Step-by-step Maths Level 5: Adding, subtracting, multiplying and dividing

Method. Split 12 into 10 and 2, then add the two results.

1. Split 12 into 10 and 2. Multiply each part by the number.
   12 × 11 = 10 × 11 + 2 × 11
2. Work out each part. Both parts are facts we know.
   10 × 11 = 110, and 2 × 11 = 22
3. Add the two parts. The total is the answer.
   110 + 22 = 132
4. Write the answer. This is the answer to 12 times the number.
   132

Check. Check: 132 ÷ 12 = 11.

Watch out. Do not add 12 to the number only once. Work out both parts.

Method. Multiply the number in each group by the number of groups, or the amount by the number of times.

1. Find the scaling number. Lee has the number of times as many as Sam.
   4 times 53
2. Multiply. Scaling is multiplying by the number of times.
   53 × 4 = 212
3. Write the answer. This is the number of stamps Lee has.
   212 stamps

Check. Check: 212 is more than the first number, because we multiplied by 4.

Watch out. Check the units: the answer should be in the same unit as the question (pencils, pence, stamps).

Method. Check by using the inverse: subtract for a sum, or add back for a difference.

1. Take one number away from the answer. If the sum is correct, taking one number away gives the other.
   918,021 - 746,477 = 171,544
2. Compare with the other number. Is 171,544 the same as 171,544?
   Same
3. Write Yes or No. Yes if the check works; No if it does not.
   Yes

Check. The correct answer is 918,021.

Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.

Method. Subtract each column from the right. Exchange 1 from the column on the left when the top digit is too small.

1. Subtract the ones column. 3 is less than 7, so exchange 1 tens for 10 ones.
   13 - 7 = 6
2. Subtract the tens column. 6 is less than 7. The hundreds is 0, so exchange across them: take 1 thousands and make each zero a 9, which gives 10 tens.
   16 - 7 = 9
3. Subtract the hundreds column. 9 is at least 6, so subtract directly.
   9 - 6 = 3
4. Subtract the thousands column. 4 is at least 3, so subtract directly.
   4 - 3 = 1
5. Subtract the ten thousands column. 8 is at least 1, so subtract directly.
   8 - 1 = 7
6. Subtract the hundred thousands column. 5 is at least 4, so subtract directly.
   5 - 4 = 1
7. Write the answer. Read the digits from the left, and put commas in.
   171,396

Check. Check: 413,677 + 171,396 = 585,073.

Watch out. When you exchange across zeros, change each zero to 9 as you pass it. Do not leave a zero in the middle.

Method. Round up when the remainder still needs one more group.

1. Divide to find the full buses and what is left. 27 children in buses of 5.
   27 ÷ 5 = 5 remainder 2
2. Decide what the remainder means. The children left over still need a bus, so round up.
   2 children left over need 1 more bus
3. Write the answer. Round up when the leftover people still need a place.
   6 buses

Check. Check: 6 buses hold 30 children, which is at least 27.

Watch out. Round up when the leftover still needs a place. Do not give the answer with a remainder.

Method. Multiply each digit by the number, starting with the ones. Carry the tens to the next column.

1. Multiply the ones digit by 9. Write 4 in the ones column and carry 5 to the tens column.
   6 × 9 = 54
2. Multiply the tens digit by 9. Write 6 in the tens column and carry 8 to the hundreds column.
   9 × 9 = 81, plus 5 carried = 86
3. Write the carried number in front. The carried number is the next digit on the left.
   8 in the hundreds column
4. Write the answer. Read the digits from the bottom of each column, from the left.
   864

Check. Check: 864 ÷ 9 = 96.

Watch out. Add any carried number before you write the digit of that column. Carry the tens; do not write them as a digit.

Method. Check by using the inverse: subtract for a sum, or add back for a difference.

1. Take one number away from the answer. If the sum is correct, taking one number away gives the other.
   882,437 - 597,023 = 285,414
2. Compare with the other number. Is 285,414 the same as 275,414?
   Not the same
3. Write Yes or No. Yes if the check works; No if it does not.
   No

Check. The correct answer is 872,437.

Watch out. A check that does not work means the answer is wrong. Find the mistake in the columns.

Method. Find the multiple of the divisor that is just below the number. It tells you the whole number of groups.

1. Count in 7s. Keep counting until you go past 51.
   7, 14, ... up to 51
2. Find the largest multiple. It is the last multiple that is not more than the number.
   49 is not more than 51
3. Write the multiple. This is the number that helps you find the division.
   49

Check. Check: 7 × 7 = 49, and 49 is not more than 51.

Watch out. Do not pick the multiple just above the number; it is more than the number.

Method. Multiply each digit by the number, starting with the ones. Carry the tens to the next column.

1. Multiply the ones digit by 3. Write 5 in the ones column and carry 1 to the tens column.
   5 × 3 = 15
2. Multiply the tens digit by 3. Write 4 in the tens column.
   1 × 3 = 3, plus 1 carried = 4
3. Multiply the hundreds digit by 3. Write 6 in the hundreds column.
   2 × 3 = 6
4. Write the answer. Read the digits from the bottom of each column, from the left.
   645

Check. Check: 645 ÷ 3 = 215.

Watch out. Add any carried number before you write the digit of that column. Carry the tens; do not write them as a digit.

Method. Multiply each digit by the number, starting with the ones. Carry the tens to the next column.

1. Multiply the ones digit by 7. Write 0 in the ones column.
   0 × 7 = 0
2. Multiply the tens digit by 7. Write 7 in the tens column.
   1 × 7 = 7
3. Write the answer. Read the digits from the bottom of each column, from the left.
   70

Check. Check: 70 ÷ 7 = 10.

Watch out. Add any carried number before you write the digit of that column. Carry the tens; do not write them as a digit.

Method. Use the short method. Share the tens first, then the ones with what is left.

1. The tens are fewer than the number of groups. 3 tens cannot make a group of 9 on their own, so use all of 30.
   30 is the number to share
2. Share 30 into groups of 9. How many groups of 9 are in 30?
   30 ÷ 9 = 3 r 3
3. Write the answer. The remainder is what is left over; write it after the answer.
   3 remainder 3

Check. Check: 3 × 9 + 3 = 30.

Watch out. The remainder must be smaller than the number you divide by.

Method. Use the inverse operation to find the missing number.

1. Add the number taken away to the answer. Adding undoes subtracting.
   207,536 + 84,766
2. Work out the sum. Use the column method.
   207,536 + 84,766 = 292,302
3. Check by subtracting. The answer must go back to the missing number.
   292,302 - 84,766 = 207,536
4. Write the missing number. This is the number in the box.
   292,302

Check. Check: put 292,302 into the sum and it works.

Watch out. To find a missing number in a sum, subtract. To find it in a difference, add.

Method. Divide, then check: answer × divisor + remainder = the number we started with.

1. Find how many groups of 5 are in 15. Use the times table of the divisor.
   3 groups of 5 = 15
2. Find what is left over. The remainder is the number left after the groups.
   15 - 15 = 0
3. Check with multiplication and addition. Multiply the answer by the divisor, then add the remainder.
   3 × 5 + 0 = 15 + 0 = 15
4. Write the answer. This is the answer with its remainder.
   3

Check. Check: 3 × 5 + 0 = 15.

Watch out. If the check does not give the number you started with, divide again.

Method. Count in 6s to find 11 groups of 6. The order of the numbers does not matter.

1. Count in 6s, 11 times. Each count adds one more group of 6.
   6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66
2. The last number we said is the answer. We counted 11 groups of 6.
   6 × 11 = 66
3. Write the answer. The order of the two numbers does not change the answer.
   66

Check. Check: 11 × 6 is also 66.

Watch out. Learn the 6 times table in order, then say it in any order.

Method. Division with equal groups: the number of boxes is the total divided by the box size.

1. Find the total and the size of each box. The total is how many; the box holds a certain number.
   108 cups in boxes of 12
2. Think: how many groups of the box size make the total? This is a division, which we can check with the table.
   12 × ___ = 108
3. Count the boxes. Count in the box size until you reach the total.
   12, 24, 36, 48, 60, 72, 84, 96, 108
4. Write the answer. This is the number of boxes.
   9

Check. Check: 9 boxes of 12 hold 9 × 12 = 108 cups.

Watch out. Check the answer makes sense: each box holds the same number.

Method. Check with multiplication and addition: groups × divisor + remainder must equal the number, and the remainder must be smaller than the divisor.

1. Multiply the whole number of groups by the divisor, then add the remainder. Check with multiplication and addition.
   5 × 6 + 0 = 30
2. Compare with 30. The result must be 30, and the remainder must be smaller than 6.
   30 is the same as 30; 0 is smaller than 6
3. Write Yes or No. Yes if the check works; No if it does not.
   Yes

Check. The correct answer is 5.

Watch out. A remainder that is as big as the divisor means the answer is wrong. Find the mistake if the check does not work.