

You can also write the remainder over the divisor, as a fraction. Suppose the 2 left over could be cut up. Shared by the 7 bags, each bag would get 2/7 more. So the answer is 745 2/7. Use a fraction only when the thing can be cut, like rope or water. Beads cannot be cut, so for beads 745 r 2 is the honest answer.

The calculation 5 217 divided by 7 as a bus stop. The answer is 745 with the fraction 2 over 7 written after it. Small carried digits 5, 3 and 3 sit in front of the digits 2, 1 and 7.7521757343527
The same calculation, with the remainder written as a fraction.

Worked example

Work out 4 806 ÷ 6.

  1. 6 goes into 4 no times. Carry the 4, so the next number is 48.
  2. 6 goes into 48 eight times, because 6 × 8 = 48. Nothing is left over.
  3. The next digit is 0. 6 goes into 0 no times, so write 0 above it.
  4. 6 goes into 6 once, because 6 × 1 = 6. The answer is 801.

Answer: 801

Worked example

Work out 3 458 ÷ 5.

  1. 5 goes into 3 no times. Carry the 3, so the next number is 34.
  2. 5 goes into 34 six times, because 5 × 6 = 30. The leftover is 34 − 30 = 4.
  3. Carry the 4, so the next number is 45. 5 goes into 45 nine times, because 5 × 9 = 45. Nothing is left over.
  4. 5 goes into 8 once, because 5 × 1 = 5. The leftover is 8 − 5 = 3.
  5. The answer is 691 with remainder 3.

Answer: 691 remainder 3

Worked example

Work out 9 175 ÷ 6. Write the remainder over the divisor, as a fraction.

  1. 6 goes into 9 once, because 6 × 1 = 6. The leftover is 9 − 6 = 3.
  2. Carry the 3, so the next number is 31. 6 goes into 31 five times, because 6 × 5 = 30. The leftover is 31 − 30 = 1.
  3. Carry the 1, so the next number is 17. 6 goes into 17 twice, because 6 × 2 = 12. The leftover is 17 − 12 = 5.
  4. Carry the 5, so the next number is 55. 6 goes into 55 nine times, because 6 × 9 = 54. The leftover is 55 − 54 = 1.
  5. The remainder 1 over the divisor 6 is 1/6. The answer is 1 529 1/6.

Answer: 1 529 1/6

See it three ways

With things

Share 23 stones between 3 people. Give each person one stone at a time. Stop when you cannot give one to everyone.

As a picture

Three groups of 7 dots, one for each person, and below them 2 dots with the label 2 left over.2 left over

In symbols

23 ÷ 3 = 7 remainder 2. If the 2 stones could be cut into thirds, each person would get 7 and 2/3.

Try it

Share 20 stones between 3 cups

You need: 20 small stones, beans or buttons, 3 cups or bowls.

  1. Put one stone in each cup. Keep going, one at a time.
  2. Stop when there are not enough stones to give one to every cup.
  3. Count the stones in each cup. Count the stones left in your hand.

What you will see: Each cup has 6 stones, and 2 stones are left over. So 20 ÷ 3 = 6 remainder 2.

If you cannot try it, this is what happens.
Three groups of 6 dots, one for each cup, and below them 2 dots with the label 2 left over.2 left over

A wrong idea many people have

Start dividing from the right, as you do when you add.

Why it feels right: Adding, taking away and multiplying all start with the ones place.

The test that breaks it: Try 72 ÷ 3 from the right. 2 ÷ 3 is 0, with 2 left over. There is no place to the right to carry the 2 to. From the left, 7 ÷ 3 = 2 remainder 1. Carry the 1 to make 12, and 12 ÷ 3 = 4. So 72 ÷ 3 = 24.

What is true: Start from the left. What is left over in one place is exchanged for ten of the next place. So you carry it to the right.

Key words

short division
dividing by a 1 digit number in the bus stop layout.
divisor
the number you divide by.
remainder
what is left over when a number does not divide exactly.

Back to your guess

The answer is C: Start from the left. Say how many 7s go into each number, and carry what is left over. The answer is 745 with 2 left over.

Division starts from the left, with the biggest place. A leftover is worth ten of the next place. So you carry it on. 5 217 ÷ 7 = 745 with 2 left over.

Check yourself

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

  1. Work out 3 648 ÷ 4.

    See the answer

    912

    4 goes into 3 no times. 4 goes into 36 nine times, with nothing left. 4 goes into 4 once. 4 goes into 8 twice. The answer is 912.

  2. Work out 4 523 ÷ 6. Give the answer and the remainder.

    See the answer

    753 remainder 5

    6 goes into 45 seven times, leaving 3. 6 goes into 32 five times, leaving 2. 6 goes into 23 three times, leaving 5. The answer is 753 remainder 5.

  3. What is 8 125 ÷ 5?

    Tap your answer.

    1. Not this one. You may have divided each digit on its own. The 3 left over from 8 must be carried to make 31, not dropped. Go to M3.2.4. Try another.

    2. Yes, that is right. 5 goes into 8 once, leaving 3. 5 goes into 31 six times, leaving 1. 5 goes into 12 twice, leaving 2. 5 goes into 25 five times. The answer is 1 625.

    3. Not this one. This is only 8 000 ÷ 5. The last 125 still needs dividing, and 125 ÷ 5 = 25. Go to M3.2.4. Try another.

    4. Not this one. You may have dropped the carried 1 after 31 ÷ 5 = 6. Then 2 ÷ 5 gives 0, not 2. Check by multiplying back: 5 × 1 605 = 8 025, not 8 125. Go to M3.2.4. Try another.

    See the answer

    1 625

    5 goes into 8 once, leaving 3. 5 goes into 31 six times, leaving 1. 5 goes into 12 twice, leaving 2. 5 goes into 25 five times. The answer is 1 625.

    • If you chose A: You may have divided each digit on its own. The 3 left over from 8 must be carried to make 31, not dropped. Go to M3.2.4.
    • If you chose C: This is only 8 000 ÷ 5. The last 125 still needs dividing, and 125 ÷ 5 = 25. Go to M3.2.4.
    • If you chose D: You may have dropped the carried 1 after 31 ÷ 5 = 6. Then 2 ÷ 5 gives 0, not 2. Check by multiplying back: 5 × 1 605 = 8 025, not 8 125. Go to M3.2.4.

  4. In 5 217 ÷ 7, the first digit is 5, which is smaller than 7. What do you do?

    The bus stop for 5 217 divided by 7 with nothing worked out yet. The divisor 7 is outside the bracket and 5 217 is inside it.75217
    Tap your answer.

    1. Not this one. The 5 is 5 thousands, so it is worth 5 000. It must be carried on, not thrown away. Go to M3.2.4. Try another.

    2. Not this one. 7 goes into 5 no times, so 5 is not an answer digit. The 5 is still worth dividing, so carry it. Go to M3.2.4. Try another.

    3. Not this one. The number inside the bracket is the one that is divided. The number outside is the divisor. Go to M3.2.4. Try another.

    4. Yes, that is right. 7 goes into 5 no times, so the 5 is carried to make 52. Then work out 52 ÷ 7.

    See the answer

    Carry it to the next digit. Then divide 52 by 7.

    7 goes into 5 no times, so the 5 is carried to make 52. Then work out 52 ÷ 7.

    • If you chose A: The 5 is 5 thousands, so it is worth 5 000. It must be carried on, not thrown away. Go to M3.2.4.
    • If you chose B: 7 goes into 5 no times, so 5 is not an answer digit. The 5 is still worth dividing, so carry it. Go to M3.2.4.
    • If you chose C: The number inside the bracket is the one that is divided. The number outside is the divisor. Go to M3.2.4.

  5. Read the bus stop for 3 458 ÷ 5. What is the remainder?

    The bus stop for 3 458 divided by 5. The answer written above the bracket is 691 r 3.5345836491r 3
    Tap your answer.

    1. Yes, that is right. 5 × 691 = 3 455, and 3 458 − 3 455 = 3. The remainder is 3.

    2. Not this one. The last digit of the number is 8, but that is not the remainder. The remainder is what is left after the last step. Go to M3.2.4. Try another.

    3. Not this one. 691 is the answer to the division, the number in each share. The remainder is the small number after the r. Go to M3.2.4. Try another.

    4. Not this one. There is a leftover. 5 × 691 = 3 455, and 3 458 is 3 more. Go to M3.2.4. Try another.

    See the answer

    3

    5 × 691 = 3 455, and 3 458 − 3 455 = 3. The remainder is 3.

    • If you chose B: The last digit of the number is 8, but that is not the remainder. The remainder is what is left after the last step. Go to M3.2.4.
    • If you chose C: 691 is the answer to the division, the number in each share. The remainder is the small number after the r. Go to M3.2.4.
    • If you chose D: There is a leftover. 5 × 691 = 3 455, and 3 458 is 3 more. Go to M3.2.4.

  6. What is 1 000 ÷ 7? Write the remainder over the divisor, as a fraction.

    Tap your answer.

    1. Not this one. The fraction needs the remainder on top, not 1. 7 × 142 = 994, and 1 000 − 994 = 6. Go to M3.2.4. Try another.

    2. Not this one. The fraction is upside down. The remainder goes on top, and the divisor goes on the bottom. Go to M3.2.4. Try another.

    3. Yes, that is right. 7 × 142 = 994, and 1 000 − 994 = 6. The remainder over the divisor is 6/7.

    4. Not this one. You may have read 142 r 6 as 142.6, which means 6 tenths. The remainder is 6 out of 7 parts, so the fraction is 6/7. Go to M3.2.4. Try another.

    See the answer

    142 6/7

    7 × 142 = 994, and 1 000 − 994 = 6. The remainder over the divisor is 6/7.

    • If you chose A: The fraction needs the remainder on top, not 1. 7 × 142 = 994, and 1 000 − 994 = 6. Go to M3.2.4.
    • If you chose B: The fraction is upside down. The remainder goes on top, and the divisor goes on the bottom. Go to M3.2.4.
    • If you chose D: You may have read 142 r 6 as 142.6, which means 6 tenths. The remainder is 6 out of 7 parts, so the fraction is 6/7. Go to M3.2.4.

  7. A rope is 1 500 cm long. Ravi cuts it into pieces of 7 cm. How many full pieces does he get?

    Tap your answer.

    1. Not this one. The last piece would need 7 cm, but only 2 cm are left. A short piece is not a full piece. Go to M3.2.4. Try another.

    2. Yes, that is right. 1 500 ÷ 7 = 214 remainder 2. So there are 214 full pieces, and 2 cm of rope is left over.

    3. Not this one. The remainder is how many centimetres are left over, not how many pieces are cut. Go to M3.2.4. Try another.

    4. Not this one. The question asks for full pieces only. The remainder is the leftover rope in centimetres. Go to M3.2.4. Try another.

    See the answer

    214

    1 500 ÷ 7 = 214 remainder 2. So there are 214 full pieces, and 2 cm of rope is left over.

    • If you chose A: The last piece would need 7 cm, but only 2 cm are left. A short piece is not a full piece. Go to M3.2.4.
    • If you chose C: The remainder is how many centimetres are left over, not how many pieces are cut. Go to M3.2.4.
    • If you chose D: The question asks for full pieces only. The remainder is the leftover rope in centimetres. Go to M3.2.4.

  8. Work out 2 000 ÷ 7. Write the remainder over the divisor, as a fraction. Do not simplify it.

    See the answer

    285 5/7

    7 × 285 = 1 995, and 2 000 − 1 995 = 5. The remainder 5 over the divisor 7 is 5/7.

  9. From an earlier chapter

    Work out 136 × 25.

    See the answer

    3 400

    Row 1: 136 × 5 = 680. Row 2: 136 × 20 = 2 720. Add the rows. 680 + 2 720 = 3 400.

Keep it

Copy this into your exercise book.

Divisor outside, number inside. Start at the left. See how many times the divisor goes into each number. Write it above, and carry what is left over. The last leftover is the remainder.

Draw: Draw the bus stop for 5 217 ÷ 7. Write 745 above it, the small carried digits, and r 2 at the end.

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

No internet tomorrow? Save the whole book as one file (792 KB). It opens on any phone.