Year 6 Long Multiplication, Long Division and BODMAS: check-up and SATs-style questions

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. 35 is 3 tens and 5 ones. We multiply by the ones first, then by the tens.
   35 = 30 + 5
2. Multiply by the ones, which is 5. Start on the right and multiply each digit. Carry any tens to the next column, as in short multiplication.
   Ones: 6 × 5 = 30. Write 0 and carry 3.
   Tens: 4 × 5 = 20, and 20 + 3 = 23. Write 3 and carry 2.
   Hundreds: 8 × 5 = 40, and 40 + 2 = 42. Write 2 and carry 4.
   Thousands: nothing left to multiply. Write the carried 4.
   846 × 5 = 4,230
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: 6 × 3 = 18. Write 8 and carry 1.
   Tens: 4 × 3 = 12, and 12 + 1 = 13. Write 3 and carry 1.
   Hundreds: 8 × 3 = 24, and 24 + 1 = 25. Write 5 and carry 2.
   Thousands: nothing left to multiply. Write the carried 2.
   846 × 3 = 2,538. With the 0 place holder this row is 25,380
4. Add the two rows. Add the two rows, one column at a time, and carry where you need to.

       846
      × 35
   -------
      4230
   + 25380
   -------
     29610

   
   4,230 + 25,380 = 29,610

Check. Estimate: 800 × 40 = 32,000, and 29,610 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.

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 27. How many 26s fit into 27? 1, because 26 × 1 = 26. Subtract: 27 − 26 = 1.
2. Find quotient digit 2. Bring down the next digit and see how many times the divisor fits.
   Bring down the 1 to make 11. How many 26s fit into 11? 0, because 26 × 0 = 0. Subtract: 11 − 0 = 11.
3. Find quotient digit 3. Bring down the next digit and see how many times the divisor fits.
   Bring down the 4 to make 114. How many 26s fit into 114? 4, because 26 × 4 = 104. Subtract: 114 − 104 = 10.
4. Write the answer. The quotient goes on top. The number left at the end is the remainder.

         104
   26 ) 2714
        26
        --
         1
          0
          -
         11
         104
         ---
          10

   
   Answer 104 r 10

Check. 26 × 104 = 2,704, and 2,704 + 10 = 2,714. ✓

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

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, 9, goes outside it.
   9 ) 5,628
2. Divide the digit 5 (digit 1 of 4). Start with the first digit on the left.
   5 ÷ 9 = 0 remainder 5. Write 0 on top and carry the remainder 5.
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 5 to the next digit to make 56 = 56. 56 ÷ 9 = 6 remainder 2. Write 6 on top and carry 2.
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 2 to the next digit to make 22 = 22. 22 ÷ 9 = 2 remainder 4. Write 2 on top and carry 4.
5. Divide the digit 8 (digit 4 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 48 = 48. 48 ÷ 9 = 5 remainder 3. Write 5 on top and carry 3.
6. Read the answer from the bus stop layout. The quotient is the top row of numbers. The last remainder is left over.

   		5	2	4
   9	5	6	2	8
   	0	6	2	5

   
   Answer 625 r 3

Check. 9 × 625 = 5,625, and 5,625 + 3 = 5,628. ✓

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

Method. Divide by 4, then divide the answer by 6. Combine the two remainders to find the true remainder.

1. Write 24 as two numbers to multiply. Dividing by 24 is the same as dividing by 4 and then by 6.
   24 = 4 × 6
2. Divide by 4 first. Use short division and keep the remainder.

   		0	0	3
   4	4	0	3	2
   	1	0	0	8

   
   4,032 ÷ 4 = 1008 remainder 0
3. Divide the answer, 1008, by 6. Use short division again.

   		1	4	4
   6	1	0	0	8
   	0	1	6	8

   
   1,008 ÷ 6 = 168 remainder 0
4. Work out the total remainder. The 0 left over from the second division is worth 4 each in the first division. So multiply 0 by 4, then add the first remainder 0.
   4 × 0 + 0 = 0 + 0 = 0
5. Write the answer.
   168

Check. 24 × 168 = 4,032, and 4,032 + 0 = 4,032. ✓

Watch out. The second remainder is worth 4 in the first division, so you must multiply it by 4. Do not just add the two remainders.

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 104 into parts. Parts that are easy to multiply by 7 make the working simpler.
   104 = 100 + 4
2. Multiply each part by 7. Multiply each part on its own.
   7 × 100 = 700
   7 × 4 = 28
3. Add the answers. Add the partial answers together.
   700 + 28 = 728

Check. Estimate: 7 × 100 = 700, which is close to 728. ✓

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

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, 40. 39 is 1 less than 40, and 40 is easy to multiply by.
   39 = 40 − 1
2. Multiply 12 by 40. Multiply by the round number first.
   12 × 40 = 480
3. Correct for the difference. 39 is 1 less than 40, so we take away 1 lots of 12.
   12 × 1 = 12, so 480 − 12 = 468

Check. The estimate 10 × 40 = 400 is close to 468. ✓

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

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.
   4 + 6 = 10
2. Replace the brackets with their answer. The brackets are done, so the calculation is simpler now.
   ( 4 + 6 ) × 3 − 8 ÷ 2 = 10 × 3 − 8 ÷ 2
3. Multiply and divide, from left to right. Multiplication and division come before addition and subtraction.
   10 × 3 − 8 ÷ 2 = 30 − 8 ÷ 2
4. Multiply and divide, from left to right. Multiplication and division come before addition and subtraction.
   30 − 8 ÷ 2 = 30 − 4
5. Add and subtract, from left to right. Work from the left. Addition and subtraction are done last.
   30 − 4 = 26
6. 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.

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.
   5 + 6 × 4 − 2 = 5 + 24 − 2
2. Add and subtract, from left to right. Work from the left. Addition and subtraction are done last.
   5 + 24 − 2 = 29 − 2
3. Add and subtract, from left to right. Work from the left. Addition and subtraction are done last.
   29 − 2 = 27
4. Write the answer.
   27

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.

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.
   3,872 is about 4,000. 49 is about 50.
2. Multiply the rounded numbers. This gives an estimate. It is close to the exact answer, but not the same.
   4,000 × 50 = 200,000
3. Decide whether the estimate is sensible. An estimate that is close to the exact answer is a good check.
   Exact answer 189,728, and 200,000 is close to it

Check. The exact answer is 189,728, which is close to 200,000. ✓

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.

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 6
2. Decide whether to round up or down. 5 or more rounds up. Less than 5 rounds down.
   6 is 5 or more, so round up
3. Write the rounded number. Replace the digits after the hundred thousands column with zeros.
   8,465,300 to the nearest 100,000 = 8,500,000

Check. 8,500,000 is the closest number to 8,465,300 that ends in 5 zeros. ✓

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

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. 13 is 1 ten and 3 ones. We multiply by the ones first, then by the tens.
   13 = 10 + 3
2. Multiply by the ones, which is 3. Start on the right and multiply each digit. Carry any tens to the next column, as in short multiplication.
   Ones: 6 × 3 = 18. Write 8 and carry 1.
   Tens: 4 × 3 = 12, and 12 + 1 = 13. Write 3 and carry 1.
   Hundreds: nothing left to multiply. Write the carried 1.
   46 × 3 = 138
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: 6 × 1 = 6. Write 6.
   Tens: 4 × 1 = 4. Write 4.
   46 × 1 = 46. With the 0 place holder this row is 460
4. Add the two rows. Add the two rows, one column at a time, and carry where you need to.

      46
    × 13
   -----
     138
   + 460
   -----
     598

   
   138 + 460 = 598

Check. Estimate: 50 × 10 = 500, and 598 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.

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 24. How many 15s fit into 24? 1, because 15 × 1 = 15. Subtract: 24 − 15 = 9.
2. Find quotient digit 2. Bring down the next digit and see how many times the divisor fits.
   Bring down the 9 to make 99. How many 15s fit into 99? 6, because 15 × 6 = 90. Subtract: 99 − 90 = 9.
3. Find quotient digit 3. Bring down the next digit and see how many times the divisor fits.
   Bring down the 0 to make 90. How many 15s fit into 90? 6, because 15 × 6 = 90. Subtract: 90 − 90 = 0.
4. Write the answer. The quotient goes on top. The number left at the end is the remainder.

         166
   15 ) 2490
        15
        --
         9
         90
         --
          9
          90
          --
           0

   
   Answer 166

Check. 15 × 166 = 2,490, with no remainder. ✓

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

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 ) 4,071
2. Divide the digit 4 (digit 1 of 4). Start with the first digit on the left.
   4 ÷ 6 = 0 remainder 4. Write 0 on top and carry the remainder 4.
3. Divide the digit 0 (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 40 = 40. 40 ÷ 6 = 6 remainder 4. Write 6 on top and carry 4.
4. Divide the digit 7 (digit 3 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 47 = 47. 47 ÷ 6 = 7 remainder 5. Write 7 on top and carry 5.
5. Divide the digit 1 (digit 4 of 4). Each remainder is smaller than the divisor. Carry it to the next digit to the right.
   Carry the remainder 5 to the next digit to make 51 = 51. 51 ÷ 6 = 8 remainder 3. Write 8 on top and carry 3.
6. Read the answer from the bus stop layout. The quotient is the top row of numbers. The last remainder is left over.

   		4	4	5
   6	4	0	7	1
   	0	6	7	8

   
   Answer 678 r 3

Check. 6 × 678 = 4,068, and 4,068 + 3 = 4,071. ✓

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

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.
   6 × 7 + 8 × 4 ÷ 2 − 5 = 42 + 8 × 4 ÷ 2 − 5
2. Multiply and divide, from left to right. Multiplication and division come before addition and subtraction.
   42 + 8 × 4 ÷ 2 − 5 = 42 + 32 ÷ 2 − 5
3. Multiply and divide, from left to right. Multiplication and division come before addition and subtraction.
   42 + 32 ÷ 2 − 5 = 42 + 16 − 5
4. Add and subtract, from left to right. Work from the left. Addition and subtraction are done last.
   42 + 16 − 5 = 58 − 5
5. Add and subtract, from left to right. Work from the left. Addition and subtraction are done last.
   58 − 5 = 53
6. Write the answer.
   53

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.

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 4
2. Decide whether to round up or down. 5 or more rounds up. Less than 5 rounds down.
   4 is less than 5, so round down
3. Write the rounded number. Replace the digits after the hundred thousands column with zeros.
   3,749,500 to the nearest 100,000 = 3,700,000

Check. 3,700,000 is the closest number to 3,749,500 that ends in 5 zeros. ✓

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

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.
   8 + 6 = 14
2. Replace the brackets with their answer. The brackets are done, so the calculation is simpler now.
   ( 8 + 6 ) ÷ 2 − 1 = 14 ÷ 2 − 1
3. Multiply and divide, from left to right. Multiplication and division come before addition and subtraction.
   14 ÷ 2 − 1 = 7 − 1
4. Add and subtract, from left to right. Work from the left. Addition and subtraction are done last.
   7 − 1 = 6
5. Write the answer.
   6

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.

Method. Work out the total number of seats, then take away the tickets sold.

1. Work out the total number of seats. Multiply the rows by the seats in each row.
   26 × 18 = 468
2. Take away the tickets sold. The empty seats are the seats that have not been sold.
   468 − 412 = 56

Check. 412 + 56 = 468. ✓

Watch out. Check your answer: the tickets sold and the empty seats must add up to the total.

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 100. How many 42s fit into 100? 2, because 42 × 2 = 84. Subtract: 100 − 84 = 16.
2. Find quotient digit 2. Bring down the next digit and see how many times the divisor fits.
   Bring down the 0 to make 160. How many 42s fit into 160? 3, because 42 × 3 = 126. Subtract: 160 − 126 = 34.
3. Write the answer. The quotient goes on top. The number left at the end is the remainder.

          23
   42 ) 1000
         84
         --
         16
         126
         ---
          34

   
   Answer 23 r 34

Check. 42 × 23 = 966, and 966 + 34 = 1,000. ✓

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

Method. Work out the total cakes, then divide by the box size. Only full boxes count.

1. Work out the cakes in the first part. The first part is 48 cakes each hour for 8 hours.
   48 × 8 = 384 cakes
2. Work out the cakes in the second part. The second part is 35 cakes each hour for 6 hours.
   35 × 6 = 210 cakes
3. Add the two parts. Add to find all the cakes.
   384 + 210 = 594 cakes
4. Divide by 12 to find the full boxes. Only full boxes can be made, so the answer is the quotient.

         49
   12 ) 594
        48
        --
        11
        108
        ---
          6

   
   6 cakes are left over, so 49 full boxes

Check. 12 × 49 = 588, and 588 + 6 = 594. ✓

Watch out. Read the last words carefully. The question asks for full boxes, so the cakes left over are not counted.

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.
   2 + 3 × 6 − 4 = 2 + 18 − 4
2. Add and subtract, from left to right. Work from the left. Addition and subtraction are done last.
   2 + 18 − 4 = 20 − 4
3. Add and subtract, from left to right. Work from the left. Addition and subtraction are done last.
   20 − 4 = 16
4. Write the answer.
   16

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.