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This page sets out the method for multiplication & division in the order you would use it, then walks through the full lesson explanation. For three complete worked examples with the slips that cost marks, see the Multiplication & division topic page.
When you see a multiplication like or on a non-calculator paper, split each number into hundreds, tens and units.
When you see 'divide', 'share equally' or 'how many … in …', set up a bus stop with the number you are dividing inside.
4 × 6 means 4 groups of 6, or 6 groups of 4. Both totals are 24.
A row of 4 added six times is the same total as a row of 6 added four times.
An array of 3 rows of 5 is 15. You can count 5 + 5 + 5 or 3 × 5.
Turning the array on its side does not change how many squares there are.
18 ÷ 3 asks how many each person gets if 18 is shared into 3 equal piles. Each pile has 6.
If it does not share exactly, there is a remainder. At this stage, stick to exact shares.
18 ÷ 3 can also ask “how many 3s are in 18?”. There are 6.
Sharing and grouping use the same division, but the story sounds different.
If you know 4 × 5 = 20, then 4 × 6 is one more 4, so 24.
Doubling a fact doubles the product. Double 3 × 4 = 12 is 6 × 4 = 24.
If 7 × 8 = 56, then 56 ÷ 8 = 7 and 56 ÷ 7 = 8.
Use that to check a division: multiply the answer by the divisor.
Set the one-digit number under the units. Multiply each digit of the three-digit number by it, starting at the right. 346 × 7: units 6 × 7 = 42, write 2 and carry 4. Tens: 4 × 7 = 28, plus the carry 4, makes 32. Write 2 and carry 3. Hundreds: 3 × 7 = 21, plus 3, makes 24. The product is 2422.
The carry is tens of that column, so it joins the next column. It is not written as an extra digit in the answer. A place-value check is 300 × 7 = 2100, 40 × 7 = 280 and 6 × 7 = 42. Then 2100 + 280 + 42 = 2422. If those three parts do not add to your column answer, a carry slipped.
Judge the size with a nearby product. 350 × 7 = 2450, so an answer near 2400 can be right. An answer near 24000 has an extra zero. An answer near 350 has added the 7 instead of multiplying.
Method: 1. Multiply the units, then the tens, then the hundreds. 2. Carry when a column is 10 or more. 3. Add the carried digit into the next column before you move on.
A grid keeps the four parts separate. For 48 × 36, split 48 into 40 and 8, and 36 into 30 and 6. The cells are 40 × 30 = 1200, 40 × 6 = 240, 8 × 30 = 240 and 8 × 6 = 48. Add them: 1200 + 240 = 1440, 1440 + 240 = 1680, 1680 + 48 = 1728.
The same work in columns uses two rows. First row: 48 × 6 = 288. Second row: 48 × 30 = 1440. The second row ends in a zero because it is tens. Then 288 + 1440 = 1728. Forgetting the zero on the tens row is the usual slip. It makes that row ten times too small, so the total is far too small.
Estimate before you multiply. 50 × 40 = 2000, so the true product should sit a little under 2000. 1728 does. 17280 has an extra zero. The estimate catches a shifted row.
Method: 1. Multiply by the units digit of the second number first. 2. Multiply by the tens digit and write that row one place to the left. 3. Add the two rows to get the product.
Ignore the points and multiply the digits as whole numbers. Then count how many digits sat after the points in the question, and put that many digits after the point in the answer. 2.4 × 1.3: ignore the points and work out 24 × 13 = 312. There were two decimal places in total, so the product is 3.12.
Zeros after the point still count. 0.3 × 0.4 is 3 × 4 = 12 with two decimal places, so 0.12. It is not 1.2 and it is not 0.7. A quick size check: both factors are less than 1, so the product must be smaller than either of them. 0.12 is smaller than 0.3 and smaller than 0.4.
Money uses the same count. Three books at £6.75 is 3 × 675 pence, or 3 × 6.75. There are two decimal places in 6.75 and none in 3, so the answer has two decimal places: £20.25. Check the pence: 3 × 75 = 225 pence, which is £2.25, and 3 × 6 = 18, so 18 + 2.25 = 20.25.
Method: 1. Ignore the decimal points and multiply the digits as whole numbers. 2. Count the decimal places in the original numbers and put that many in the answer. 3. Check the size by estimating with whole numbers.
A decimal divisor is awkward because the times table is in tenths. Scale both numbers by the same power of 10 until the divisor is a whole number. The answer stays the same, because you have multiplied the dividend and the divisor by the same amount. 4.68 ÷ 0.12 becomes 468 ÷ 12 after multiplying both by 100.
Then divide the whole numbers. 12 × 30 = 360 and 468 - 360 = 108. 12 × 9 = 108, so 30 + 9 = 39. A ribbon of 4.68 metres cut into pieces of 0.12 metres gives 39 pieces. Check by multiplying back: 39 × 0.12 should be 4.68. Two decimal places in 0.12 means 39 × 12 = 468, then two places give 4.68.
A remainder is what will not make another whole group. 186 biscuits in boxes of 18: 18 × 10 = 180, so 10 boxes use 180 and 6 biscuits are left. If the question asks how many full boxes, the answer is 10 remainder 6. If every biscuit must go in a box, you need 11 boxes, and the last one is not full. Round up only when a leftover still needs a whole group. Seats on a bus, coaches for a trip, and boxes that must hold every item are round-up stories. Full groups with a remainder stated are not.
Method: 1. Multiply the divisor and the dividend by the same power of 10 until the divisor is whole. 2. Then divide. 3. The answer stays the same because both numbers were scaled together.
Multiplying by 10, 100 or 1000 moves every digit one, two or three places to the left. Dividing moves them to the right. The decimal point does not move; the digits do. So 0.45 × 100 = 45 and 5200 ÷ 10 = 520. Exam questions then use multiples such as 200 or 0.3, and the trick is to split the number into a simple part and a power of ten.
For 0.45 × 200, write 200 = 2 × 100. Then 0.45 × 2 = 0.9 and 0.9 × 100 = 90. For 840 ÷ 0.4, remember that dividing by a number smaller than one makes the answer bigger. Multiply both numbers by 10 to get 8400 ÷ 4 = 2100. Both methods avoid long working and are easy to check by estimating the size.
Method: 1. Split the multiplier into a single digit and a power of ten. 2. Multiply by the single digit first, then shift the digits for the power of ten. 3. For a decimal divisor, multiply both numbers by the same power of ten. 4. Estimate the size of the answer to check it is sensible.
Tip: Dividing by 0.1 is the same as multiplying by 10. Dividing by 0.5 is the same as multiplying by 2.
Short division is a compact way of dividing by a single digit. Long division is used for divisors such as 52. Write out the multiples of the divisor first: 52, 104, 156, 208, 260 and so on. For 2184 ÷ 52, ask how many 52s fit into 218. That is 4, since 4 × 52 = 208. Subtract to get 10, bring down the 4 to make 104, and 104 ÷ 52 = 2.
The answer is 42. A question that says "show your working" gives the method marks for the multiples list and the subtractions, so set them out neatly in columns. If the division does not come out exactly, there is a remainder. You can write it as "42 remainder 5", as a fraction, or as a decimal by continuing after the point with zeros, so read the question wording to see which form it wants.
Method: 1. List the multiples of the divisor, at least to the fifth multiple. 2. Divide the first digits, write the answer above and subtract. 3. Bring down the next digit and repeat until every digit is used. 4. Write any remainder in the form the question asks for, then check by multiplying back.
A classic exam question gives a product, for example 23 × 47 = 1081, and asks for other calculations that use the same digits, such as 2.3 × 0.47. You are not meant to start again. Use the fact, then fix the place value. Count the decimal places in the question: 2.3 has one and 0.47 has two, so the answer has three, which gives 1.081.
The same fact lets you divide. If 23 × 47 = 1081, then 1081 ÷ 4.7 must be 230, because 4.7 is 47 divided by 10, so the answer is ten times as big as 1081 ÷ 47 = 23. Command words here are "Use this to work out" or "Use the information above". Showing that you used the fact, rather than a different method, is what earns the marks.
Method: 1. Write down the digits of the given fact. 2. Compare each number in the new question with the number in the fact. 3. Decide how many places the answer has moved, or how many decimal places it needs. 4. Check with an estimate that the size is right.
Tip: If the new numbers are ten times smaller, the product is ten times smaller, but a quotient may be ten times bigger. Think carefully about which.
Real questions combine multiplying and dividing in a story: "A school hires 13 coaches at £450 each. 150 pupils share the total cost equally. How much does each pupil pay?" Split the problem into two stages and write each as a line of working. First 13 × 450 = 5850. Then 5850 ÷ 150 = 39.
Name each number, because it keeps you from dividing by the wrong thing. Write "cost of coaches" next to 5850. A good final line is "each pupil pays £39". Money answers must be written with a pound sign and two decimal places if they are not whole pounds, so £39.00 or £39 are both fine here. If remainders appear in a context, think about whether the extra people need another vehicle, which means rounding up.
Method: 1. Underline what is being multiplied and what is being shared. 2. Write the first calculation with a short label. 3. Use that answer in the second calculation. 4. Give the answer in context, with units, and decide whether a remainder needs rounding up.
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