Year 4 · Chapter 4

Year 4 Short Multiplication and Short Division: worked examples and questions with answers

By the end of this chapter you will be able to

Multiply and divide whole numbers by 10 and 100 by moving the digits, not by “adding a zero”
Multiply a two-digit or three-digit number by a one-digit number using partitioning, the grid method and the distributive law
Use the expanded column method, then short multiplication with carrying
Divide using place value and known facts, then use short division (the bus stop method) for 2-digit and 3-digit numbers
Work out and use remainders, and decide what a remainder means in a story
Solve scaling problems (“4 times as many”) and correspondence problems (3 hats and 5 tops make 15 outfits)
Multiply three numbers, and solve word problems using bar models

You should already know

Times tables up to 12 × 12 and the matching division facts (Chapter 3)
Place value in numbers up to 10,000 (Chapter 1)
Column addition with carrying (Chapter 2)
Multiplying by 10 in the 10 times table, and using known facts like 6 × 7 = 42 to work out 60 × 7

Where this leads

Fractions of amounts and dividing quantities (Chapter 5)
Money, decimals, and dividing by 10 and 100 to make decimals (Chapter 6)
Perimeter and area (Chapter 7)
Long multiplication and long division in Years 5 and 6
Multiplying and dividing by 1,000 in Year 5

National Curriculum for England, Key Stage 2 mathematics, Year 4. Content: Multiply and divide whole numbers by 10 and 100, Multiply two-digit and three-digit numbers by a one-digit number using a formal written layout, Use the distributive law to multiply two-digit numbers by one digit, and multiply together three numbers, Divide using place value and known facts, including short division with and without remainders for 2- and 3-digit numbers by 1-digit numbers, and interpret remainders, Solve integer scaling problems and harder correspondence problems such as n objects are connected to m objects.

Printable worksheets. Every lesson in this chapter has a free worksheet and an answer sheet to download and print.

4.1 Multiplying and dividing by 10 and 100: Worksheet (PDF) · Answers (PDF) · lesson page
4.2 Partitioning, the grid method and the distributive law: Worksheet (PDF) · Answers (PDF) · lesson page
4.3 Column multiplication: expanded and short: Worksheet (PDF) · Answers (PDF) · lesson page
4.4 Dividing with place value and short division: Worksheet (PDF) · Answers (PDF) · lesson page
4.5 Division with remainders: Worksheet (PDF) · Answers (PDF) · lesson page
4.6 Scaling, correspondence and multiplying three numbers: Worksheet (PDF) · Answers (PDF) · lesson page
4.7 Word problems with bar models: Worksheet (PDF) · Answers (PDF) · lesson page
Check-up and SATs-style questions: Worksheet (PDF) · Answers (PDF) · lesson page

Exercise P Do you remember?

Non-calculator

FWork out 7 × 8.

Done

FWork out 72 ÷ 9.

Done

FWork out 240 + 42.

Done

FYou know that 3 × 4 = 12. Use this to work out 3 × 40.

Done

FA box holds 6 eggs. How many eggs are in 8 boxes?

Done

4.1

Multiplying and dividing by 10 and 100

Year 4
Multiply and divide whole numbers by 10 and 100

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Multiplying and dividing by 10 and 100Coming soon

Each column in a number is worth ten times the column on its right. Ones, tens, hundreds, thousands: every step to the left makes the value ten times bigger. That is why multiplying and dividing by 10 and 100 is so quick. You only have to move the digits.

Multiplying by 10. Take 34. It is 3 tens and 4 ones. Multiply by 10 and every digit becomes ten times bigger. 3 tens become 3 hundreds. 4 ones become 4 tens. Every digit moves one place to the left. That leaves the ones place empty, so we fill it with a 0. So 34 × 10 = 340.

34 × 10 = 340. Each digit moves one place to the left. The orange 0 holds the empty ones place.

Multiplying by 100 makes every digit 100 times bigger. Every digit moves two places to the left and two zeros fill the gaps. 52 × 100 = 5,200. It is the same as multiplying by 10 and then by 10 again.

52 × 100 = 5,200. Each digit moves two places to the left.

Dividing is the opposite. Dividing by 10 makes every digit ten times smaller, so each digit moves one place to the right. Dividing by 100 moves each digit two places to the right. In this chapter the numbers divide exactly, which means they end in enough zeros. 4,500 ÷ 100 = 45.

4,500 ÷ 100 = 45. Each digit moves two places to the right. The two zeros at the end fall off.

The zero is a place holder. You may have heard “just add a zero”. That is not the real rule. The digits move, and the zero only holds the empty place so that the other digits stay in the right columns. This matters: 305 × 10 = 3,050, and when you divide, only the zeros at the end fall off: 3,050 ÷ 10 = 305.

Key words. A digit is one of the symbols 0 to 9. A place holder is a zero that keeps the other digits in the right columns. Place value is what each digit is worth.

Key ideas

× 10: every digit moves 1 place to the left. Put a 0 in the ones place.
× 100: every digit moves 2 places to the left. Put 0 in the tens and ones places.
÷ 10: every digit moves 1 place to the right. (The number must end in 0.)
÷ 100: every digit moves 2 places to the right. (The number must end in 00.)
× 100 is the same as × 10 and then × 10 again.
Multiplying and dividing undo each other: 34 × 10 = 340 and 340 ÷ 10 = 34.

Worked example 4.1.1

Work out 36 × 10.

Key words. A digit is one of the symbols 0 to 9. A place holder is a zero that keeps the other digits in the right columns. Place value tells you what each digit is worth.

Method. To multiply by 10, move every digit 1 place to the left and put a zero in the empty place on the right. Do not just “add a zero”: it is the digits that move.

Say what each digit is worth. The digits have different values because of their columns.
3 tens
6 ones
Make every digit 10 times bigger. Multiplying by 10 makes each digit 10 times bigger. That moves it 1 place to the left.
3 tens become 3 hundreds
6 ones become 6 tens
Show it in a place-value table. The 1 gap on the right is empty, so we put a zero there.
Write the answer. The zero place holders keep every digit in the right column.
36 × 10 = 360

Check. Divide back: 360 ÷ 10 = 36. ✓

Watch out. Move the digits, then fill the empty place with a zero. Check the answer is 10 times bigger.

Worked example 4.1.2

Work out 47 × 100.

Key words. A digit is one of the symbols 0 to 9. A place holder is a zero that keeps the other digits in the right columns. Place value tells you what each digit is worth.

Method. To multiply by 100, move every digit 2 places to the left and put 2 zeros in the empty places on the right. Do not just “add a zero”: it is the digits that move.

Say what each digit is worth. The digits have different values because of their columns.
4 tens
7 ones
Make every digit 100 times bigger. Multiplying by 100 makes each digit 100 times bigger. That moves it 2 places to the left.
4 tens become 4 thousands
7 ones become 7 hundreds
Show it in a place-value table. The 2 gaps on the right are empty, so we put zeros there.
Write the answer. The zero place holders keep every digit in the right column.
47 × 100 = 4,700

Check. Divide back: 4,700 ÷ 100 = 47. ✓

Watch out. Move the digits, then fill the empty places with zeros. Check the answer is 100 times bigger.

Worked example 4.1.3

Work out 520 ÷ 10.

Key words. A digit is one of the symbols 0 to 9. A place holder is a zero that keeps the other digits in the right columns. Place value tells you what each digit is worth.

Method. To divide by 10, move every digit 1 place to the right. A number that ends in 1 zero will divide exactly.

Look at the end of the number. Are there at least 1 zero? Every digit moves 1 place to the right, so the ones column must be able to hold the moved digits. A number that ends in 1 zero divides exactly.
520 ends in 1 zero. ✓
Make every digit 10 times smaller. Dividing by 10 makes each digit 10 times smaller. That moves it 1 place to the right.
5 hundreds become 5 tens
2 tens become 2 ones
Show it in a place-value table. The last 1 zero falls off the end.
Th H T O
520 5 2 0
÷ 10 → 1 place right
52 5 2
Write the answer.
520 ÷ 10 = 52

Check. Multiply back: 52 × 10 = 520. ✓

Watch out. Move the digits 1 place to the right. Check the answer is 10 times smaller.

Worked example 4.1.4

Work out 3,400 ÷ 100.

Key words. A digit is one of the symbols 0 to 9. A place holder is a zero that keeps the other digits in the right columns. Place value tells you what each digit is worth.

Method. To divide by 100, move every digit 2 places to the right. A number that ends in 2 zeros will divide exactly.

Look at the end of the number. Are there at least 2 zeros? Every digit moves 2 places to the right, so the ones column must be able to hold the moved digits. A number that ends in 2 zeros divides exactly.
3,400 ends in 2 zeros. ✓
Make every digit 100 times smaller. Dividing by 100 makes each digit 100 times smaller. That moves it 2 places to the right.
3 thousands become 3 tens
4 hundreds become 4 ones
Show it in a place-value table. The last 2 zeros fall off the end.
Th H T O
3,400 3 4 0 0
÷ 100 → 2 places right
34 3 4
Write the answer.
3,400 ÷ 100 = 34

Check. Multiply back: 34 × 100 = 3,400. ✓

Watch out. Move the digits 2 places to the right. Check the answer is 100 times smaller.

Worked example 4.1.5

Work out 305 × 10. Take care with the zero in the middle.

Key words. A digit is one of the symbols 0 to 9. A place holder is a zero that keeps the other digits in the right columns. Place value tells you what each digit is worth.

Method. To multiply by 10, move every digit 1 place to the left and put a zero in the empty place on the right. Do not just “add a zero”: it is the digits that move.

Say what each digit is worth. The digits have different values because of their columns.
3 hundreds
5 ones
Make every digit 10 times bigger. Multiplying by 10 makes each digit 10 times bigger. That moves it 1 place to the left.
3 hundreds become 3 thousands
5 ones become 5 tens
Show it in a place-value table. The 1 gap on the right is empty, so we put a zero there.
Th H T O
305 3 0 5
× 10 ← 1 place left
3,050 3 0 5 0
Write the answer. The zero place holders keep every digit in the right column.
305 × 10 = 3,050

Check. Divide back: 3,050 ÷ 10 = 305. ✓

Watch out. Move the digits, then fill the empty place with a zero. Check the answer is 10 times bigger.

Worked example 4.1.6

Find the missing number. 45 × ? = 4,500

Key words. A digit is one of the symbols 0 to 9. A place holder is a zero that keeps the other digits in the right columns. Place value tells you what each digit is worth.

Method. Find out how far the digits move. One place to the left is × 10. Two places to the left is × 100.

Compare the two numbers. 45 is on one side, 4,500 is the answer.
45 → 4,500
Count how many places the digits moved. Each digit of 45 moved to the left. Count the places.
Th H T O
45 4 5
× 100 ← 2 places left
4,500 4 5 0 0
Say what the number is. Digits that move 2 places to the left mean × 100.
45 × 100 = 4,500

Check. 45 × 100 = 4,500. ✓

Watch out. Put your answer back in the box and read the whole sentence. It must be true.

Worked example 4.1.7

A packet holds 100 stickers. A shop has 28 packets. How many stickers does the shop have?

Key words. A digit is one of the symbols 0 to 9. A place holder is a zero that keeps the other digits in the right columns. Place value tells you what each digit is worth.

Method. To multiply by 100, move every digit 2 places to the left and put 2 zeros in the empty places on the right. Do not just “add a zero”: it is the digits that move.

Choose the operation. Each packet holds 100 stickers, so we multiply the number of packets by 100.
28 × 100
Say what each digit is worth. The digits have different values because of their columns.
2 tens
8 ones
Make every digit 100 times bigger. Multiplying by 100 makes each digit 100 times bigger. That moves it 2 places to the left.
2 tens become 2 thousands
8 ones become 8 hundreds
Show it in a place-value table. The 2 gaps on the right are empty, so we put zeros there.
Th H T O
28 2 8
× 100 ← 2 places left
2,800 2 8 0 0
Write the answer. Include the unit.
2,800 stickers

Check. Divide back: 2,800 ÷ 100 = 28. ✓

Watch out. Move the digits, then fill the empty places with zeros. Check the answer is 100 times bigger.

Watch out

Do not just “add a zero” or “take away a zero”. Think about the digits moving. To multiply by 100, move two places (two zeros). To divide by 10, take off one zero only: 600 ÷ 10 = 60, not 6. And a zero inside a number stays: 305 × 10 = 3,050.

4.2

Partitioning, the grid method and the distributive law

Year 4
Multiply two-digit and three-digit numbers by a one-digit number, The distributive law

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Partitioning, the grid method and the distributive lawComing soon

Big multiplications are easier when you partition the number. Partition means split it into parts: tens and ones, or hundreds, tens and ones. Imagine 6 packs with 34 stickers in each pack. 34 is 30 + 4. So 6 packs hold 6 lots of 30 and 6 lots of 4. 6 × 30 = 180 and 6 × 4 = 24. Add them: 180 + 24 = 204.

The area model. The big rectangle is split into a 30 part and a 4 part. Its area is 180 + 24 = 204.

An area model is a rectangle split into parts. The length of each part is across the top, and the number on the side is the same for every part. The area of each part is a partial product. Add the partial products to get the total.

The distributive law. Splitting a number before you multiply always works. We write it like this:

34 × 6 = (30 + 4) × 6 = (30 × 6) + (4 × 6) = 180 + 24 = 204

That is the distributive law. You can do it in your head for two-digit numbers: multiply the tens, multiply the ones, then add.

Near a round number? If a number is just below a multiple of ten, you can multiply the round number and take some away. 29 × 6: 30 × 6 = 180, but 29 is one less than 30, so take away one lot of 6. 180 − 6 = 174.

The grid method for three-digit numbers. Do exactly the same with hundreds, tens and ones. 347 = 300 + 40 + 7. Put the parts in a grid and multiply each by 6.

347 × 6: three parts, 1,800 + 240 + 42 = 2,082.

×	300	40	7
6	300 × 61,800	40 × 6240	7 × 642

1,800 + 240 + 42 = 2,082. A zero digit makes no part: 305 = 300 + 5, so there are only two boxes.

Key words. To partition a number is to split it into parts, one for each digit. A partial product is the answer to one part of the multiplication. The grid method puts the parts in a table.

Key ideas

Partition the number: 34 = 30 + 4 and 347 = 300 + 40 + 7.
Multiply each part by the one-digit number. Use a times table fact and put the zeros back.
Add the partial products: this is the distributive law, (30 + 4) × 6 = 30 × 6 + 4 × 6.
Near a round number? 29 × 6 = 30 × 6 − 6.
A zero digit makes no part. Never skip a part that is not zero.

Worked example 4.2.1

Use the distributive law to work out 34 × 6 in your head.

Key words. The distributive law says that (30 + 4) × 6 is the same as 30 × 6 + 4 × 6. You can split a number and multiply the parts.

Method. Partition the two-digit number into tens and ones. Multiply each part by the one-digit number and add the two answers. This is called the distributive law.

Partition the number into tens and ones. Tens and ones are easy to multiply.
34 = 30 + 4
Write it as two multiplications. This is the distributive law: multiplying a sum by 6 is the same as multiplying each part by 6 and adding.
34 × 6 = (30 + 4) × 6 = (30 × 6) + (4 × 6)
Multiply the tens part. 3 × 6 = 18, so 3 tens × 6 = 18 tens.
30 × 6 = 180
Multiply the ones part. Use a times table fact.
4 × 6 = 24
Add the two answers.
180 + 24 = 204

Check. Estimate: 34 is about 30 (to the nearest 10). 30 × 6 = 180. Our answer, 204, is a bit more than 180, which is right because we rounded down. ✓

Watch out. Both parts must be multiplied by 6. A common mistake is to multiply only the tens, or only the ones.

Worked example 4.2.2

Work out 58 × 7 by partitioning 58 into tens and ones.

Key words. The distributive law says that (30 + 4) × 6 is the same as 30 × 6 + 4 × 6. You can split a number and multiply the parts.

Method. Partition the two-digit number into tens and ones. Multiply each part by the one-digit number and add the two answers. This is called the distributive law.

Partition the number into tens and ones. Tens and ones are easy to multiply.
58 = 50 + 8
Write it as two multiplications. This is the distributive law: multiplying a sum by 7 is the same as multiplying each part by 7 and adding.
58 × 7 = (50 + 8) × 7 = (50 × 7) + (8 × 7)
Multiply the tens part. 5 × 7 = 35, so 5 tens × 7 = 35 tens.
50 × 7 = 350
Multiply the ones part. Use a times table fact.
8 × 7 = 56
Add the two answers.
350 + 56 = 406

Check. Estimate: 58 is about 60 (to the nearest 10). 60 × 7 = 420. Our answer, 406, is a bit less than 420, which is right because we rounded up. ✓

Watch out. Both parts must be multiplied by 7. A common mistake is to multiply only the tens, or only the ones.

Worked example 4.2.3

Work out 49 × 6. Use a round number that is close to 49.

Key words. To compensate is to make up for a change. Here we multiply an easy number, then correct the answer.

Method. 49 is close to the round number 50. Multiply 50 first, then take away the extra bit.

Spot a round number close by. 49 is just 1 below 50.
49 = 50 − 1
Multiply the round number by 6. Round numbers are easy to multiply.
50 × 6 = 300
Work out the extra bit: 1 × 6. We multiplied 50 by 6, but 49 is 1 less, so we have to take away 1 lots of 6.
1 × 6 = 6
Fix the answer. Take it away (or add it) to get to the real answer.
300 − 6 = 294

Check. Partition instead: 49 = 40 + 9, and 240 + 54 = 294. ✓

Watch out. Take away 1 lots of 6, not just 1.

Worked example 4.2.4

Use the grid method to work out 47 × 8.

Key words. To partition a number is to split it into parts, one for each digit (347 = 300 + 40 + 7). A partial product is the answer to one part of a multiplication. The grid method (or area method) puts the parts in a table and multiplies each part.

Method. Split the number into hundreds, tens and ones (partition it). Multiply each part by the one-digit number, then add the answers.

Partition the number you are multiplying. Split it into hundreds, tens and ones. Each part is easy to multiply.
47 = 40 + 7
Draw the grid and write the parts across the top. The number on the side is 8. Every part gets multiplied by 8.

× 40 7
8 ? ?
Multiply each part. Use a times table fact, then put the zeros back.
7 × 8 = 56
40 is 4 tens. 4 × 8 = 32, so 4 tens × 8 = 32 tens = 320
Write each answer in the grid. These are the partial products.
× 40 7
8 40 × 8320 7 × 856
Add the partial products. Add them one at a time, from the biggest. Keep a running total.
320 + 56 = 376
Write the answer.
47 × 8 = 376

Check. Estimate: 47 is about 50 (to the nearest 10). 50 × 8 = 400. Our answer, 376, is a bit less than 400, which is right because we rounded up. ✓

Watch out. Do not forget a part. Count the parts: every digit that is not 0 needs its own box. And keep the zeros: 40 × 6 is 240, not 24.

Worked example 4.2.5

Use the grid method to work out 325 × 4.

Method. Split the number into hundreds, tens and ones (partition it). Multiply each part by the one-digit number, then add the answers.

Partition the number you are multiplying. Split it into hundreds, tens and ones. Each part is easy to multiply.
325 = 300 + 20 + 5
Draw the grid and write the parts across the top. The number on the side is 4. Every part gets multiplied by 4.

× 300 20 5
4 ? ? ?
Multiply each part. Use a times table fact, then put the zeros back.
5 × 4 = 20
20 is 2 tens. 2 × 4 = 8, so 2 tens × 4 = 8 tens = 80
300 is 3 hundreds. 3 × 4 = 12, so 3 hundreds × 4 = 12 hundreds = 1,200
Write each answer in the grid. These are the partial products.
× 300 20 5
4 300 × 41,200 20 × 480 5 × 420
Add the partial products. Add them one at a time, from the biggest. Keep a running total.
1,200 + 80 = 1,280
1,280 + 20 = 1,300
Write the answer.
325 × 4 = 1,300

Check. Estimate: 325 is about 300 (to the nearest 100). 300 × 4 = 1,200. Our answer, 1,300, is a bit more than 1,200, which is right because we rounded down. ✓

Watch out. Do not forget a part. Count the parts: every digit that is not 0 needs its own box. And keep the zeros: 40 × 6 is 240, not 24.

Worked example 4.2.6

Use the grid method to work out 406 × 7. Take care with the zero.

Method. Split the number into hundreds, tens and ones (partition it). Multiply each part by the one-digit number, then add the answers.

Partition the number you are multiplying. Split it into hundreds, tens and ones. Each part is easy to multiply. The 0 in the number means there are none of that place, so it has no part.
406 = 400 + 6
Draw the grid and write the parts across the top. The number on the side is 7. Every part gets multiplied by 7.

× 400 6
7 ? ?
Multiply each part. Use a times table fact, then put the zeros back.
6 × 7 = 42
400 is 4 hundreds. 4 × 7 = 28, so 4 hundreds × 7 = 28 hundreds = 2,800
Write each answer in the grid. These are the partial products.
× 400 6
7 400 × 72,800 6 × 742
Add the partial products. Add them one at a time, from the biggest. Keep a running total.
2,800 + 42 = 2,842
Write the answer.
406 × 7 = 2,842

Check. Estimate: 406 is about 400 (to the nearest 100). 400 × 7 = 2,800. Our answer, 2,842, is a bit more than 2,800, which is right because we rounded down. ✓

Watch out. Do not forget a part. Count the parts: every digit that is not 0 needs its own box. And keep the zeros: 40 × 6 is 240, not 24.

Worked example 4.2.7

Find the missing number. 52 × 7 = ? + 14

Method. Partition the number into tens and ones, multiply each part by the one-digit number, and add the answers.

Partition the number. Split 52 into tens and ones.
52 = 50 + 2
Write what each part makes. 50 × 7 = 350 and 2 × 7 = 14.
(50 × 7) + (2 × 7) = 350 + 14
Find the missing number. The tens part of 52 is 50. Multiply it by 7.
50 × 7 = 350

Check. Total: 350 + 14 = 364, and 52 × 7 = 364. ✓

Watch out. Read the whole sentence with your answer in the box. It must be true.

Watch out

Multiply the whole value of each part. In 52 × 6, the 5 is worth 50, so work out 50 × 6 = 300, not 5 × 6 = 30. Also remember every part: if you only multiply the tens, you have forgotten the ones.

4.3

Column multiplication: expanded and short

Year 4
Multiply two-digit and three-digit numbers by a one-digit number using a formal written layout

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Column multiplication: expanded and shortComing soon

The grid method works well, but it takes a lot of space. Column methods are tidier and quicker. We meet two of them. The first is the expanded column method. It is the grid written in a column. Multiply the ones, then the tens, then the hundreds. Write each answer on its own line, then add the lines.

Here is 347 × 6. The lines are 7 × 6 = 42, 40 × 6 = 240 and 300 × 6 = 1,800.

	347
×	6
	42	7 × 6
	240	40 × 6
	1,800	300 × 6
	2,082

Notice that you multiply the whole value of each digit: the 4 is worth 40, so you work out 40 × 6.

The second method is short multiplication. It is the same sum, set out in a smaller space. You multiply one column at a time, starting with the ones. If the answer is 10 or more, you write the ones digit and carry the tens digit to the next column. You then add the carried digit after you have multiplied that column.

Carrying: 10 ones make 1 ten. So 42 ones is 4 tens and 2 ones. We write the 2 and carry the 4 tens.

Let us do 347 × 6 again. 7 × 6 = 42: write 2, carry 4. 4 × 6 = 24, and 24 + 4 = 28: write 8, carry 2. 3 × 6 = 18, and 18 + 2 = 20: write 0, carry 2. There are no more digits, so write the carried 2. The answer is 2,082, the same as the expanded method.

	Th	H	T	O
		3	4	7
×				6
	2	0	8	2
	2	2	4

Estimate first. 347 is about 300, and 300 × 6 = 1,800. So the answer should be a bit more than 1,800. 2,082 is sensible.

Key words. Short multiplication is a written method with a one-digit multiplier. To carry is to move the tens digit of an answer into the next column. The expanded method shows every partial product on its own line.

Key ideas

Expanded method: multiply each part (7 × 6, 40 × 6, 300 × 6), then add the lines.
Short multiplication: start with the ones, multiply each column, carry the tens digit.
Add the carried digit AFTER you multiply the next column.
Estimate first: round and multiply. Your answer should be close.
A zero digit: 0 × the number is 0, but still add the carried digit.

Worked example 4.3.1

Work out 38 × 6 using the expanded column method.

Key words. The expanded method writes each partial product on its own line, so you can see every step. It leads to the short method.

Method. Multiply each digit of the number by the one-digit number, starting with the ones. Write each answer on its own line. Then add the lines.

Write the numbers in a column. The number to multiply goes on top. The one-digit number goes underneath, on the right.
38
× 6
48 8 × 6
180 30 × 6
Multiply the ones. Always start on the right.
8 × 6 = 48
Multiply the tens. The digit 3 is worth 30, so multiply 30 by 6, not just 3.
30 is 3 tens. 3 × 6 = 18, so 3 tens × 6 = 18 tens = 180
Add the lines together. Draw a line and add up the partial products.
38
× 6
48 8 × 6
180 30 × 6
228

48 + 180 = 228
Write the answer.
38 × 6 = 228

Check. Estimate: 38 is about 40 (to the nearest 10). 40 × 6 = 240. Our answer, 228, is a bit less than 240, which is right because we rounded up. ✓

Watch out. Multiply the whole value of each digit: the 4 in 347 is worth 40, so work out 40 × 6, not 4 × 6.

Worked example 4.3.2

Work out 274 × 3 using the expanded column method.

Key words. The expanded method writes each partial product on its own line, so you can see every step. It leads to the short method.

Method. Multiply each digit of the number by the one-digit number, starting with the ones. Write each answer on its own line. Then add the lines.

Write the numbers in a column. The number to multiply goes on top. The one-digit number goes underneath, on the right.
274
× 3
12 4 × 3
210 70 × 3
600 200 × 3
Multiply the ones. Always start on the right.
4 × 3 = 12
Multiply the tens. The digit 7 is worth 70, so multiply 70 by 3, not just 7.
70 is 7 tens. 7 × 3 = 21, so 7 tens × 3 = 21 tens = 210
Multiply the hundreds. The digit 2 is worth 200, so multiply 200 by 3, not just 2.
200 is 2 hundreds. 2 × 3 = 6, so 2 hundreds × 3 = 6 hundreds = 600
Add the lines together. Draw a line and add up the partial products.
274
× 3
12 4 × 3
210 70 × 3
600 200 × 3
822

12 + 210 = 222
222 + 600 = 822
Write the answer.
274 × 3 = 822

Check. Estimate: 274 is about 300 (to the nearest 100). 300 × 3 = 900. Our answer, 822, is a bit less than 900, which is right because we rounded up. ✓

Watch out. Multiply the whole value of each digit: the 4 in 347 is worth 40, so work out 40 × 6, not 4 × 6.

Worked example 4.3.3

Work out 38 × 6 using short multiplication.

Key words. Short multiplication is a written method. You write the number to multiply on top, the one-digit number below it, and multiply one column at a time, starting with the ones. When a column makes 10 or more, you carry the tens digit to the next column and add it there.

Method. Write the numbers in columns. Multiply each digit by the one-digit number, starting with the ones. If an answer is 10 or more, write the ones digit and carry the tens digit to the next column.

Write the numbers in columns. Ones go under ones, tens under tens and hundreds under hundreds. The one-digit number goes under the ones.
H T O
3 8
× 6
Multiply the ones. Always start with the ones, on the right.
8 × 6 = 48
48 is 4 tens and 8 ones. Write 8 in the ones column and carry 4 to the tens column.
H T O
3 8
× 6
8
4
Multiply the tens. Now the next column to the left. Multiply, then add the 4 we carried.
3 × 6 = 18, and 18 + 4 (carried) = 22
22 is 2 hundreds and 2 tens. Write 2 in the tens column and carry 2 to the hundreds column.
H T O
3 8
× 6
2 8
2 4
Write the carried digit. There are no digits left to multiply in the hundreds column, but we carried 2 into it.
Nothing to multiply. Write 2 in the hundreds column.
H T O
3 8
× 6
2 2 8
2 4
Read the answer. Check that the answer has about the right number of digits, then write it.
H T O
3 8
× 6
2 2 8
2 4

38 × 6 = 228

Check. Estimate: 38 is about 40 (to the nearest 10). 40 × 6 = 240. Our answer, 228, is a bit less than 240, which is right because we rounded up. ✓

Watch out. Do not forget the carried digits. Write each one small, then add it in the next column AFTER you have multiplied.

Worked example 4.3.4

Work out 347 × 6 using short multiplication.

Write the numbers in columns. Ones go under ones, tens under tens and hundreds under hundreds. The one-digit number goes under the ones.
Th H T O
3 4 7
× 6
Multiply the ones. Always start with the ones, on the right.
7 × 6 = 42
42 is 4 tens and 2 ones. Write 2 in the ones column and carry 4 to the tens column.
Th H T O
3 4 7
× 6
2
4
Multiply the tens. Now the next column to the left. Multiply, then add the 4 we carried.
4 × 6 = 24, and 24 + 4 (carried) = 28
28 is 2 hundreds and 8 tens. Write 8 in the tens column and carry 2 to the hundreds column.
Th H T O
3 4 7
× 6
8 2
2 4
Multiply the hundreds. Now the next column to the left. Multiply, then add the 2 we carried.
3 × 6 = 18, and 18 + 2 (carried) = 20
20 is 2 thousands and 0 hundreds. Write 0 in the hundreds column and carry 2 to the thousands column.
Th H T O
3 4 7
× 6
0 8 2
2 2 4
Write the carried digit. There are no digits left to multiply in the thousands column, but we carried 2 into it.
Nothing to multiply. Write 2 in the thousands column.
Th H T O
3 4 7
× 6
2 0 8 2
2 2 4
Read the answer. Check that the answer has about the right number of digits, then write it.
Th H T O
3 4 7
× 6
2 0 8 2
2 2 4

347 × 6 = 2,082

Check. Estimate: 347 is about 300 (to the nearest 100). 300 × 6 = 1,800. Our answer, 2,082, is a bit more than 1,800, which is right because we rounded down. ✓

Watch out. Do not forget the carried digits. Write each one small, then add it in the next column AFTER you have multiplied.

Worked example 4.3.5

Work out 528 × 4 using short multiplication. There is a carry in two columns.

Write the numbers in columns. Ones go under ones, tens under tens and hundreds under hundreds. The one-digit number goes under the ones.
Th H T O
5 2 8
× 4
Multiply the ones. Always start with the ones, on the right.
8 × 4 = 32
32 is 3 tens and 2 ones. Write 2 in the ones column and carry 3 to the tens column.
Multiply the tens. Now the next column to the left. Multiply, then add the 3 we carried.
2 × 4 = 8, and 8 + 3 (carried) = 11
11 is 1 hundred and 1 ten. Write 1 in the tens column and carry 1 to the hundreds column.
Multiply the hundreds. Now the next column to the left. Multiply, then add the 1 we carried.
5 × 4 = 20, and 20 + 1 (carried) = 21
21 is 2 thousands and 1 hundred. Write 1 in the hundreds column and carry 2 to the thousands column.
Write the carried digit. There are no digits left to multiply in the thousands column, but we carried 2 into it.
Nothing to multiply. Write 2 in the thousands column.
Read the answer. Check that the answer has about the right number of digits, then write it.
Th H T O
5 2 8
× 4
2 1 1 2
2 1 3

528 × 4 = 2,112

Check. Estimate: 528 is about 500 (to the nearest 100). 500 × 4 = 2,000. Our answer, 2,112, is a bit more than 2,000, which is right because we rounded down. ✓

Watch out. Do not forget the carried digits. Write each one small, then add it in the next column AFTER you have multiplied.

Worked example 4.3.6

Work out 407 × 5. Take care with the zero in the tens place.

Write the numbers in columns. Ones go under ones, tens under tens and hundreds under hundreds. The one-digit number goes under the ones.
Th H T O
4 0 7
× 5
Multiply the ones. Always start with the ones, on the right.
7 × 5 = 35
35 is 3 tens and 5 ones. Write 5 in the ones column and carry 3 to the tens column.
Multiply the tens. Now the next column to the left. Multiply, then add the 3 we carried.
0 × 5 = 0, and 0 + 3 (carried) = 3
3 is less than 10, so write 3 in the tens column. There is nothing to carry.
Multiply the hundreds. Now the next column to the left.
4 × 5 = 20
20 is 2 thousands and 0 hundreds. Write 0 in the hundreds column and carry 2 to the thousands column.
Write the carried digit. There are no digits left to multiply in the thousands column, but we carried 2 into it.
Nothing to multiply. Write 2 in the thousands column.
Read the answer. Check that the answer has about the right number of digits, then write it.
Th H T O
4 0 7
× 5
2 0 3 5
2 3

407 × 5 = 2,035

Check. Estimate: 407 is about 400 (to the nearest 100). 400 × 5 = 2,000. Our answer, 2,035, is a bit more than 2,000, which is right because we rounded down. ✓

Watch out. A zero digit still needs a step: 0 × the number is 0, but do not forget to add the carried digit.

Worked example 4.3.7

First estimate, then work out 623 × 4.

Key words. To estimate is to find an answer that is close, usually by rounding first. ≈ means “is about equal to”.

Method. Round the big number and multiply to get an estimate. Then work out the exact answer with short multiplication and check that it is close to the estimate.

Round 623 to the nearest 100. Rounded numbers are easy to multiply.
623 ≈ 600 (the tens digit is 2, so round down)
Work out the estimate. Multiply the rounded number by the one-digit number.
600 × 4 = 2,400
Write the numbers in columns. Ones go under ones, tens under tens and hundreds under hundreds. The one-digit number goes under the ones.
Th H T O
6 2 3
× 4
Multiply the ones. Always start with the ones, on the right.
3 × 4 = 12
12 is 1 ten and 2 ones. Write 2 in the ones column and carry 1 to the tens column.
Multiply the tens. Now the next column to the left. Multiply, then add the 1 we carried.
2 × 4 = 8, and 8 + 1 (carried) = 9
9 is less than 10, so write 9 in the tens column. There is nothing to carry.
Multiply the hundreds. Now the next column to the left.
6 × 4 = 24
24 is 2 thousands and 4 hundreds. Write 4 in the hundreds column and carry 2 to the thousands column.
Write the carried digit. There are no digits left to multiply in the thousands column, but we carried 2 into it.
Nothing to multiply. Write 2 in the thousands column.
Compare the exact answer with the estimate. They should be close.
Th H T O
6 2 3
× 4
2 4 9 2
2 1

Exact: 2,492. Estimate: 2,400. They are close. ✓

Check. Estimate: 623 is about 600 (to the nearest 100). 600 × 4 = 2,400. Our answer, 2,492, is a bit more than 2,400, which is right because we rounded down. ✓

Watch out. An estimate is not the exact answer. It is a quick way to catch big mistakes.

Worked example 4.3.8

Priya says that 623 × 4 = 292. Use an estimate to show that she must be wrong. Then find the correct answer.

Method. Estimate the answer to see if Priya’s answer is sensible. Then find the correct answer.

Round 623 to the nearest 100. Rounded numbers are easy to multiply.
623 ≈ 600 (the tens digit is 2, so round down)
Work out the estimate. 6 × 4 = 24, so 6 hundreds × 4 = 24 hundreds.
600 × 4 = 2,400
Compare with Priya’s answer. The exact answer should be close to 2,400.
Priya’s answer, 292, is far too small. It is not sensible.
Work out the correct answer. Use short multiplication.
Th H T O
6 2 3
× 4
2 4 9 2
2 1

623 × 4 = 2,492

Check. Estimate: 623 is about 600 (to the nearest 100). 600 × 4 = 2,400. Our answer, 2,492, is a bit more than 2,400, which is right because we rounded down. ✓

Watch out. If your answer is not close to the estimate, go back and check each column. Priya has lost a digit.

Watch out

Carry, then add. In 347 × 6, the 4 is multiplied by 6 first (24), and then you add the carried 4 (28). Do not add the carry before multiplying. Also write the carried digit small, in the next column, so you do not lose it. And keep every digit of the answer, including a 0.

4.4

Dividing with place value and short division

Year 4
Divide using place value and known facts, Short division (the bus stop method) for 2- and 3-digit numbers by 1-digit numbers

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Dividing with place value and short divisionComing soon

Dividing means sharing equally, or making equal groups. You already know many division facts from your times tables. For bigger numbers, use place value and known facts.

Try 84 ÷ 4. Partition 84 into 80 + 4. 80 ÷ 4 = 20 (because 8 ÷ 4 = 2, so 8 tens ÷ 4 = 2 tens). 4 ÷ 4 = 1. Add the answers: 20 + 1 = 21. Or use a known fact with zeros: 24 ÷ 6 = 4, so 2,400 ÷ 6 = 400 (24 hundreds ÷ 6 = 4 hundreds).

Sometimes the tens do not share out exactly. For 52 ÷ 4, split 52 into chunks that you can divide: 52 = 40 + 12. 40 ÷ 4 = 10 and 12 ÷ 4 = 3. So 52 ÷ 4 = 10 + 3 = 13.

Why does that work? Share 52 (5 tens and 2 ones) between 4. Each share gets 1 ten, and 1 ten is left over. You cannot share 1 ten between 4, so exchange it for 10 ones. Now there are 10 + 2 = 12 ones. Each share gets 3 ones. Each share has 1 ten and 3 ones, which is 13.

52 = 5 tens and 2 ones

exchange 1 ten for 10 ones

Sharing 52 into 4 equal shares.

The bus stop method (short division) writes this down neatly. Put the number you divide inside the bus stop and the number you divide by outside. Work from the left. For 52 ÷ 4: 5 ÷ 4 = 1 remainder 1. Write 1 above the 5. Carry the remainder 1 to the next digit: it makes 12. 12 ÷ 4 = 3. Write 3 above the 2.

	1	3
4	5	¹2

The answer is 13.

Three-digit numbers work the same way, one digit at a time. For 756 ÷ 6: 7 ÷ 6 = 1 remainder 1, carry 1. 15 ÷ 6 = 2 remainder 3, carry 3. 36 ÷ 6 = 6. The answer is 126.

	1	2	6
6	7	¹5	³6

If the first digit is too small, write 0 above it and carry it. For 285 ÷ 5: 2 ÷ 5 = 0 remainder 2, carry 2. 28 ÷ 5 = 5 remainder 3, carry 3. 35 ÷ 5 = 7. The answer is 57 (we do not write the 0 at the front).

	0	5	7
5	2	²8	³5

Key words. Short division (the bus stop method) is a written way to divide. A chunk is a part of the number that is easy to divide. To carry is to pass the leftover to the next digit.

Key ideas

Use place value: 84 ÷ 4 = 80 ÷ 4 + 4 ÷ 4 = 20 + 1.
Use known facts: 24 ÷ 6 = 4, so 2,400 ÷ 6 = 400.
Bus stop: start on the LEFT. Divide the first digit. Carry any remainder to the next digit as extra tens, ones...
If a digit is too small to divide, write 0 above it and carry it.
Check by multiplying back: 13 × 4 = 52.
Answers have a zero when a step has nothing to share: 618 ÷ 6 = 103.

Worked example 4.4.1

Use the fact 36 ÷ 9 = 4 to work out 3,600 ÷ 9.

Key words. A known fact is a fact you can already recall. A derived fact is a new fact you work out from a known fact.

Method. Use the fact you know, 36 ÷ 9 = 4, and place value. 3,600 is 36 hundreds, so the answer is 4 hundreds.

Spot the fact you know. Take away the zeros and look at the digits that are left.
36 ÷ 9 = 4
Say the big number in hundreds. 3,600 is 36 hundreds.
3,600 = 36 hundreds
Divide, using the known fact. 36 hundreds ÷ 9 = 4 hundreds, because 36 ÷ 9 = 4.
36 ÷ 9 = 4, so 36 hundreds ÷ 9 = 4 hundreds
Change 4 hundreds back to a number. Put the zeros back.
4 hundreds = 400
Write the answer.
3,600 ÷ 9 = 400

Check. Multiply back: 400 × 9 = 3,600. ✓

Watch out. Keep the zeros. 420 ÷ 7 is 60, not 6.

Worked example 4.4.2

Use place value to work out 84 ÷ 4.

Key words. To divide using place value is to split the number into parts that you can divide using times table facts, then add the answers. A remainder is what is left over.

Method. Split 84 into chunks that you can divide by 4 using times table facts. Divide each chunk, then add the answers.

Split the number into chunks that are easy to divide. Use multiples of 4 like 40, 80, 120... and the 4 times table. Take the biggest chunk first.
84 = 80 + 4
Divide each chunk. Use a times table fact and place value.
80 ÷ 4 = 20 (because 20 × 4 = 80)
4 ÷ 4 = 1 (because 1 × 4 = 4)
Add the answers. Add up the chunk answers.
20 + 1 = 21
Write the answer.
84 ÷ 4 = 21

Check. Multiply back: 21 × 4 = 20 × 4 + 1 × 4 = 80 + 4 = 84. ✓

Watch out. Make sure the chunks add back to 84: 80 + 4 = 84.

Worked example 4.4.3

Use chunks to work out 78 ÷ 6.

Key words. To divide using place value is to split the number into parts that you can divide using times table facts, then add the answers. A remainder is what is left over.

Method. Split 78 into chunks that you can divide by 6 using times table facts. Divide each chunk, then add the answers.

Split the number into chunks that are easy to divide. Use multiples of 6 like 60, 120, 180... and the 6 times table. Take the biggest chunk first.
78 = 60 + 18
Divide each chunk. Use a times table fact and place value.
60 ÷ 6 = 10 (because 10 × 6 = 60)
18 ÷ 6 = 3 (because 3 × 6 = 18)
Add the answers. Add up the chunk answers.
10 + 3 = 13
Write the answer.
78 ÷ 6 = 13

Check. Multiply back: 13 × 6 = 10 × 6 + 3 × 6 = 60 + 18 = 78. ✓

Watch out. Make sure the chunks add back to 78: 60 + 18 = 78.

Worked example 4.4.4

Use the bus stop method to work out 52 ÷ 4.

Key words. Short division (the bus stop method) is a written way to divide. The number you divide goes inside the bus stop and the number you divide by goes outside. A remainder is what is left over when a number does not divide exactly.

Method. Draw the bus stop. Divide one digit at a time, starting on the left. If there is a remainder, carry it to the next digit as extra ones, tens or hundreds.

Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

4 5 2
Divide the tens. Always start on the left, with the biggest place value.
5 ÷ 4 = 1, remainder 1
Write 1 above the 5.
Exchange the remainder: 1 ten becomes 10 ones. Carry 1 to the next digit.
1
4 5 ¹2
Divide the ones. Move to the next digit on the right. Add the 1 that was carried.
The carried 1 makes 10 ones. With the 2 ones there are 10 + 2 = 12 ones.
12 ÷ 4: 3 × 4 = 12. Write 3 above the 2.
Nothing is left over, so there is no remainder.
1 3
4 5 ¹2
Read the answer. Read the digits above the line from left to right.
1 3
4 5 ¹2

52 ÷ 4 = 13

Check. Multiply back: 13 × 4 = 10 × 4 + 3 × 4 = 40 + 12 = 52. ✓

Watch out. Start on the left, with the biggest place value. Carry any remainder to the next digit.

Worked example 4.4.5

Use the bus stop method to work out 96 ÷ 3.

Method. Draw the bus stop. Divide one digit at a time, starting on the left. If there is a remainder, carry it to the next digit as extra ones, tens or hundreds.

Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

3 9 6
Divide the tens. Always start on the left, with the biggest place value.
9 ÷ 3 = 3
Write 3 above the 9.
Divide the ones. Move to the next digit on the right.
Nothing was carried, so there are just 6 ones.
6 ÷ 3: 2 × 3 = 6. Write 2 above the 6.
Nothing is left over, so there is no remainder.
Read the answer. Read the digits above the line from left to right.
3 2
3 9 6

96 ÷ 3 = 32

Check. Multiply back: 32 × 3 = 30 × 3 + 2 × 3 = 90 + 6 = 96. ✓

Watch out. Start on the left, with the biggest place value. Carry any remainder to the next digit.

Worked example 4.4.6

Use the bus stop method to work out 756 ÷ 6.

Method. Draw the bus stop. Divide one digit at a time, starting on the left. If there is a remainder, carry it to the next digit as extra ones, tens or hundreds.

Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

6 7 5 6
Divide the hundreds. Always start on the left, with the biggest place value.
7 ÷ 6 = 1, remainder 1
Write 1 above the 7.
Exchange the remainder: 1 hundred becomes 10 tens. Carry 1 to the next digit.
1
6 7 ¹5 6
Divide the tens. Move to the next digit on the right. Add the 1 that was carried.
The carried 1 makes 10 tens. With the 5 tens there are 10 + 5 = 15 tens.
15 ÷ 6: 2 × 6 = 12, and 15 − 12 = 3. Write 2 above the 5.
Carry the remainder 3 to the next digit.
1 2
6 7 ¹5 ³6
Divide the ones. Move to the next digit on the right. Add the 3 that was carried.
The carried 3 makes 30 ones. With the 6 ones there are 30 + 6 = 36 ones.
36 ÷ 6: 6 × 6 = 36. Write 6 above the 6.
Nothing is left over, so there is no remainder.
1 2 6
6 7 ¹5 ³6
Read the answer. Read the digits above the line from left to right.
1 2 6
6 7 ¹5 ³6

756 ÷ 6 = 126

Check. Multiply back: 126 × 6 = 100 × 6 + 20 × 6 + 6 × 6 = 600 + 120 + 36 = 756. ✓

Watch out. Start on the left, with the biggest place value. Carry any remainder to the next digit.

Worked example 4.4.7

Use the bus stop method to work out 285 ÷ 5. The first digit is smaller than 5.

Method. Draw the bus stop. Divide one digit at a time, starting on the left. If there is a remainder, carry it to the next digit as extra ones, tens or hundreds.

Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

5 2 8 5
Divide the hundreds. Always start on the left, with the biggest place value.
2 ÷ 5: 2 is smaller than 5, so we cannot make a whole hundred for each share. Write 0 (this leading 0 is not part of the answer).
Exchange the 2 hundreds for 20 tens and carry them to the next digit.
Divide the tens. Move to the next digit on the right. Add the 2 that was carried.
The carried 2 makes 20 tens. With the 8 tens there are 20 + 8 = 28 tens.
28 ÷ 5: 5 × 5 = 25, and 28 − 25 = 3. Write 5 above the 8.
Carry the remainder 3 to the next digit.
Divide the ones. Move to the next digit on the right. Add the 3 that was carried.
The carried 3 makes 30 ones. With the 5 ones there are 30 + 5 = 35 ones.
35 ÷ 5: 7 × 5 = 35. Write 7 above the 5.
Nothing is left over, so there is no remainder.
Read the answer. Ignore any 0 at the front of the answer.
0 5 7
5 2 ²8 ³5

285 ÷ 5 = 57

Check. Multiply back: 57 × 5 = 50 × 5 + 7 × 5 = 250 + 35 = 285. ✓

Watch out. If the first digit is smaller than 5, write 0 above it (not in the final answer) and carry the digit to the next place.

Worked example 4.4.8

Work out 618 ÷ 6. There will be a zero in the answer.

Method. Draw the bus stop. Divide one digit at a time, starting on the left. If there is a remainder, carry it to the next digit as extra ones, tens or hundreds.

Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

6 6 1 8
Divide the hundreds. Always start on the left, with the biggest place value.
6 ÷ 6 = 1
Write 1 above the 6.
Divide the tens. Move to the next digit on the right.
Nothing was carried, so there are just 1 tens.
1 ÷ 6: 1 is smaller than 6, so write 0.
Carry the remainder 1 to the next digit.
Divide the ones. Move to the next digit on the right. Add the 1 that was carried.
The carried 1 makes 10 ones. With the 8 ones there are 10 + 8 = 18 ones.
18 ÷ 6: 3 × 6 = 18. Write 3 above the 8.
Nothing is left over, so there is no remainder.
Read the answer. Read the digits above the line from left to right.
1 0 3
6 6 1 ¹8

618 ÷ 6 = 103

Check. Multiply back: 103 × 6 = 100 × 6 + 3 × 6 = 600 + 18 = 618. ✓

Watch out. Start on the left, with the biggest place value. Carry any remainder to the next digit.

Watch out

Start on the left (the opposite of addition and multiplication!). And do not lose the remainder: carry it to the next digit. When a step has nothing to share (like 1 ÷ 6) you must still write 0 in the answer, or the digits slip into the wrong places: 618 ÷ 6 = 103, not 13.

4.5

Division with remainders

Year 4
Short division with remainders for 2- and 3-digit numbers by 1-digit numbers, Interpret remainders in context

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Division with remaindersComing soon

Sometimes a number does not divide exactly. Take 29 ÷ 4. The 4 times table goes 4, 8, 12, 16, 20, 24, 28, 32. 4 × 7 = 28, so 7 groups of 4 fit into 29, but there is 1 left over. The number left over is called the remainder. We write 29 ÷ 4 = 7 remainder 1, or 7 r 1.

29 split into 7 groups of 4 with 1 left over.

The remainder is always smaller than the number you divide by. If it was 4 or more, you could make another group. So when you divide by 4, the remainder can only be 0, 1, 2 or 3.

The bus stop method with a remainder. Divide one digit at a time and carry each remainder, just as before. The last remainder is the answer’s remainder. Try 297 ÷ 4. 2 ÷ 4 = 0 remainder 2, so write 0 and carry 2. 29 ÷ 4 = 7 remainder 1: write 7, carry 1. 17 ÷ 4 = 4 remainder 1: write 4. The last remainder is 1.

	0	7	4	r 1
4	2	²9	¹7

297 ÷ 4 = 74 remainder 1.

Check with multiplication. Multiply the answer by the number you divided by, then add the remainder. 74 × 4 = 296, and 296 + 1 = 297. ✓

What does the remainder mean in a story? Read the question carefully. 50 pupils go on a trip. Each minibus holds 8 pupils. 50 ÷ 8 = 6 remainder 2.

50 pupils: 6 full minibuses and 2 pupils left over.

The question asks…	What to do with the remainder	Answer for 50 pupils, 8 in a minibus
How many are left over?	The answer is the remainder.	2 pupils
How many full minibuses?	Ignore the remainder (round down).	6 minibuses
How many minibuses are needed for everyone?	The leftovers need a minibus too. Add 1 (round up).	7 minibuses
Share equally; how many each and how many left?	Give both parts.	6 each, 2 left over

Key words. A remainder is what is left over after dividing. Round up means choose the next whole number. Round down means ignore the leftover part.

Key ideas

A remainder is what is left over: 29 ÷ 4 = 7 remainder 1.
The remainder is always smaller than the number you divide by.
Bus stop: carry each remainder; the last remainder goes at the end.
Check: (answer × divisor) + remainder = the number you started with.
Read the story: left over (the remainder), full groups (ignore it), enough for everyone (round up), or both parts.

Worked example 4.5.1

Work out 29 ÷ 4.

Method. Draw the bus stop. Divide one digit at a time, starting on the left. If there is a remainder, carry it to the next digit as extra ones, tens or hundreds.

Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

4 2 9
Divide the tens. Always start on the left, with the biggest place value.
2 ÷ 4: 2 is smaller than 4, so we cannot make a whole ten for each share. Write 0 (this leading 0 is not part of the answer).
Exchange the 2 tens for 20 ones and carry them to the next digit.
Divide the ones. Move to the next digit on the right. Add the 2 that was carried.
The carried 2 makes 20 ones. With the 9 ones there are 20 + 9 = 29 ones.
29 ÷ 4: 7 × 4 = 28, and 29 − 28 = 1. Write 7 above the 9.
The remainder is 1.
Read the answer. Ignore any 0 at the front of the answer.
0 7 r 1
4 2 ²9

29 ÷ 4 = 7 remainder 1

Check. Multiply back: 7 × 4 = 28
28 + 1 (the remainder) = 29. ✓

Watch out. The first digit is smaller than 4: write 0 and carry it. The remainder is always smaller than 4.

Worked example 4.5.2

Work out 85 ÷ 6. Give your answer with a remainder.

Method. Draw the bus stop. Divide one digit at a time, starting on the left. If there is a remainder, carry it to the next digit as extra ones, tens or hundreds.

Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

6 8 5
Divide the tens. Always start on the left, with the biggest place value.
8 ÷ 6 = 1, remainder 2
Write 1 above the 8.
Exchange the remainder: 2 tens become 20 ones. Carry 2 to the next digit.
1
6 8 ²5
Divide the ones. Move to the next digit on the right. Add the 2 that was carried.
The carried 2 makes 20 ones. With the 5 ones there are 20 + 5 = 25 ones.
25 ÷ 6: 4 × 6 = 24, and 25 − 24 = 1. Write 4 above the 5.
The remainder is 1.
1 4 r 1
6 8 ²5
Read the answer. Read the digits above the line from left to right.
1 4 r 1
6 8 ²5

85 ÷ 6 = 14 remainder 1

Check. Multiply back: 14 × 6 = 10 × 6 + 4 × 6 = 60 + 24 = 84
84 + 1 (the remainder) = 85. ✓

Watch out. The remainder must always be smaller than the number you divide by (6). If it is not, you can divide again.

Worked example 4.5.3

Work out 297 ÷ 4. Give your answer with a remainder.

Method. Draw the bus stop. Divide one digit at a time, starting on the left. If there is a remainder, carry it to the next digit as extra ones, tens or hundreds.

Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

4 2 9 7
Divide the hundreds. Always start on the left, with the biggest place value.
2 ÷ 4: 2 is smaller than 4, so we cannot make a whole hundred for each share. Write 0 (this leading 0 is not part of the answer).
Exchange the 2 hundreds for 20 tens and carry them to the next digit.
0
4 2 ²9 7
Divide the tens. Move to the next digit on the right. Add the 2 that was carried.
The carried 2 makes 20 tens. With the 9 tens there are 20 + 9 = 29 tens.
29 ÷ 4: 7 × 4 = 28, and 29 − 28 = 1. Write 7 above the 9.
Carry the remainder 1 to the next digit.
0 7
4 2 ²9 ¹7
Divide the ones. Move to the next digit on the right. Add the 1 that was carried.
The carried 1 makes 10 ones. With the 7 ones there are 10 + 7 = 17 ones.
17 ÷ 4: 4 × 4 = 16, and 17 − 16 = 1. Write 4 above the 7.
The remainder is 1.
0 7 4 r 1
4 2 ²9 ¹7
Read the answer. Ignore any 0 at the front of the answer.
0 7 4 r 1
4 2 ²9 ¹7

297 ÷ 4 = 74 remainder 1

Check. Multiply back: 74 × 4 = 70 × 4 + 4 × 4 = 280 + 16 = 296
296 + 1 (the remainder) = 297. ✓

Watch out. The first digit is smaller than 4: write 0 and carry it. The remainder is always smaller than 4.

Worked example 4.5.4

Work out 803 ÷ 5. Look out for a 0 in the answer.

Method. Draw the bus stop. Divide one digit at a time, starting on the left. If there is a remainder, carry it to the next digit as extra ones, tens or hundreds.

Draw the bus stop. The number we are dividing goes inside. The number we divide by goes outside, on the left.

5 8 0 3
Divide the hundreds. Always start on the left, with the biggest place value.
8 ÷ 5 = 1, remainder 3
Write 1 above the 8.
Exchange the remainder: 3 hundreds become 30 tens. Carry 3 to the next digit.
Divide the tens. Move to the next digit on the right. Add the 3 that was carried.
The carried 3 makes 30 tens. With the 0 tens there are 30 + 0 = 30 tens.
30 ÷ 5: 6 × 5 = 30. Write 6 above the 0.
Divide the ones. Move to the next digit on the right.
Nothing was carried, so there are just 3 ones.
3 ÷ 5: 3 is smaller than 5, so write 0.
The remainder is 3.
Read the answer. Read the digits above the line from left to right.
1 6 0 r 3
5 8 ³0 3

803 ÷ 5 = 160 remainder 3

Check. Multiply back: 160 × 5 = 100 × 5 + 60 × 5 = 500 + 300 = 800
800 + 3 (the remainder) = 803. ✓

Watch out. The remainder must always be smaller than the number you divide by (5). If it is not, you can divide again.

Worked example 4.5.5

50 pupils go on a trip. Each minibus holds 8 pupils. How many minibuses are needed so that everyone has a seat?

Key words. A remainder is what is left over after you divide. What you do with it depends on the story: ignore it, round up, or say how many are left over.

Method. Divide to find how many groups of the number fit. Then think about what the remainder means in the story.

Choose the operation. We are putting 50 into equal groups of 8, so we divide.
50 ÷ 8
Work it out with the bus stop. Divide one digit at a time and keep the remainder.
50 ÷ 8 = 6 remainder 2
0 6 r 2
8 5 ⁵0
Decide what the remainder means. Every one must be in a group. The remainder still needs a group of its own, so we need one more group. We round UP.
6 minibuses are full. The last 2 pupils need another one: 6 + 1 = 7
Write the answer. Answer the question that was asked.
7 minibuses.

Check. Multiply back: 6 × 8 = 48, and 48 + 2 = 50. ✓

Watch out. Do not forget the leftovers. They still need somewhere to go, so we round up, not down.

Worked example 4.5.6

A farmer packs eggs in boxes of 6. She has 100 eggs. How many full boxes can she make?

Key words. A remainder is what is left over after you divide. What you do with it depends on the story: ignore it, round up, or say how many are left over.

Method. Divide to find how many groups of the number fit. Then think about what the remainder means in the story.

Choose the operation. We are putting 100 into equal groups of 6, so we divide.
100 ÷ 6
Work it out with the bus stop. Divide one digit at a time and keep the remainder.
100 ÷ 6 = 16 remainder 4
0 1 6 r 4
6 1 ¹0 ⁴0
Decide what the remainder means. The question asks how many FULL groups there are. The remainder does not make a full group, so we ignore it.
16 boxes and 4 eggs left over. We only count the whole ones: 16.
Write the answer. Answer the question that was asked.
16 boxes.

Check. Multiply back: 16 × 6 = 96, and 96 + 4 = 100. ✓

Watch out. The remainder is not a full group. Do not add 1 here: that is only for when everything must fit.

Worked example 4.5.7

A teacher puts 53 crayons into bags of 6. How many crayons are left over?

Key words. A remainder is what is left over after you divide. What you do with it depends on the story: ignore it, round up, or say how many are left over.

Method. Divide to find how many groups of the number fit. Then think about what the remainder means in the story.

Choose the operation. We are putting 53 into equal groups of 6, so we divide.
53 ÷ 6
Work it out with the bus stop. Divide one digit at a time and keep the remainder.
53 ÷ 6 = 8 remainder 5
0 8 r 5
6 5 ⁵3
Decide what the remainder means. The question asks how many are LEFT OVER, so the answer is the remainder.
remainder = 5
Write the answer. Answer the question that was asked.
There are 5 crayons left over.

Check. Multiply back: 8 × 6 = 48, and 48 + 5 = 53. ✓

Watch out. The answer is the remainder, not the number you get when you divide.

Worked example 4.5.8

38 stickers are shared equally between 5 children. How many does each child get, and how many stickers are left over?

Key words. A remainder is what is left over after you divide. What you do with it depends on the story: ignore it, round up, or say how many are left over.

Method. Divide to find how many groups of the number fit. Then think about what the remainder means in the story.

Choose the operation. We are putting 38 into equal groups of 5, so we divide.
38 ÷ 5
Work it out with the bus stop. Divide one digit at a time and keep the remainder.
38 ÷ 5 = 7 remainder 3
0 7 r 3
5 3 ³8
Decide what the remainder means. The items are shared equally. The remainder is what cannot be shared out, so it is left over.
7 each, remainder 3
Write the answer. Answer the question that was asked.
Each gets 7 and 3 stickers are left over.

Check. Multiply back: 7 × 5 = 35, and 35 + 3 = 38. ✓

Watch out. Write both parts: how many each, and how many are left over.

Worked example 4.5.9

Find the missing number. 47 ÷ 5 = 9 remainder ?

Method. A division with a remainder can be checked with multiplication: (answer × divisor) + remainder = the number you started with.

Multiply the answer by the divisor. 9 lots of 5 make the biggest multiple of 5 that fits into 47.
9 × 5 = 45
Take it away from the number. What is left over is the remainder.
47 − 45 = 2
Check the remainder is small enough. A remainder must be smaller than 5.
2 is smaller than 5. ✓

Check. 9 × 5 + 2 = 45 + 2 = 47. ✓

Watch out. Check the remainder is smaller than the divisor.

Watch out

Do not always round up, or always ignore the remainder. Read the question: “how many full boxes?” ignores the remainder, but “how many boxes do we need?” rounds up. And a remainder must be smaller than the number you divide by. If it is not, you can still make another group.

4.6

Scaling, correspondence and multiplying three numbers

Year 4
Integer scaling problems, Correspondence problems: n objects are connected to m objects, Multiply together three numbers

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Scaling, correspondence and multiplying three numbersComing soon

Scaling. Alex has 6 marbles. Ben has 4 times as many marbles as Alex. “4 times as many” means 4 equal groups, and each group is the same size as Alex’s 6. So Ben has 4 × 6 = 24 marbles. We can draw it as a bar model: Alex has one box, and Ben has four boxes the same size.

Ben has 4 times as many as Alex.

The same picture helps you go backwards. If Ben has 24 marbles, which is 4 times as many as Alex, then 4 boxes make 24, so one box is 24 ÷ 4 = 6. Alex has 6.

Be careful: “4 times as many” is multiplying by 4. “4 more” is adding 4. They are very different! Alex has 6, so 4 more is 10 but 4 times as many is 24. We use scaling for lengths, heights, masses and money too: “3 times as long”, “5 times as heavy”, “twice the price”.

Correspondence. Priya has 3 hats and 5 tops. She picks one hat and one top. How many different outfits can she make? Each hat can go with any of the 5 tops. That is 5 outfits for the first hat, 5 for the second and 5 for the third. Make a table: each box is one outfit.

hat \ top	stripy	spotty	plain	checked	yellow
red hat	1	2	3	4	5
blue hat	6	7	8	9	10
green hat	11	12	13	14	15

There are 3 rows of 5 boxes. It is an array! So the number of outfits is 3 × 5 = 15. We say that 3 objects are connected to 5 objects in 3 × 5 ways. It works for roads, meals, sandwiches and lots more. If Priya also has 2 scarves, each of the 15 outfits can have either scarf: 15 × 2 = 30.

Multiplying three numbers. You can multiply three numbers in any order. Look for two numbers that make an easy answer, like a pair that makes 10 or 100, and multiply those first. 5 × 17 × 2: 5 × 2 = 10, then 17 × 10 = 170. Much easier than 5 × 17 first!

Key words. Scaling means making an amount a number of times bigger or smaller. A combination is one choice from each group put together.

Key ideas

“n times as many” means n equal groups: multiply the small amount by n.
To find the small amount, divide the big amount by n.
“More than” means add. “Times as many” means multiply.
If every one of n things can go with every one of m things, there are n × m combinations.
Three groups of choices: multiply all three numbers.
You can multiply three numbers in any order. Make 10 or 100 first if you can.

Worked example 4.6.1

Maya has 14 stickers. Her dad has 6 times as many stickers as Maya. How many stickers does her dad have?

Key words. Scaling means making something a number of times bigger. “3 times as many” means 3 equal groups, so you multiply by 3. To go back, divide by 3.

Method. 6 times as many means 6 equal groups, each the same size as the smaller amount. Multiply the smaller amount by 6.

Draw a bar model. Maya has one box. Dad has 6 boxes the same size.
Choose the operation. 6 equal boxes: multiply the size of one box by 6.
14 × 6
Work it out. Use a written method if you need to.
14 × 6 = 84
T O
1 4
× 6
8 4
2
Write the answer. Include the unit.
84 stickers

Check. Divide back: 84 ÷ 6 = 14. ✓

Watch out. “6 times as many” is multiplying by 6. It is not adding 6.

Worked example 4.6.2

Zainab has £25. Liam has 4 times as much money as Zainab. How much money does Liam have?

Key words. Scaling means making something a number of times bigger. “3 times as many” means 3 equal groups, so you multiply by 3. To go back, divide by 3.

Method. 4 times as many means 4 equal groups, each the same size as the smaller amount. Multiply the smaller amount by 4.

Draw a bar model. Zainab has one box. Liam has 4 boxes the same size.
Choose the operation. 4 equal boxes: multiply the size of one box by 4.
£25 × 4
Work it out. Use a written method if you need to.
£25 × 4 = £100
H T O
2 5
× 4
1 0 0
1 2
Write the answer. Include the unit.
£100

Check. Divide back: £100 ÷ 4 = £25. ✓

Watch out. “4 times as many” is multiplying by 4. It is not adding 4.

Worked example 4.6.3

Noah has 96 books. He has 4 times as many books as Priya. How many books does Priya have?

Key words. Scaling means making something a number of times bigger. “3 times as many” means 3 equal groups, so you multiply by 3. To go back, divide by 3.

Method. Noah has 4 times as many as Priya. So Noah’s amount is split into 4 equal boxes and Priya has one box. Divide by 4.

Draw a bar model. Noah has 4 equal boxes. Priya has one box the same size.
Choose the operation. We know the total of the 4 boxes and want the size of one box: divide by 4.
96 ÷ 4
Work it out. Use the bus stop method.
96 ÷ 4 = 24
2 4
4 9 ¹6
Write the answer. Include the unit.
24 books

Check. Multiply back: 24 × 4 = 96. ✓

Watch out. The one with “times as many” has the bigger amount. To find the smaller amount, divide, do not multiply.

Worked example 4.6.4

Priya has 3 hats and 5 tops. She wears one hat and one top. How many different outfits can she make?

Key words. A combination is one choice from each group put together. To find how many combinations there are, multiply the numbers of choices.

Method. Each choice from one group can be paired with every choice from the other group. So multiply the numbers of choices together.

Take just one hat. Pick the first hat. It can go with any of the 5 tops, so that makes 5 outfits.
1 hat → 5 outfits
Do the same for every hat. All the hats have the same choices. There are 3 hats, so there are 3 groups of 5 outfits.
5 + 5 + 5 = 15
Show every choice in a table. Each box is one outfit: one hat with one top. Count the boxes: it is an array.
hat \ top top 1 top 2 top 3 top 4 top 5
hat 1 1 2 3 4 5
hat 2 6 7 8 9 10
hat 3 11 12 13 14 15
Multiply. 3 rows of 5 is a multiplication.
3 × 5 = 15
Write the answer. Say what the number means.
15 different outfits

Check. Swap the numbers: 5 × 3 = 15. ✓

Watch out. Do not add the numbers. 3 + 5 = 8 would only count each item once. Every hat goes with every top, so multiply.

Worked example 4.6.5

An ice cream shop has 4 flavours and 6 toppings. A customer chooses one flavour and one topping. How many different ice creams can be made?

Key words. A combination is one choice from each group put together. To find how many combinations there are, multiply the numbers of choices.

Method. Each choice from one group can be paired with every choice from the other group. So multiply the numbers of choices together.

Take just one flavour. Pick the first flavour. It can go with any of the 6 toppings, so that makes 6 ice creams.
1 flavour → 6 ice creams
Do the same for every flavour. All the flavours have the same choices. There are 4 flavours, so there are 4 groups of 6 ice creams.
6 + 6 + 6 + 6 = 24
Show every choice in a table. Each box is one ice cream: one flavour with one topping. Count the boxes: it is an array.
flavour \ topping topping 1 topping 2 topping 3 topping 4 topping 5 topping 6
flavour 1 1 2 3 4 5 6
flavour 2 7 8 9 10 11 12
flavour 3 13 14 15 16 17 18
flavour 4 19 20 21 22 23 24
Multiply. 4 rows of 6 is a multiplication.
4 × 6 = 24
Write the answer. Say what the number means.
24 different ice creams

Check. Swap the numbers: 6 × 4 = 24. ✓

Watch out. Do not add the numbers. 3 + 5 = 8 would only count each item once. Every flavour goes with every topping, so multiply.

Worked example 4.6.6

There are 3 roads from Ashby to Bridge and 4 roads from Bridge to Carton. How many different routes are there from Ashby to Carton through Bridge?

Key words. A combination is one choice from each group put together. To find how many combinations there are, multiply the numbers of choices.

Method. Each choice from one group can be paired with every choice from the other group. So multiply the numbers of choices together.

Take just one road. Pick the first road. It can go with any of the 4 roads, so that makes 4 routes.
1 road → 4 routes
Do the same for every road. All the roads have the same choices. There are 3 roads, so there are 3 groups of 4 routes.
4 + 4 + 4 = 12
Multiply. 3 rows of 4 is a multiplication.
3 × 4 = 12
Write the answer. Say what the number means.
12 different routes

Check. Swap the numbers: 4 × 3 = 12. ✓

Watch out. Do not add the numbers. 3 + 5 = 8 would only count each item once. Every road goes with every road, so multiply.

Worked example 4.6.7

Amira has 3 skirts, 4 tops and 2 pairs of shoes. How many different outfits can she make with one of each?

Key words. A combination is one choice from each group put together. To find how many combinations there are, multiply the numbers of choices.

Method. Each choice from one group can be paired with every choice from the other group. So multiply the numbers of choices together.

Find how many choices there are for the first two. Each of the 3 skirts can go with each of the 4 tops.
3 × 4 = 12
Now bring in the pairs of shoes. Each of those 12 choices can go with each of the 2 pairs of shoes.
12 × 2 = 24
Write the answer. Say what the number means.
24 different outfits

Check. Multiply in a different order: 4 × 2 = 8, then 8 × 3 = 24. ✓

Watch out. Do not add the numbers. 3 + 5 = 8 would only count each item once. Every skirt goes with every top, so multiply.

Worked example 4.6.8

A menu lets you choose one starter and one main. There are 36 different meals and 4 starters. How many mains are there?

Method. Each of the 4 starters goes with every main, so the 36 meals are 4 equal groups. Division undoes multiplication.

Write it as a multiplication with a missing number. 4 starters × the number of mains = 36 meals.
4 × ? = 36
Divide to find the missing number. Share the 36 meals between the 4 starters.
36 ÷ 4 = 9
0 9
4 3 ³6
Write the answer.
9 mains

Check. Multiply back: 4 × 9 = 36. ✓

Watch out. Use the opposite operation: if the total is known, divide by the number you know.

Worked example 4.6.9

Work out 5 × 17 × 2.

Key words. You can multiply three numbers in any order and the answer is the same.

Method. You can multiply three numbers in any order. Look for two numbers that make an easy answer (like a pair that makes 10 or 100) and multiply those first.

Look for a pair that is easy to multiply. 5 × 2 = 10, and multiplying by 10 is easy.
5 × 2 = 10
Multiply by the last number, 17. Multiplying by 10 moves every digit 1 place to the left.
10 × 17: move the digits of 17 1 place to the left.
10 × 17 = 170
Write the answer.
5 × 17 × 2 = 170

Check. Try a different order: 34 × 5 = 170. ✓

Watch out. You can multiply in any order. Pick the pair that gives an easy answer first.

Worked example 4.6.10

Work out 25 × 4 × 6.

Key words. You can multiply three numbers in any order and the answer is the same.

Method. You can multiply three numbers in any order. Look for two numbers that make an easy answer (like a pair that makes 10 or 100) and multiply those first.

Look for a pair that is easy to multiply. 25 × 4 = 100, and multiplying by 100 is easy.
25 × 4 = 100
Multiply by the last number, 6. Multiplying by 100 moves every digit 2 places to the left.
100 × 6: move the digits of 6 2 places to the left.
100 × 6 = 600
Write the answer.
25 × 4 × 6 = 600

Check. Try a different order: 24 × 25 = 600. ✓

Watch out. You can multiply in any order. Pick the pair that gives an easy answer first.

Watch out

“Times as many” is not “more”. 4 times as many as 6 is 24 (multiply). 4 more than 6 is 10 (add). For outfits, do not add the numbers: 3 hats and 5 tops make 3 × 5 = 15 outfits, not 3 + 5 = 8, because every hat goes with every top.

4.7

Word problems with bar models

Year 4
Solve problems involving multiplication and division, including scaling and correspondence

Printable: Worksheet (PDF) · Answers (PDF) · Open this lesson on its own page

Video lesson: Word problems with bar modelsComing soon

Real problems often take more than one step. Use these five steps every time.

Read the problem. What do you know? What do you need to find out?
Draw a bar model.
Choose which operations to use.
Work it out, one step at a time, with a written method.
Check your answer. Does it make sense? Does it answer the question?

A bar model is a picture of the numbers. For equal groups, draw equal boxes: the problem is a multiplication or a division. For a whole made of parts, draw the parts side by side: add or subtract. To compare two amounts, draw two bars, one above the other.

Equal groups: 6 × 24

Parts: 144 − 105

Two bar models. The first needs a multiplication, the second needs a subtraction.

Some problems use two operations. A baker makes 6 trays of 24 buns, then sells 105. First multiply (6 × 24 = 144 buns baked). Then subtract (144 − 105 = 39 buns left). Always check the question at the end: the answer to the first step is not always the answer to the question.

Key words. A bar model is a drawing of boxes that shows the numbers in a problem. A multi-step problem needs more than one calculation.

Key ideas

Read, draw, choose, work, check.
Equal groups: multiply to find the total, divide to find the size of a group or the number of groups.
Whole made of parts: add the parts, or subtract a part from the whole.
Write the working for each step and the unit with each answer.
Check: does the answer fit the story and answer the question?

Worked example 4.7.1

A baker bakes 6 trays of 24 buns. She sells 105 of the buns. How many buns are left?

Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.

Draw a bar model. The whole is the buns baked. 105 are sold. We want the part that is left.
Step 1: Find how many buns the baker makes. 6 trays with 24 buns on each, so multiply.
24 × 6 = 144
H T O
2 4
× 6
1 4 4
1 2
Step 2: Find how many buns are left. She sells 105 buns, so take them away from the total.
144 − 105 = 39
H T O
1 34 144
− 1 0 5
3 9
Write the answer. Include the unit and make sure it answers the question.
39 buns left

Check. Add back: 39 + 105 = 144, and 144 ÷ 6 = 24. ✓

Watch out. The answer to step 1 (144) is not the answer to the question. Keep going to step 2.

Worked example 4.7.2

4 tickets for a show cost £36 altogether. How much do 7 tickets cost?

Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.

Draw a bar model. 4 equal tickets share the £36. We find one ticket first.
Step 1: Find the cost of 1 ticket. 4 tickets cost £36 altogether, so share £36 into 4 equal parts.
£36 ÷ 4 = £9
0 9
4 3 ³6
Step 2: Find the cost of 7 tickets. 7 tickets cost 7 times as much as 1 ticket.
£9 × 7 = £63
Write the answer. Include the unit and make sure it answers the question.
£63

Check. Another way: 7 tickets is 4 tickets plus 3 tickets: £36 + 3 × £9 = £36 + £27 = £63. ✓

Watch out. Find the cost of ONE first. Then multiply by the number of tickets you need.

Worked example 4.7.3

Jack has 138 stickers. Mia has 114 stickers. They put all their stickers into 6 equal packs. How many stickers are in each pack?

Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.

Draw a bar model. All 252 stickers are shared into 6 equal packs.
Step 1: Find how many stickers Jack and Mia have altogether. Put the two amounts together: add.
138 + 114 = 252
H T O
1 3 8
+ 1 1 4
2 5 2
1
Step 2: Share the stickers equally between 6 packs. Equal sharing means divide.
252 ÷ 6 = 42
0 4 2
6 2 ²5 ¹2
Write the answer. Include the unit and make sure it answers the question.
42 stickers in each pack

Check. Multiply back: 42 × 6 = 252, and 138 + 114 = 252. ✓

Watch out. Add first, then divide. If you divide 138 by 6 you forget Mia’s stickers.

Worked example 4.7.4

Sam buys 5 pens at 85p each. He pays with a £5 note. How much change does he get? (£1 = 100p)

Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.

Draw a bar model. The whole is the £5 note (500p). Part of it pays for the pens. The rest is the change.
Step 1: Find the cost of 5 pens. 5 pens at 85p each, so multiply.
85p × 5 = 425p
H T O
8 5
× 5
4 2 5
4 2
Step 2: Find the change. The change is the money paid take away the cost.
500p − 425p = 75p
H T O
45 90 100
− 4 2 5
7 5
Write the answer. Include the unit and make sure it answers the question.
75p change

Check. Add back: 425p + 75p = 500p. ✓

Watch out. Keep the units the same: £5 is 500p. Work in pence all the way through.

Worked example 4.7.5

52 pupils go on a trip. Each minibus holds 8 pupils. Each minibus costs £35 to hire. How much does it cost to hire enough minibuses for everyone?

Method. First find how many minibuses are needed (divide, and think about the remainder). Then multiply by the cost of one minibus.

Draw a bar model. The 52 pupils are put into groups of 8, with some left over.
Step 1: divide the pupils by 8. Each minibus holds 8 pupils.
52 ÷ 8 = 6 remainder 4
0 6 r 4
8 5 ⁵2
Decide what to do with the remainder. 4 pupils are left over, but they still need to travel. We need one more minibus: round up.
6 + 1 = 7 minibuses
Step 2: find the cost. Each minibus costs £35 to hire, so multiply 7 by 35.
£35 × 7 = £245
H T O
3 5
× 7
2 4 5
2 3
Write the answer.
£245

Check. 7 × 8 = 56, which is enough for 52 pupils, but 6 × 8 = 48 is not enough. ✓

Watch out. Everyone must have a seat, so round the number of minibuses UP.

Worked example 4.7.6

Amir has 3 times as many cards as Bea. Together they have 84 cards. How many cards does each have?

Method. Read the problem, draw a bar model, then decide which operations to use. Work out one step at a time.

Draw a bar model. Bea has one box. Amir has three boxes the same size. Together they have 84.
Step 1: Find how many equal boxes there are altogether. Bea has 1 box and Amir has 3 boxes the same size.
1 + 3 = 4
Step 2: Find the size of one box. The 4 equal boxes share the 84 cards, so divide.
84 ÷ 4 = 21
2 1
4 8 4
Step 3: Find how many cards Amir has. Amir has 3 boxes.
21 × 3 = 63
T O
2 1
× 3
6 3
Write the answer. Include the unit and make sure it answers the question.
Bea 21 cards, Amir 63 cards

Check. Add: 21 + 63 = 84. And 63 is 3 times 21. ✓

Watch out. Amir has 3 times as many as Bea, so there are 4 boxes in the whole bar (1 + 3), not 3.

Watch out

The answer to the first step is not always the answer to the question. Look back at what the question asks for. Also keep your units the same: pounds with pounds, pence with pence. And do not add when the problem says “times as many”.

Chapter summary

Skill	Method
× and ÷ by 10, 100	Digits move 1 place (10) or 2 places (100): left to multiply, right to divide. Zeros are place holders. 35 × 100 = 3,500; 3,500 ÷ 10 = 350.
Distributive law	34 × 6 = (30 + 4) × 6 = 30 × 6 + 4 × 6 = 180 + 24 = 204. Near a round number: 29 × 6 = 30 × 6 − 6.
Grid method	Partition into hundreds, tens and ones. Multiply each part. Add the partial products. Keep the zeros.
Expanded column	Multiply each digit by its value: 7 × 6, 40 × 6, 300 × 6. Write each on its own line. Add.
Short multiplication	Start with the ones. Write the ones digit of each answer, carry the tens digit and add it in the next column. Estimate to check.
Division with place value	Split into chunks: 84 ÷ 4 = 80 ÷ 4 + 4 ÷ 4. Use known facts: 2,400 ÷ 6 = 400.
Short division (bus stop)	Start on the left. Divide each digit, carry the remainder to the next digit. Write 0 when a step is too small. Check by multiplying back.
Remainders	The remainder is what is left over and is smaller than the divisor. Check: answer × divisor + remainder = start.
Remainders in context	Left over: the remainder. Full groups: ignore it. Enough for everyone: round up. Sharing: say both parts.
Scaling	“n times as many”: multiply. To go back, divide. Do not confuse with “n more” (add).
Correspondence	n things each connected to m things make n × m combinations (3 hats and 5 tops: 15 outfits). Three groups: multiply all three.
Three numbers	Multiply in any order. Look for a pair that makes 10 or 100 first.
Word problems	Read, draw a bar model, choose the operations, work one step at a time, check the answer fits the question.

Can you…?

I can multiply and divide whole numbers by 10 and 100 by moving the digits.
I can use partitioning and the distributive law to multiply a two-digit number by a one-digit number in my head.
I can use the grid method and the expanded method to multiply two-digit and three-digit numbers by one digit.
I can use short multiplication with carrying, and check with an estimate.
I can divide using place value and known facts.
I can use the bus stop method to divide two-digit and three-digit numbers by one digit, with and without a remainder.
I can decide what a remainder means in a story.
I can solve problems with “n times as many”.
I can solve problems like 3 hats and 5 tops (correspondence).
I can multiply three numbers by choosing an easy pair first.
I can solve multi-step word problems with a bar model.

Key terms

Digit: One of the symbols 0 to 9 used to write numbers.
Place holder: A zero that keeps other digits in the right columns, as in 340.
Partition: To split a number into parts, such as 347 = 300 + 40 + 7.
Partial product: The answer to one part of a multiplication, such as 40 × 6 = 240.
Distributive law: Splitting a number and multiplying each part: (30 + 4) × 6 = 30 × 6 + 4 × 6.
Grid method: Putting the parts of a number in a table and multiplying each part.
Carry: Moving the tens digit of an answer into the next column.
Short multiplication: A written method for multiplying by a one-digit number, column by column.
Short division: The bus stop method for dividing a number by a one-digit number.
Divisor: The number you divide by.
Remainder: What is left over when a number does not divide exactly.
Exchange: Swapping 1 of a place for 10 of the next place down, such as 1 ten for 10 ones.
Scaling: Making an amount a number of times bigger or smaller, such as 4 times as many.
Correspondence: Pairing each object in one group with each object in another group to count the combinations.
Combination: One choice from each group put together, such as one hat and one top.
Estimate: A quick answer that is close, found by rounding.
Bar model: A drawing of boxes that shows the numbers in a problem.
Inverse: The opposite operation. Multiplying and dividing undo each other.

Total 18 marks. Allow about 18 minutes.

← Chapter 3: Times tables to 12 × 12 Chapter 5: Fractions →

Year 4 Short Multiplication and Short Division: worked examples and questions with answers

By the end of this chapter you will be able to

You should already know

Where this leads

Exercise P Do you remember?

Multiplying and dividing by 10 and 100

Key ideas

Watch out

Exercise 4A Multiplying and dividing by 10 and 100

Partitioning, the grid method and the distributive law

Key ideas

Watch out

Exercise 4B Partitioning, the grid method and the distributive law

Column multiplication: expanded and short

Key ideas

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Exercise 4C Column multiplication: expanded and short

Dividing with place value and short division

Key ideas

Watch out

Exercise 4D Dividing with place value and short division

Division with remainders

Key ideas

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Exercise 4E Division with remainders

Scaling, correspondence and multiplying three numbers

Key ideas

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Exercise 4F Scaling, correspondence and three numbers

Word problems with bar models

Key ideas

Watch out

Exercise 4G Word problems with bar models

Chapter summary

Can you…?

Key terms

Exercise 4Z Check-up (non-calculator)

Exercise 4X SATs-style questions

hat \ top	top 1	top 2	top 3	top 4	top 5
hat 1	1	2	3	4	5
hat 2	6	7	8	9	10
hat 3	11	12	13	14	15

flavour \ topping	topping 1	topping 2	topping 3	topping 4	topping 5	topping 6
flavour 1	1	2	3	4	5	6
flavour 2	7	8	9	10	11	12
flavour 3	13	14	15	16	17	18
flavour 4	19	20	21	22	23	24

T-shirt \ pair of shorts	pair of shorts 1	pair of shorts 2	pair of shorts 3	pair of shorts 4	pair of shorts 5	pair of shorts 6
T-shirt 1	1	2	3	4	5	6
T-shirt 2	7	8	9	10	11	12
T-shirt 3	13	14	15	16	17	18
T-shirt 4	19	20	21	22	23	24

starter \ main	main 1	main 2	main 3	main 4
starter 1	1	2	3	4
starter 2	5	6	7	8
starter 3	9	10	11	12
starter 4	13	14	15	16
starter 5	17	18	19	20

type of bread \ filling	filling 1	filling 2	filling 3	filling 4	filling 5
type of bread 1	1	2	3	4	5
type of bread 2	6	7	8	9	10
type of bread 3	11	12	13	14	15
type of bread 4	16	17	18	19	20
type of bread 5	21	22	23	24	25
type of bread 6	26	27	28	29	30

shirt \ pair of trousers	pair of trousers 1	pair of trousers 2	pair of trousers 3	pair of trousers 4	pair of trousers 5	pair of trousers 6
shirt 1	1	2	3	4	5	6
shirt 2	7	8	9	10	11	12
shirt 3	13	14	15	16	17	18
shirt 4	19	20	21	22	23	24