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Primary Maths · Year 6Chapter 2 · The four operations and order of operations
Year 6 · Chapter 2

Year 6 Long Multiplication, Long Division and BODMAS: worked examples and questions with answers

By the end of this chapter you will be able to

  • Multiply a number of up to four digits by a two-digit number using long multiplication, and show the zero place holder
  • Divide a number of up to four digits by a two-digit number using long division, and write the remainder
  • Decide whether a remainder should be a whole number, a fraction or rounded, depending on the problem
  • Divide by a one-digit number using short division (the bus stop method), and use short division for some two-digit divisors
  • Use mental methods, such as partitioning, near round numbers, halving and doubling, and zeros, with large numbers and mixed operations
  • Use the order of operations, including brackets, to work out calculations
  • Estimate answers by rounding to one significant figure and check that an answer is sensible
  • Solve multi-step problems and give the answer to a suitable degree of accuracy

You should already know

  • Multiplying 2- and 3-digit numbers by a one-digit number, and by 10 and 100 (Year 5, Chapter 4)
  • Short division by a one-digit number and the idea of a remainder (Year 5, Chapter 5)
  • Place value of numbers up to 10,000,000 and rounding (Chapter 1)

Where this leads

  • Common factors and multiples, which use division and multiplication facts (Chapter 3)
  • Fractions and decimals, including dividing a whole number by a fraction (Chapters 4 and 5)
  • Algebra and the order of operations in formulae (Chapter 8)
  • Estimating and checking calculations in measures and area (Chapter 9)

National Curriculum for England, Key Stage 2 mathematics, Year 6. Content: Multiply multi-digit numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication, Divide numbers up to 4 digits by a two-digit whole number using long division, and interpret remainders as whole numbers, fractions, or by rounding, Divide numbers up to 4 digits by a one-digit number using short division, and by a two-digit number where appropriate, Perform mental calculations, including with mixed operations and large numbers, Use the order of operations to carry out calculations involving the four operations, Solve addition, subtraction, multiplication and division multi-step problems, Use estimation to check answers and determine an appropriate degree of accuracy.

Exercise P  Do you remember?

Non-calculator
1
FWork out 46 × 7.
2
FWork out 3,240 ÷ 9.
3
FRound 6,482 to the nearest hundred.
4
FWork out 430 × 10.
5
FWhat is the value of the 7 in 4,732?
2.1

Long multiplication

Year 6
Multiply numbers up to 4 digits by a two-digit whole number using the formal written method of long multiplication, Use estimation to check answers
Video lesson: Long multiplicationComing soon

A two-digit number is multiplied in two rows. First multiply the top number by the ones digit. Then multiply by the tens digit, but write a 0 place holder first. This is because the tens digit is worth ten times more. Finally, add the two rows together. This is the long multiplication method.

An area model shows why the method works. The number 23 is split into 20 and 3, and 16 is split into 10 and 6. Each box is a partial product. The four partial products add up to the whole answer.

2031020030612018
23 × 16 split into parts: 200 + 30 + 120 + 18 = 368. The long method adds the same parts in two rows.

Estimate before you calculate. Round each number to one significant figure and multiply. For 2,345 × 16, the estimate is 2,000 × 20 = 40,000. The exact answer should be close to the estimate. If your answer is far away, check your working.

Key ideas

  • Multiply by the ones digit first, then by the tens digit. Each row is a partial product.
  • Write a 0 place holder at the start of the tens row: 2,345 × 10 = 23,450, not 2,345.
  • Carry tens into the next column in each row, and add them after you multiply the column.
  • Add the two rows at the end. Estimate first, so you can see if the answer is sensible.
Worked example 2.1.1
Work out 2,345 × 16.

Key words. Long multiplication is a written method for multiplying by a number with two digits. We multiply by the ones digit on one row, then by the tens digit on the next row. The zero place holder keeps the tens digit in the tens column.

Method. Use long multiplication. Multiply by the ones digit, then by the tens digit with a 0 place holder, then add the two rows.

  1. Split the second number into tens and ones. 16 is 1 ten and 6 ones. We multiply by the ones first, then by the tens.
    16 = 10 + 6
  2. Multiply by the ones, which is 6. Start on the right and multiply each digit. Carry any tens to the next column, as in short multiplication.
    Ones: 5 × 6 = 30. Write 0 and carry 3.
    Tens: 4 × 6 = 24, and 24 + 3 = 27. Write 7 and carry 2.
    Hundreds: 3 × 6 = 18, and 18 + 2 = 20. Write 0 and carry 2.
    Thousands: 2 × 6 = 12, and 12 + 2 = 14. Write 4 and carry 1.
    Ten thousands: nothing left to multiply. Write the carried 1.
    2,345 × 6 = 14,070
  3. Write a 0 place holder, then multiply by the tens, which is 1. The 1 stands for 10. The 0 in the ones column holds the place, so the answer is ten times bigger.
    Ones: 5 × 1 = 5. Write 5.
    Tens: 4 × 1 = 4. Write 4.
    Hundreds: 3 × 1 = 3. Write 3.
    Thousands: 2 × 1 = 2. Write 2.
    2,345 × 1 = 2,345. With the 0 place holder this row is 23,450
  4. Add the two rows. Add the two rows, one column at a time, and carry where you need to.
       2345
   × 16
-------
  14070
+ 23450
-------
  37520

    14,070 + 23,450 = 37,520

Check. Estimate: 2,000 × 20 = 40,000, and 37,520 is close to it. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it, the answer is ten times too small.

Worked example 2.1.2
Work out 618 × 27.

Key words. Long multiplication is a written method for multiplying by a number with two digits. We multiply by the ones digit on one row, then by the tens digit on the next row. The zero place holder keeps the tens digit in the tens column.

Method. Use long multiplication. Multiply by the ones digit, then by the tens digit with a 0 place holder, then add the two rows.

  1. Split the second number into tens and ones. 27 is 2 tens and 7 ones. We multiply by the ones first, then by the tens.
    27 = 20 + 7
  2. Multiply by the ones, which is 7. Start on the right and multiply each digit. Carry any tens to the next column, as in short multiplication.
    Ones: 8 × 7 = 56. Write 6 and carry 5.
    Tens: 1 × 7 = 7, and 7 + 5 = 12. Write 2 and carry 1.
    Hundreds: 6 × 7 = 42, and 42 + 1 = 43. Write 3 and carry 4.
    Thousands: nothing left to multiply. Write the carried 4.
    618 × 7 = 4,326
  3. Write a 0 place holder, then multiply by the tens, which is 2. The 2 stands for 20. The 0 in the ones column holds the place, so the answer is ten times bigger.
    Ones: 8 × 2 = 16. Write 6 and carry 1.
    Tens: 1 × 2 = 2, and 2 + 1 = 3. Write 3.
    Hundreds: 6 × 2 = 12. Write 2 and carry 1.
    Thousands: nothing left to multiply. Write the carried 1.
    618 × 2 = 1,236. With the 0 place holder this row is 12,360
  4. Add the two rows. Add the two rows, one column at a time, and carry where you need to.
        618
   × 27
-------
   4326
+ 12360
-------
  16686

    4,326 + 12,360 = 16,686

Check. Estimate: 600 × 30 = 18,000, and 16,686 is close to it. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it, the answer is ten times too small.

Worked example 2.1.3
A stadium has 4,072 seats in each block. There are 34 blocks. How many seats are there altogether?

Key words. Long multiplication is a written method for multiplying by a number with two digits. We multiply by the ones digit on one row, then by the tens digit on the next row. The zero place holder keeps the tens digit in the tens column.

Method. Use long multiplication. Multiply by the ones digit, then by the tens digit with a 0 place holder, then add the two rows.

  1. Split the second number into tens and ones. 34 is 3 tens and 4 ones. We multiply by the ones first, then by the tens.
    34 = 30 + 4
  2. Multiply by the ones, which is 4. Start on the right and multiply each digit. Carry any tens to the next column, as in short multiplication.
    Ones: 2 × 4 = 8. Write 8.
    Tens: 7 × 4 = 28. Write 8 and carry 2.
    Hundreds: 0 × 4 = 0, and 0 + 2 = 2. Write 2.
    Thousands: 4 × 4 = 16. Write 6 and carry 1.
    Ten thousands: nothing left to multiply. Write the carried 1.
    4,072 × 4 = 16,288
  3. Write a 0 place holder, then multiply by the tens, which is 3. The 3 stands for 30. The 0 in the ones column holds the place, so the answer is ten times bigger.
    Ones: 2 × 3 = 6. Write 6.
    Tens: 7 × 3 = 21. Write 1 and carry 2.
    Hundreds: 0 × 3 = 0, and 0 + 2 = 2. Write 2.
    Thousands: 4 × 3 = 12. Write 2 and carry 1.
    Ten thousands: nothing left to multiply. Write the carried 1.
    4,072 × 3 = 12,216. With the 0 place holder this row is 122,160
  4. Add the two rows. Add the two rows, one column at a time, and carry where you need to.
        4072
    × 34
--------
   16288
+ 122160
--------
  138448

    16,288 + 122,160 = 138,448

Check. Estimate: 4,000 × 30 = 120,000, and 138,448 is close to it. ✓

Watch out. Do not forget the 0 place holder on the second row. Without it, the answer is ten times too small.

Watch out

Do not forget the 0 place holder in the tens row. Without it, the answer is ten times too small. Also, do not add the rows before you have multiplied every column.

Exercise 2A  Long multiplication with two-digit numbers

Non-calculator
1
FWork out 127 × 23.
2
FWork out each multiplication.
(a)1,046 × 12
(b)2,305 × 14
(c)3,098 × 21
(d)408 × 36
3
FWork out 2,675 × 47.
4
FWork out each multiplication. Estimate each answer first.
(a)56 × 38
(b)904 × 65
5
PSA cinema has 32 rows with 26 seats in each row. How many seats are there altogether?
6
PSA box holds 36 eggs. A café orders 125 boxes. How many eggs is that?
7
PSEach packet has 18 stickers. Ella buys 46 packets. How many stickers does she have?
8
PSA school trip costs £13 for each of the 214 pupils. How much money is paid altogether? Give your answer in pounds.
9
RSam works out 35 × 12. He writes 35 for the tens row, adds the rows and gets 105. Explain Sam's mistake and find the correct answer.
10
RMia says that 1,234 × 20 = 24,680, because she multiplies 1,234 by 2 and then adds a zero. Is she right? Explain your answer.
11
FWork out 809 × 52.
12
RThe answer to a 2-digit number multiplied by 37 is 1,406. Find the 2-digit number.
2.2

Long division with two-digit divisors

Year 6
Divide numbers up to 4 digits by a two-digit whole number using the formal written method of long division, Interpret remainders as whole numbers, fractions or by rounding, as appropriate for the context
Video lesson: Long division with two-digit divisorsComing soon

In long division we share a number into equal groups, one digit of the answer at a time. First find how many times the divisor fits into the first digits. Write that digit on top. Multiply it by the divisor and subtract. Then bring down the next digit and repeat.

Sometimes a number does not share out exactly. The part left over is the remainder. A remainder must always be smaller than the divisor. What you do with the remainder depends on the problem: you might write it as a whole number, as a fraction, or round the answer up or down.

4 full groups of 12 and 2 left over
50 counters in groups of 12: 4 full groups and 2 left over. So 50 ÷ 12 = 4 remainder 2. The orange counters are the remainder.

Key ideas

  • Long division is repeated subtraction, written neatly. Each digit of the answer is worked out with multiplication.
  • Estimate the first digit by thinking about the divisor. Round it to a friendly number, such as 23 is about 20.
  • The remainder is always smaller than the divisor. If it is not, the answer is one more group.
  • Check with multiplication: divisor × quotient + remainder = the number you started with.
  • Decide what the remainder means: round up (for example, buses needed), round down (full boxes), or write it as a fraction (sharing).
Worked example 2.2.1
Work out 2,048 ÷ 23. Write your answer with the remainder.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, the quotient is the answer and the remainder is what is left over.

Method. Set out the division. Find how many times the divisor fits into the first part of the number, write that digit on top, subtract, and bring down the next digit.

  1. Find quotient digit 1. Start with the first digits that are big enough to hold the divisor.
    Use 204. How many 23s fit into 204? 8, because 23 × 8 = 184. Subtract: 204 − 184 = 20.
  2. Find quotient digit 2. Bring down the next digit and see how many times the divisor fits.
    Bring down the 8 to make 208. How many 23s fit into 208? 9, because 23 × 9 = 207. Subtract: 208 − 207 = 1.
  3. Write the answer. The quotient goes on top. The number left at the end is the remainder.
           89
23 ) 2048
     184
     ---
      20
      207
      ---
        1

    Answer 89 r 1

Check. 23 × 89 = 2,047, and 2,047 + 1 = 2,048. ✓

Watch out. Keep going until the number left is smaller than the divisor. A remainder must always be smaller than the divisor.

Worked example 2.2.2
Work out 1,597 ÷ 14.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, the quotient is the answer and the remainder is what is left over.

Method. Set out the division. Find how many times the divisor fits into the first part of the number, write that digit on top, subtract, and bring down the next digit.

  1. Find quotient digit 1. Start with the first digits that are big enough to hold the divisor.
    Use 15. How many 14s fit into 15? 1, because 14 × 1 = 14. Subtract: 15 − 14 = 1.
  2. Find quotient digit 2. Bring down the next digit and see how many times the divisor fits.
    Bring down the 9 to make 19. How many 14s fit into 19? 1, because 14 × 1 = 14. Subtract: 19 − 14 = 5.
  3. Find quotient digit 3. Bring down the next digit and see how many times the divisor fits.
    Bring down the 7 to make 57. How many 14s fit into 57? 4, because 14 × 4 = 56. Subtract: 57 − 56 = 1.
  4. Write the answer. The quotient goes on top. The number left at the end is the remainder.
          114
14 ) 1597
     14
     --
      1
      14
      --
       5
       56
       --
        1

    Answer 114 r 1

Check. 14 × 114 = 1,596, and 1,596 + 1 = 1,597. ✓

Watch out. Keep going until the number left is smaller than the divisor. A remainder must always be smaller than the divisor.

Worked example 2.2.3
Work out 5,000 ÷ 36. Write your answer as a quotient and a remainder.

Key words. In long division, the dividend is the number being divided, the divisor is the number we divide by, the quotient is the answer and the remainder is what is left over.

Method. Set out the division. Find how many times the divisor fits into the first part of the number, write that digit on top, subtract, and bring down the next digit.

  1. Find quotient digit 1. Start with the first digits that are big enough to hold the divisor.
    Use 50. How many 36s fit into 50? 1, because 36 × 1 = 36. Subtract: 50 − 36 = 14.
  2. Find quotient digit 2. Bring down the next digit and see how many times the divisor fits.
    Bring down the 0 to make 140. How many 36s fit into 140? 3, because 36 × 3 = 108. Subtract: 140 − 108 = 32.
  3. Find quotient digit 3. Bring down the next digit and see how many times the divisor fits.
    Bring down the 0 to make 320. How many 36s fit into 320? 8, because 36 × 8 = 288. Subtract: 320 − 288 = 32.
  4. Write the answer. The quotient goes on top. The number left at the end is the remainder.
          138
36 ) 5000
     36
     --
     14
     108
     ---
      32
      288
      ---
       32

    Answer 138 r 32

Check. 36 × 138 = 4,968, and 4,968 + 32 = 5,000. ✓

Watch out. Keep going until the number left is smaller than the divisor. A remainder must always be smaller than the divisor.

Watch out

Do not stop when you run out of digits. Keep going until the number left is smaller than the divisor. A remainder bigger than the divisor is always a mistake. Also, do not give the remainder before you have checked the answer with multiplication.

Exercise 2B  Long division and remainders

Non-calculator
1
FWork out 1,365 ÷ 15.
2
FWork out each division.
(a)1,728 ÷ 24
(b)2,640 ÷ 32
(c)3,412 ÷ 17
(d)4,500 ÷ 45
3
FWork out 2,097 ÷ 26.
4
FWork out 6,839 ÷ 43.
5
PSA minibus can carry 14 people. 100 pupils need to travel to a museum. How many minibuses are needed?
6
PSA bakery packs 500 biscuits in boxes of 12. How many full boxes can it fill?
7
PSShare 500 stickers equally between 12 children. How many stickers does each child get? Write your answer as a mixed number.
8
PS£1,000 is shared equally between 36 people. How much does each person get? Give your answer to the nearest penny.
9
RRuby says, “500 ÷ 12 = 40 remainder 20.” Explain why Ruby is wrong, then find the correct answer.
10
RSam says, “500 ÷ 12 = 41 remainder 8, because 12 × 41 + 8 = 500.” Is Sam right? Explain how you know.
11
FWork out 9,876 ÷ 12.
12
RWhen a number is divided by 25, the quotient is 38 and the remainder is 17. What is the number?
2.3

Short division (the bus stop method)

Year 6
Divide numbers up to 4 digits by a one-digit number using the formal written method of short division, and interpret remainders, Divide by a two-digit number using short division where appropriate
Video lesson: Short division (the bus stop method)Coming soon

Short division is the quickest written method for dividing by a one-digit number. We work from left to right. Each digit is divided by the divisor. If there is a remainder, it is carried to the next digit on the right, where it is worth ten times more. The answer goes on top, one digit above each digit of the number.

For a two-digit divisor, short division is useful when the divisor is a product of two one-digit numbers. For example, 12 is 3 × 4. Divide by 3, then divide the answer by 4. The remainders must then be combined so that the final remainder is correct.

Key ideas

  • Work from left to right. Divide each digit, and carry any remainder to the next digit.
  • A carried remainder is always smaller than the divisor.
  • The answer is written on top. The last remainder is written after the answer as r.
  • For a two-digit divisor such as 12, divide by 3 and then by 4. Combine the remainders with a × r2 + r1.
  • Check with multiplication, as in long division.
Worked example 2.3.1
Work out 4,625 ÷ 5.

Key words. Short division (the bus stop method) divides by a one-digit number. A small carried remainder is written above the next digit. It is the number left over, and it is worth ten times more in the next column.

Method. Use short division. Work from left to right, one digit at a time. Carry any remainder into the next digit.

  1. Set out the bus stop layout. The number goes inside the bus stop and the divisor, 5, goes outside it.
    5 ) 4,625
  2. Divide the digit 4 (digit 1 of 4). Start with the first digit on the left.
    4 ÷ 5 = 0 remainder 4. Write 0 on top and carry the remainder 4.
  3. Divide the digit 6 (digit 2 of 4). Each remainder is smaller than the divisor. Carry it to the next digit to the right.
    Carry the remainder 4 to the next digit to make 46 = 46. 46 ÷ 5 = 9 remainder 1. Write 9 on top and carry 1.
  4. Divide the digit 2 (digit 3 of 4). Each remainder is smaller than the divisor. Carry it to the next digit to the right.
    Carry the remainder 1 to the next digit to make 12 = 12. 12 ÷ 5 = 2 remainder 2. Write 2 on top and carry 2.
  5. Divide the digit 5 (digit 4 of 4). Each remainder is smaller than the divisor. Carry it to the next digit to the right.
    Carry the remainder 2 to the next digit to make 25 = 25. 25 ÷ 5 = 5 remainder 0. Write 5 on top and carry 0.
  6. Read the answer from the bus stop layout. The quotient is the top row of numbers. The last remainder is left over.
    412
    54625
    0925

    Answer 925

Check. 5 × 925 = 4,625. ✓

Watch out. A carried remainder must be smaller than the divisor. If you get a remainder that is too big, check your times table.

Worked example 2.3.2
Work out 3,814 ÷ 6. Write the remainder.

Key words. Short division (the bus stop method) divides by a one-digit number. A small carried remainder is written above the next digit. It is the number left over, and it is worth ten times more in the next column.

Method. Use short division. Work from left to right, one digit at a time. Carry any remainder into the next digit.

  1. Set out the bus stop layout. The number goes inside the bus stop and the divisor, 6, goes outside it.
    6 ) 3,814
  2. Divide the digit 3 (digit 1 of 4). Start with the first digit on the left.
    3 ÷ 6 = 0 remainder 3. Write 0 on top and carry the remainder 3.
  3. Divide the digit 8 (digit 2 of 4). Each remainder is smaller than the divisor. Carry it to the next digit to the right.
    Carry the remainder 3 to the next digit to make 38 = 38. 38 ÷ 6 = 6 remainder 2. Write 6 on top and carry 2.
  4. Divide the digit 1 (digit 3 of 4). Each remainder is smaller than the divisor. Carry it to the next digit to the right.
    Carry the remainder 2 to the next digit to make 21 = 21. 21 ÷ 6 = 3 remainder 3. Write 3 on top and carry 3.
  5. Divide the digit 4 (digit 4 of 4). Each remainder is smaller than the divisor. Carry it to the next digit to the right.
    Carry the remainder 3 to the next digit to make 34 = 34. 34 ÷ 6 = 5 remainder 4. Write 5 on top and carry 4.
  6. Read the answer from the bus stop layout. The quotient is the top row of numbers. The last remainder is left over.
    323
    63814
    0635

    Answer 635 r 4

Check. 6 × 635 = 3,810, and 3,810 + 4 = 3,814. ✓

Watch out. A carried remainder must be smaller than the divisor. If you get a remainder that is too big, check your times table.

Worked example 2.3.3
Work out 3,596 ÷ 12 by dividing by 3 and then by 4. Find the remainder.

Method. Divide by 3, then divide the answer by 4. Combine the two remainders to find the true remainder.

  1. Write 12 as two numbers to multiply. Dividing by 12 is the same as dividing by 3 and then by 4.
    12 = 3 × 4
  2. Divide by 3 first. Use short division and keep the remainder.
    022
    33596
    1198

    3,596 ÷ 3 = 1198 remainder 2
  3. Divide the answer, 1198, by 4. Use short division again.
    133
    41198
    0299

    1,198 ÷ 4 = 299 remainder 2
  4. Work out the total remainder. The 2 left over from the second division is worth 3 each in the first division. So multiply 2 by 3, then add the first remainder 2.
    3 × 2 + 2 = 6 + 2 = 8
  5. Write the answer.
    299 r 8

Check. 12 × 299 = 3,588, and 3,588 + 8 = 3,596. ✓

Watch out. The second remainder is worth 3 in the first division, so you must multiply it by 3. Do not just add the two remainders.

Watch out

A common mistake is to add the two remainders together. In the factor method, the second remainder is worth 3 in the first division, so it must be multiplied first. Also, do not carry a remainder that is bigger than the divisor.

Exercise 2C  Short division and remainders

Non-calculator
1
FWork out 4,839 ÷ 7.
2
FWork out each division using short division.
(a)1,248 ÷ 4
(b)5,670 ÷ 9
(c)3,568 ÷ 8
(d)2,493 ÷ 3
3
FWork out 6,215 ÷ 6.
4
FWork out 8,004 ÷ 4.
5
FWork out 3,597 ÷ 7.
6
FWork out 3,360 ÷ 12 by dividing by 3 and then by 4.
7
FWork out 4,700 ÷ 15 by dividing by 3 and then by 5. Give the remainder.
8
PSA baker shares 2,850 cupcakes equally into 6 boxes. How many cupcakes go in each box?
9
PSMia has £3,826 to share equally between 7 friends. How much does each friend get, and how much is left over?
10
PSRavi has 1,035 marbles. He puts them into bags of 8. How many full bags can he make, and how many marbles are left?
11
RRuby works out 5,372 ÷ 6 and writes 895 remainder 2. Is she right? Explain how you know.
12
RWhen a four-digit number is divided by 8, the quotient is 437 and the remainder is 5. What is the number?
2.4

Mental calculation with mixed operations and large numbers

Year 6
Perform mental calculations, including with mixed operations and large numbers, Use known facts, partitioning and place value to work out larger calculations
Video lesson: Mental calculation with mixed operations and large numbersComing soon

You do not always need a written method. Good mental methods use facts you already know. Here are four strategies:

In a calculation with mixed operations, work out each part first, then add and subtract. Remember the order of operations (section 2.5).

10025125
125 split into 100 and 25. Each part is multiplied by 8, then the answers are added.

Key ideas

  • Partition one number into easy parts, then multiply each part.
  • Use a nearby round number, then correct for the small difference.
  • Halving one number and doubling the other does not change the product.
  • Take zeros off, multiply, then put them back.
  • In mixed calculations, multiply and divide before you add and subtract.
Worked example 2.4.1
Work out 8 × 125 in your head. Show how you split the numbers.

Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.

Method. Split one number into parts, multiply each part, then add the answers.

  1. Split 125 into parts. Parts that are easy to multiply by 8 make the working simpler.
    125 = 100 + 25
  2. Multiply each part by 8. Multiply each part on its own.
    8 × 100 = 800
    8 × 25 = 200
  3. Add the answers. Add the partial answers together.
    800 + 200 = 1,000

Check. Estimate: 8 × 100 = 800, which is close to 1,000. ✓

Watch out. Make sure the parts add up to the whole number. Check that you have not missed a part.

Worked example 2.4.2
Work out 18 × 49 using 50.

Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.

Method. Multiply by a round number that is close, then correct the answer for the small difference.

  1. Use the nearby round number, 50. 49 is 1 less than 50, and 50 is easy to multiply by.
    49 = 50 − 1
  2. Multiply 18 by 50. Multiply by the round number first.
    18 × 50 = 900
  3. Correct for the difference. 49 is 1 less than 50, so we take away 1 lots of 18.
    18 × 1 = 18, so 900 − 18 = 882

Check. The estimate 20 × 50 = 1,000 is close to 882. ✓

Watch out. Correct in the right direction. If the round number was too big, take the difference away.

Worked example 2.4.3
Work out 1,440 ÷ 24 by splitting 1,440 into two easy parts.

Method. Split the number into easy parts, divide each part, then add the answers.

  1. Split the number into parts. Choose parts of 1,440 that 24 divides into exactly.
    1,440 = 1,200 + 240
  2. Divide each part by 24. Each answer is a whole number, so the parts are easy to work out.
    1,200 ÷ 24 = 50
    240 ÷ 24 = 10
  3. Add the answers. Add the answers from each part.
    50 + 10 = 60

Check. 24 × 60 = 1,440. ✓

Watch out. Pick parts that the divisor fits into exactly. Then each part is a simple division.

Worked example 2.4.4
Work out 16 × 35 by halving and doubling.

Key words. A mental method works out an answer in your head. A partition splits a number into parts, such as 36 = 30 + 6. Compensate means adjust an answer because one number was rounded.

Method. Halve one number and double the other. The product stays the same.

  1. Halve one number and double the other. Halving 16 and doubling 35 does not change the answer.
    16 × 35 = 8 × 70
  2. Work out 8 × 70. If 70 is hard to multiply, split it into easy parts and add the answers.
    8 × 70 = 560
  3. Check the answer. Check by halving and doubling in the other way, or by estimating.
    Estimate: 20 × 40 = 800, which is close to 560

Check. Check with the other order: 35 × 16 = 560. ✓

Watch out. Only halve and double in the same way. If you halve 16 you must double 35.

Watch out

When you round a number, remember to correct the answer. If you used 50 instead of 49, you used one lot too many, so take one lot away. Also, do not put the zeros in the wrong place: 4,000 × 60 is 240,000, not 24,000.

Exercise 2D  Mental calculation and mixed operations

Non-calculator
1
FWork out each multiplication. Use a method that splits one number into parts.
(a)36 × 5
(b)48 × 4
(c)73 × 6
(d)7 × 99
2
FWork out each multiplication. Use a method of your choice.
(a)9 × 46
(b)14 × 50
(c)6 × 98
3
FWork out 24 × 45 by halving and doubling.
4
FWork out 3,240 ÷ 18 by splitting 3,240 into two parts that are easy to divide.
5
FWork out each multiplication. Take off the zeros first.
(a)4,000 × 60
(b)300 × 700
(c)90 × 2,000
6
FWork out 36 × 5 + 4 × 25.
7
FWork out 60 ÷ 4 + 12 × 3.
8
PSWork out 3 × 40 + 5 × 8 − 12 ÷ 4.
9
PSWork out 25 × 36 in your head. Show how you split the numbers.
10
RPriya says 99 × 6 = 600, because 100 × 6 = 600. Is she right? Explain your answer.
11
RWork out 1,250 × 8. Explain the method you chose.
2.5

The order of operations

Year 6
Use the order of operations to carry out calculations involving the four operations, Understand that brackets change the order of a calculation
Video lesson: The order of operationsComing soon

When a calculation has more than one operation, we must follow the order of operations. Otherwise two people could work out the same calculation and get different answers.

  1. Brackets first.
  2. Multiplication and division, from left to right.
  3. Addition and subtraction, from left to right.

A way to remember the order is BIDMAS: Brackets, Indices (powers), Division and Multiplication, Addition and Subtraction. This chapter uses brackets, multiplication and division, and addition and subtraction.

Key ideas

  • Brackets come first. Work out the sum inside them.
  • Multiplication and division come before addition and subtraction, even if the addition is written first.
  • When two operations have the same rank, work from the left: 12 ÷ 3 × 2 = 8, not 2.
  • Brackets can change the answer. (2 + 3) × 4 = 20, but 2 + 3 × 4 = 14.
Worked example 2.5.1
Work out 3 + 4 × 5.

Key words. The order of operations is the order in which we work out a calculation. Brackets come first, then multiplication and division (from left to right), then addition and subtraction (from left to right).

Method. Brackets first, then multiplication and division from left to right, then addition and subtraction from left to right.

  1. Multiply and divide, from left to right. Multiplication and division come before addition and subtraction.
    3 + 4 × 5 = 3 + 20
  2. Add and subtract, from left to right. Work from the left. Addition and subtraction are done last.
    3 + 20 = 23
  3. Write the answer.
    23

Check. Read the calculation again and follow the same order. ✓

Watch out. Multiplication and division come before addition and subtraction, even when the addition is written first.

Worked example 2.5.2
Work out (3 + 4) × 5.

Key words. The order of operations is the order in which we work out a calculation. Brackets come first, then multiplication and division (from left to right), then addition and subtraction (from left to right).

Method. Brackets first, then multiplication and division from left to right, then addition and subtraction from left to right.

  1. Work out inside the brackets. Brackets come first. Inside them, multiply and divide before adding and subtracting.
    3 + 4 = 7
  2. Replace the brackets with their answer. The brackets are done, so the calculation is simpler now.
    ( 3 + 4 ) × 5 = 7 × 5
  3. Multiply and divide, from left to right. Multiplication and division come before addition and subtraction.
    7 × 5 = 35
  4. Write the answer.
    35

Check. Read the calculation again and follow the same order. ✓

Watch out. Multiplication and division come before addition and subtraction, even when the addition is written first.

Worked example 2.5.3
Work out 20 − 12 ÷ 3 + 2 × 5.

Key words. The order of operations is the order in which we work out a calculation. Brackets come first, then multiplication and division (from left to right), then addition and subtraction (from left to right).

Method. Brackets first, then multiplication and division from left to right, then addition and subtraction from left to right.

  1. Multiply and divide, from left to right. Multiplication and division come before addition and subtraction.
    20 − 12 ÷ 3 + 2 × 5 = 20 − 4 + 2 × 5
  2. Multiply and divide, from left to right. Multiplication and division come before addition and subtraction.
    20 − 4 + 2 × 5 = 20 − 4 + 10
  3. Add and subtract, from left to right. Work from the left. Addition and subtraction are done last.
    20 − 4 + 10 = 16 + 10
  4. Add and subtract, from left to right. Work from the left. Addition and subtraction are done last.
    16 + 10 = 26
  5. Write the answer.
    26

Check. Read the calculation again and follow the same order. ✓

Watch out. Multiplication and division come before addition and subtraction, even when the addition is written first.

Worked example 2.5.4
Work out 24 ÷ (2 + 6) × 3. Take care: the division and the multiplication are done from left to right.

Key words. The order of operations is the order in which we work out a calculation. Brackets come first, then multiplication and division (from left to right), then addition and subtraction (from left to right).

Method. Brackets first, then multiplication and division from left to right, then addition and subtraction from left to right.

  1. Work out inside the brackets. Brackets come first. Inside them, multiply and divide before adding and subtracting.
    2 + 6 = 8
  2. Replace the brackets with their answer. The brackets are done, so the calculation is simpler now.
    24 ÷ ( 2 + 6 ) × 3 = 24 ÷ 8 × 3
  3. Multiply and divide, from left to right. Multiplication and division come before addition and subtraction.
    24 ÷ 8 × 3 = 3 × 3
  4. Multiply and divide, from left to right. Multiplication and division come before addition and subtraction.
    3 × 3 = 9
  5. Write the answer.
    9

Check. Read the calculation again and follow the same order. ✓

Watch out. Multiplication and division come before addition and subtraction, even when the addition is written first.

Watch out

The most common mistake is to work from left to right and ignore the order. In 3 + 4 × 5, the answer is 23, not 35. Another mistake is to forget that division and multiplication are done from left to right: 24 ÷ 8 × 3 is 3 × 3 = 9, not 24 ÷ 24 = 1.

Exercise 2E  Order of operations

Non-calculator
1
FWork out each calculation. Follow the order of operations.
(a)2 + 3 × 4
(b)10 − 6 ÷ 2
(c)5 × 4 + 6
(d)18 ÷ 3 × 2
2
FWork out each calculation. Work out the brackets first.
(a)(2 + 3) × 4
(b)10 − (6 ÷ 2)
(c)3 × (4 + 5)
(d)(12 − 4) ÷ 2
3
FWork out 40 − 6 × 5 + 12 ÷ 4.
4
FWork out (20 + 4) ÷ 6 × 5.
5
FWork out 7 × 8 − 4 × 6 + 2.
6
PSTickets cost £5 each for 3 adults, and £2 each for 4 children. Write one calculation and work out the total cost.
7
PSAli pays with a £5 note. He buys 2 cakes at 80p each and a drink for 95p. Write one calculation to find his change, and work it out.
8
PSA hall has 6 rows of 12 chairs and 4 rows of 15 chairs. How many chairs are there altogether? Write one calculation.
9
PSPut one pair of brackets in this calculation to make it correct:   2 + 3 × 4 − 1 = 19.
10
PSPut one pair of brackets in this calculation to make it correct:   10 − 4 ÷ 2 + 3 = 6.
11
RSam says 5 + 3 × 2 = 16, because 5 + 3 = 8 and 8 × 2 = 16. Explain his mistake, then work out the correct answer.
12
RWork out 2 + 3 × 6, and (2 + 3) × 6. Explain why the answers are different.
2.6

Multi-step problems, estimation and degrees of accuracy

Year 6
Solve addition, subtraction, multiplication and division multi-step problems, Use estimation to check answers and determine an appropriate degree of accuracy
Video lesson: Multi-step problems, estimation and degrees of accuracyComing soon

A multi-step problem needs more than one calculation. Start by reading the question carefully and finding the numbers you need. Draw a bar model if it helps. Then decide the order of your steps, and work them out one at a time.

Always estimate first. An estimate shows you roughly what size the answer should be. Then decide how accurate the answer must be. Some answers must be whole numbers, such as the number of buses needed. Money is usually given to the nearest penny.

Key ideas

  • Read the question twice. Note the numbers and the words that tell you the operation.
  • Estimate first. Round each number to one significant figure and work out the rough answer.
  • Choose the degree of accuracy the question asks for, such as the nearest 1,000 or the nearest penny.
  • For a remainder in a problem, decide whether to round up, round down or use a fraction.
  • Check the answer is sensible, and write the unit.
Worked example 2.6.1
Estimate the answer to 4,873 × 29.

Key words. To estimate is to find an answer that is close but not exact. To estimate, round each number to one significant figure: keep the first digit and make the other digits zero. The first digit is the most important one.

Method. Round each number to one significant figure, then multiply the rounded numbers.

  1. Round each number to one significant figure. Keep the first digit and make the other digits zero. Round up if the next digit is 5 or more.
    4,873 is about 5,000. 29 is about 30.
  2. Multiply the rounded numbers. This gives an estimate. It is close to the exact answer, but not the same.
    5,000 × 30 = 150,000
  3. Decide whether the estimate is sensible. An estimate that is close to the exact answer is a good check.
    Exact answer 141,317, and 150,000 is close to it

Check. The exact answer is 141,317, which is close to 150,000. ✓

Watch out. An estimate tells you whether an exact answer is sensible. It does not replace the exact answer when a question asks for it.

£42£32Total £74
The total is made of two parts: the adults and the children.
Worked example 2.6.2
Tickets cost £14 for an adult and £8 for a child. A family has 3 adults and 4 children. What is the total cost?

Key words. To estimate is to find an answer that is close but not exact. To estimate, round each number to one significant figure: keep the first digit and make the other digits zero. The first digit is the most important one.

Method. Work out the cost of each group, then add the two costs.

  1. Work out the cost of the adults. Multiply the number of adults by the price for one adult.
    3 × £14 = £42
  2. Work out the cost of the children. Multiply the number of children by the price for one child.
    4 × £8 = £32
  3. Add the two costs. The total cost is the adults plus the children.
    £42 + £32 = £74

Check. Estimate: 3 × 14 is about 40, and 4 × 8 is about 30, so about 70. £74 is close. ✓

Watch out. Read carefully: the question asks for the total cost of the family, not the cost of one ticket.

Worked example 2.6.3
A town has 2,376,540 people. Round this to the nearest 100,000 to give an estimate of the population.

Key words. The degree of accuracy is how closely we round an answer, such as to the nearest 10, the nearest whole number or the nearest penny. Rounding a money answer to the nearest penny means two decimal places, but money is usually written in pounds and pence.

Method. Look at the digit after the place you are rounding to. Change the digits after that place to zeros.

  1. Look at the digit in the column just below hundred thousands. The rounding digit is the one to the right of the hundred thousands column.
    The digit is 7
  2. Decide whether to round up or down. 5 or more rounds up. Less than 5 rounds down.
    7 is 5 or more, so round up
  3. Write the rounded number. Replace the digits after the hundred thousands column with zeros.
    2,376,540 to the nearest 100,000 = 2,400,000

Check. 2,400,000 is the closest number to 2,376,540 that ends in 5 zeros. ✓

Watch out. Round the original number, not a number you have already rounded.

Worked example 2.6.4
£1,000 is shared equally between 3 charities. Give each share to the nearest penny.

Key words. The degree of accuracy is how closely we round an answer, such as to the nearest 10, the nearest whole number or the nearest penny. Rounding a money answer to the nearest penny means two decimal places, but money is usually written in pounds and pence.

Method. Work in pence, then round the answer to the nearest penny.

  1. Change £1,000 into pence. There are 100 pence in every pound.
    £1,000 = 100,000p
  2. Divide by 3. Share the pence equally between the three charities.
    100,000 ÷ 3 = 33,333 remainder 1
  3. Round to the nearest penny. The remainder 1 is one third of a penny more, so the exact amount is 33,333 and one third of a penny. The next digit is less than 5, so round down.
    33,333 pence, so 33,333p to the nearest penny
  4. Write the answer in pounds. Divide by 100 to change pence into pounds.
    33,333p = £333.33

Check. 3 × £333.33 = £999.99, which is close to £1,000. ✓

Watch out. Money is written to the nearest penny with two figures after the point, so £333.33, not £333.3.

Watch out

Do not round the numbers too early in a multi-step problem. Round only at the end, when the question asks for an accurate answer or an estimate. Also, an estimate is not the exact answer: 4,873 × 29 is about 150,000, but the exact answer is 141,317.

Exercise 2F  Estimation, problems and degrees of accuracy

Non-calculator
1
FEstimate each answer by rounding each number to one significant figure.
(a)3,946 × 58
(b)612 × 49
(c)2,875 × 33
2
FWhich is the best estimate for 812 × 29?   Tick one:   A  2,000   B  16,000   C  24,000   D  160,000
3
FRound 48,650 to the nearest 1,000.
4
FRound 2,375,461 to the nearest 10,000.
5
PSA shop sells 36 pencils at 45p each and 12 rulers at £1.25 each. How much money does the shop get altogether?
6
PSA bus holds 52 passengers. 1,000 people need to travel to a football match. How many buses are needed?
7
PSJo has £250. She buys 6 tickets at £17 each and a book for £14. Write one calculation to find how much money she has left, and work it out.
8
PSA school buys 24 packs of paper. Each pack has 250 sheets. The school uses 3,000 sheets. How many sheets are left?
9
PSA tank holds 1,200 litres. A pump fills it at 45 litres each minute. How many minutes does it take? Give your answer to the nearest minute.
10
RBen says 4,873 × 29 is about 1,500,000. Is he right? Explain your answer.
11
RJo says 6,249 ÷ 31 is about 200. Is this a good estimate? Explain how you know.
12
RNine friends share £100 equally. Give each share to the nearest penny. Explain why the answer must be rounded.

Chapter summary

SkillMethod
Long multiplicationMultiply by the ones digit on one row. Write a 0 place holder, multiply by the tens digit on the next row, then add the rows.
Long divisionFind how many times the divisor fits into each part of the number. Write the digit on top, multiply, subtract and bring down. The remainder is smaller than the divisor.
RemaindersChoose the answer the context needs: a whole number (round up or down), a fraction, or a rounded amount of money.
Short divisionDivide each digit from the left. Carry the remainder to the next digit. For a two-digit divisor such as 12, divide by 3, then by 4, and combine the remainders.
Mental methodsPartition, use a near round number and correct, halve and double, or take off the zeros. Multiply and divide before you add and subtract.
Order of operationsBrackets first, then multiplication and division from left to right, then addition and subtraction from left to right.
EstimationRound each number to one significant figure, then calculate. An estimate shows whether the answer is sensible.
Degrees of accuracyRound to the place the question asks for. Money is given to the nearest penny with two figures after the point.

Can you…?

  • I can multiply a number of up to four digits by a two-digit number using long multiplication.
  • I can divide a number of up to four digits by a two-digit number using long division, and write the remainder.
  • I can decide how to show a remainder: as a whole number, a fraction, or by rounding.
  • I can divide by a one-digit number using short division (the bus stop method).
  • I can use a mental method, such as partitioning or a near round number, to work out a calculation.
  • I can use the order of operations, including brackets, to work out a calculation.
  • I can estimate an answer and check whether my answer is sensible.
  • I can solve multi-step problems and give my answer to the right degree of accuracy.

Key terms

Dividend
The number being divided, such as 2,048 in 2,048 ÷ 23.
Divisor
The number we divide by, such as 23 in 2,048 ÷ 23.
Quotient
The answer to a division, before any remainder is written.
Remainder
The amount left over after dividing. It is always smaller than the divisor.
Partial product
The answer to one part of a multiplication, such as 14,070 in the long multiplication of 2,345 × 16.
Place holder
A zero that keeps each digit in its correct column, such as the zero in the second row of long multiplication.
Carry
Move a digit to the next column to the left, when a column makes ten or more.
Bus stop method
Another name for short division. The remainder is carried to the next digit.
Order of operations
The order in which we work out a calculation: brackets, then multiplication and division, then addition and subtraction.
Partition
Split a number into easy parts, such as 36 = 30 + 6.
Estimate
An answer that is close to the exact answer, found by rounding.
Significant figure
The first digit of a number that is not zero, used when rounding to estimate.
Degree of accuracy
How closely a number is rounded, such as to the nearest 100 or the nearest penny.

Exercise 2Z  Check-up

Non-calculator
1
Work out 846 × 35.
2
Work out 2,714 ÷ 26. Write the remainder.
3
Work out 5,628 ÷ 9.
4
Work out 4,032 ÷ 24 by dividing by 4, then by 6.
5
Work out 7 × 104 in your head. Show how you split the numbers.
6
Work out 12 × 39 by using 40.
7
Work out (4 + 6) × 3 − 8 ÷ 2.
8
Work out 5 + 6 × 4 − 2.
9
Estimate the answer to 3,872 × 49 by rounding each number to one significant figure.
10
Round 8,465,300 to the nearest 100,000.

Total 24 marks. Allow about 24 minutes.

Exercise 2X  SATs-style questions

Non-calculator
1
FWork out 46 × 13.[1 mark]
2
FWork out 2,490 ÷ 15.[1 mark]
3
FWork out 4,071 ÷ 6. Write your answer with the remainder.[2 marks]
4
FWork out 6 × 7 + 8 × 4 ÷ 2 − 5.[2 marks]
5
FRound 3,749,500 to the nearest 100,000.[2 marks]
6
FPut one pair of brackets in this calculation to make it correct:   8 + 6 ÷ 2 − 1 = 6.[2 marks]
7
FA cinema has 26 rows with 18 seats in each row. 412 tickets have been sold. How many seats are empty?   Show your working.[3 marks]
8
FSam shares 1,000 stickers equally between 42 children. How many stickers are left over?[3 marks]
9
FA bakery makes 48 cakes each hour for 8 hours, and 35 cakes each hour for 6 hours. It packs the cakes into boxes of 12. How many full boxes can it make?[4 marks]
10
RAli says 2 + 3 × 6 − 4 = 20. Ruby says it is 16. Who is right?   Explain your answer.[4 marks]
← Chapter 1: Place value to 10,000,000 and negative numbersChapter 3: Common factors, multiples and primes →

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