Year 6 · Chapter 2

Multiplication and division

By the end of this chapter you will be able to

Multiply a four-digit number by a one-digit number using a column method
Multiply a number by a two-digit number using long multiplication
Divide by a one-digit number and give the remainder in the form the question asks
Divide by a two-digit number using long division
Multiply and divide decimals by 10, 100 and 1000, and by a whole number

You should already know

Times tables to 12 × 12 and short multiplication (Year 5 maths)
Remainders and short division (Year 5 maths)

Where this leads

Fractions, including fractions of amounts (Chapter 3 of this book)

KS2 SATs practice for arithmetic (Year 6). Original questions in the style of the national tests. Content: Multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication (Year 5-6), Divide numbers up to 4 digits by a one-digit number using short division, interpreting remainders (Year 5), Divide numbers up to 4 digits by a two-digit whole number using long division, interpreting the remainder (Year 6), Multiply and divide mentally, using known facts, factor pairs and commutativity (Year 5), Multiply and divide whole numbers and decimals by 10, 100 and 1000 (Year 5-6), Multiply a number with up to two decimal places by a one-digit whole number, and use written division with up to two decimal places (Year 6), Solve problems combining multiplication and division, including the meaning of the equals sign (Year 5-6).

Exercise P Do you remember?

Non-calculator

FWork out 4 × 6.

Done

FWork out 36 ÷ 4.

Done

FWork out 2,305 + 1,998.

Done

FWork out 6 × 40.

Done

FRound 3,452 to the nearest 100.

Done

2.1

Multiplying by a one-digit number

Year 6
Multiply numbers up to 4 digits by a one-digit number using a formal written method (Year 5-6)

Video lesson: Multiplying by a one-digit numberComing soon

Practice material. Non-calculator practice. This book is practice in the style of the Year 6 arithmetic test. It is not the official test, and no official test item is used. Do every question without a calculator. Marks and timings are a practice guide only.

Story. A bakery makes 2,075 scones every day. Each scone sits on a tray, and the baker needs to know how many scones it makes in four days. She writes the multiplication in a column.

Rule

In short multiplication, multiply the digits one column at a time, starting from the ones. Write the ones digit of each answer and carry the tens digit to the next column.

Check with an estimate: 4,836 × 7 is about 5,000 × 7 = 35,000, so an answer near 33,852 is sensible.

The full written method for longer numbers is in Year 5 maths, chapter 2.

Key ideas

Multiply from the ones column first, and carry where needed.
Estimate first to check that the answer is the right size.
Every column of the bigger number is multiplied by the one-digit number.

Worked example 2.1.1

Work out 4,836 × 7.

Method. Multiply column by column from the right, carrying where needed.

Multiply the ones digit 6 by 7. Write the ones digit of the answer in this column and carry the tens digit to the next column.
6 × 7 + 0 = 42, write 2, carry 4
Multiply the tens digit 3 by 7. Write the ones digit of the answer in this column and carry the tens digit to the next column.
3 × 7 + 4 = 25, write 5, carry 2
Multiply the hundreds digit 8 by 7. Write the ones digit of the answer in this column and carry the tens digit to the next column.
8 × 7 + 2 = 58, write 8, carry 5
Multiply the thousands digit 4 by 7. Write the ones digit of the answer in this column and carry the tens digit to the next column.
4 × 7 + 5 = 33, write 3, carry 3
Multiply the ten thousands digit 0 by 7. Write the ones digit of the answer in this column and carry the tens digit to the next column.
0 × 7 + 3 = 3, write 3
Write the answer.
33,852

Check. Estimate: 5,000 × 7 = 35,000, close to 33,852. ✓

Watch out. Do not forget to add the carried tens to the next product.

Worked example 2.1.2

Work out 2,075 × 4.

Method. Start with the ones column, which is 5 × 4.

Multiply the ones digit 5 by 4. Write the ones digit of the answer in this column and carry the tens digit to the next column.
5 × 4 + 0 = 20, write 0, carry 2
Multiply the tens digit 7 by 4. Write the ones digit of the answer in this column and carry the tens digit to the next column.
7 × 4 + 2 = 30, write 0, carry 3
Multiply the hundreds digit 0 by 4. Write the ones digit of the answer in this column and carry the tens digit to the next column.
0 × 4 + 3 = 3, write 3
Multiply the thousands digit 2 by 4. Write the ones digit of the answer in this column and carry the tens digit to the next column.
2 × 4 + 0 = 8, write 8
Write the answer.
8,300

Check. 8,300 ÷ 4 = 2,075. ✓

Watch out. The zero in the tens column still gives a zero in the answer.

Worked example 2.1.3

Work out 3,249 × 6.

Method. Multiply each column, then add the carries.

Multiply the ones digit 9 by 6. Write the ones digit of the answer in this column and carry the tens digit to the next column.
9 × 6 + 0 = 54, write 4, carry 5
Multiply the tens digit 4 by 6. Write the ones digit of the answer in this column and carry the tens digit to the next column.
4 × 6 + 5 = 29, write 9, carry 2
Multiply the hundreds digit 2 by 6. Write the ones digit of the answer in this column and carry the tens digit to the next column.
2 × 6 + 2 = 14, write 4, carry 1
Multiply the thousands digit 3 by 6. Write the ones digit of the answer in this column and carry the tens digit to the next column.
3 × 6 + 1 = 19, write 9, carry 1
Multiply the ten thousands digit 0 by 6. Write the ones digit of the answer in this column and carry the tens digit to the next column.
0 × 6 + 1 = 1, write 1
Write the answer.
19,494

Check. 19,494 ÷ 6 = 3,249. ✓

2.2

Multiplying by a two-digit number

Year 6
Multiply numbers up to 4 digits by a one- or two-digit number using a formal written method, including long multiplication (Year 5-6), Multiply and divide mentally, using known facts, factor pairs and commutativity (Year 5)

Video lesson: Multiplying by a two-digit numberComing soon

Story. A hall has 24 chairs in each row. The caretaker has 4,836 chairs in the store, so he wants to know how many chairs there would be if he filled 24 rows of the same kind.

Rule

In long multiplication, multiply by the ones digit, then by the tens digit (which is worth ten times as much). Write each answer as a partial product, then add the partial products.

Commutativity: 24 × 4836 gives the same answer as 4836 × 24, so you can choose the easier order.

Key ideas

Multiply by the ones digit, then by the tens digit.
Each partial product for the tens digit ends in a zero.
Add the partial products to get the answer.
Estimate first: round each number to see whether the answer is about right.

Worked example 2.2.1

Work out 4,836 × 24.

Method. Work out two partial products and add them.

Multiply by the ones digit of 24, which is 4 (4). Write the partial product, and put a zero in each empty column below it.
4,836 × 4 = 19,344
Multiply by the tens digit of 24, which is 2 (20). Write the partial product, and put a zero in each empty column below it.
4,836 × 20 = 96,720
Add the partial products. Line up the place values and add.
19,344 + 96,720 = 116,064
Write the answer.
116,064

Check. About 5,000 × 24 = 120,000, close to 116,064. ✓

Watch out. The tens partial product is ten times bigger, so it has a zero in the ones column.

Worked example 2.2.2

Work out 307 × 36.

Method. Multiply by the ones digit, then the tens digit, then add.

Multiply by the ones digit of 36, which is 6 (6). Write the partial product, and put a zero in each empty column below it.
307 × 6 = 1,842
Multiply by the tens digit of 36, which is 3 (30). Write the partial product, and put a zero in each empty column below it.
307 × 30 = 9,210
Add the partial products. Line up the place values and add.
1,842 + 9,210 = 11,052
Write the answer.
11,052

Check. About 300 × 36 = 10,800, close to 11,052. ✓

Watch out. Do not forget the empty tens column: the zero in 307 must be multiplied too.

Worked example 2.2.3

Work out 2,015 × 18.

Method. Multiply by 8, then by 10, then add the two partial products.

Multiply by the ones digit of 18, which is 8 (8). Write the partial product, and put a zero in each empty column below it.
2,015 × 8 = 16,120
Multiply by the tens digit of 18, which is 1 (10). Write the partial product, and put a zero in each empty column below it.
2,015 × 10 = 20,150
Add the partial products. Line up the place values and add.
16,120 + 20,150 = 36,270
Write the answer.
36,270

Check. 36,270 ÷ 18 = 2,015. ✓

2.3

Short division and remainders

Year 6
Divide numbers up to 4 digits by a one-digit number using short division, interpreting remainders for the context (Year 5)

Video lesson: Short division and remaindersComing soon

Story. Ali has 83 stickers to share equally between 7 friends. He wants to know how many each friend gets, and how many stickers are left over.

Rule

In short division you divide one digit at a time, from the left. Any remainder is carried to the next column as ten times more.

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

In a problem, decide what the remainder means: sometimes you leave it, sometimes you write it as a fraction, and sometimes you round up so that everybody has a place.

Key ideas

Divide one digit at a time, starting from the left.
The remainder is always smaller than the number you divide by.
Check with multiplication: quotient × divisor + remainder = the starting number.
The context tells you which form of the remainder to give.

Worked example 2.3.1

Work out 7,268 ÷ 4.

Method. Divide each digit from the left, carrying the remainder.

Divide the thousands part: 7 ÷ 4. Write how many times the divisor goes in, and carry the remainder to the next column.
7 ÷ 4 = 1 remainder 3
Divide the hundreds part: 32 ÷ 4. Write how many times the divisor goes in, and carry the remainder to the next column.
32 ÷ 4 = 8 remainder 0
Divide the tens part: 6 ÷ 4. Write how many times the divisor goes in, and carry the remainder to the next column.
6 ÷ 4 = 1 remainder 2
Divide the ones part: 28 ÷ 4. Write how many times the divisor goes in, and carry the remainder to the next column.
28 ÷ 4 = 7 remainder 0
Write the answer.
1,817 remainder 0

Check. 1,817 × 4 + 0 = 7,268. ✓

Watch out. Write the zero in the answer if a column does not divide.

Worked example 2.3.2

Work out 5,931 ÷ 6, and give the remainder.

Method. Divide one digit at a time from the left.

Divide the thousands part: 5 ÷ 6. Write how many times the divisor goes in, and carry the remainder to the next column.
5 ÷ 6 = 0 remainder 5
Divide the hundreds part: 59 ÷ 6. Write how many times the divisor goes in, and carry the remainder to the next column.
59 ÷ 6 = 9 remainder 5
Divide the tens part: 53 ÷ 6. Write how many times the divisor goes in, and carry the remainder to the next column.
53 ÷ 6 = 8 remainder 5
Divide the ones part: 51 ÷ 6. Write how many times the divisor goes in, and carry the remainder to the next column.
51 ÷ 6 = 8 remainder 3
Write the answer.
988 remainder 3

Check. 988 × 6 + 3 = 5,931. ✓

Worked example 2.3.3

Ali shares 83 stickers equally between 7 friends. How many does each friend get, and how many are left?

Method. Divide the stickers by the friends, and the remainder is the number left.

Divide the tens part: 8 ÷ 7. Write how many times the divisor goes in, and carry the remainder to the next column.
8 ÷ 7 = 1 remainder 1
Divide the ones part: 13 ÷ 7. Write how many times the divisor goes in, and carry the remainder to the next column.
13 ÷ 7 = 1 remainder 6
Write the answer.
11 each, and 6 left over

Check. 11 × 7 + 6 = 83, so 7 friends get 11 each. ✓

Watch out. The remainder 6 is less than 7, so it cannot go to each friend.

2.4

Long division by a two-digit number

Year 6
Divide numbers up to 4 digits by a two-digit whole number using long division, interpreting the remainder as a whole number, a fraction or by rounding (Year 6)

Video lesson: Long division by a two-digit numberComing soon

Story. A school has 1,046 pupils. Each class has room for 25 pupils. The office needs to know how many classes are needed, and the remainder decides whether one more class is needed.

Rule

In long division, divide by the two-digit number one digit of the answer at a time. Multiply the divisor by each digit, subtract, and bring down the next digit of the starting number.

The remainder can be written in three ways: as a whole number (98 r 0), as a fraction (41 2/3), or rounded in context (a class of 25 needs 42 classes if 21 pupils are left over).

Key ideas

Each step: divide, multiply, subtract, bring down.
The remainder must be smaller than the two-digit divisor.
The question tells you which form of the remainder to give.
A fraction remainder is the remainder over the divisor, simplified.

Worked example 2.4.1

Work out 1,568 ÷ 16.

Method. Divide one digit of the answer at a time.

Bring down the thousands digit to make 1. 1 is less than 16, so this digit of the answer is 0.
1 < 16, so 0 with remainder 1
Bring down the hundreds digit to make 15. 15 is less than 16, so this digit of the answer is 0.
15 < 16, so 0 with remainder 15
Bring down the tens digit to make 156. 16 goes into 156 9 times, and the remainder is what is left.
16 × 9 = 144, 156 - 144 = 12
Bring down the ones digit to make 128. 16 goes into 128 8 times, and the remainder is what is left.
16 × 8 = 128, 128 - 128 = 0
Write the answer.
98

Check. 16 × 98 = 1,568. ✓

Watch out. The remainder here is 0, so the division is exact.

Worked example 2.4.2

Work out 1,000 ÷ 24. Give the remainder as a fraction.

Method. Divide as usual. The remainder becomes a fraction over 24.

Bring down the thousands digit to make 1. 1 is less than 24, so this digit of the answer is 0.
1 < 24, so 0 with remainder 1
Bring down the hundreds digit to make 10. 10 is less than 24, so this digit of the answer is 0.
10 < 24, so 0 with remainder 10
Bring down the tens digit to make 100. 24 goes into 100 4 times, and the remainder is what is left.
24 × 4 = 96, 100 - 96 = 4
Bring down the ones digit to make 40. 24 goes into 40 1 times, and the remainder is what is left.
24 × 1 = 24, 40 - 24 = 16
Write the remainder as a fraction. The remainder is 16 out of 24. Simplify it.
16/24 = 2/3
Write the answer.
41 2/3

Check. 41 + 2/3 = 41 2/3, and 24 × 41 + 16 = 1,000. ✓

Worked example 2.4.3

A school has 1,046 pupils and each class holds 25 pupils. How many classes are needed?

Method. Divide to find the full classes, then decide what the remainder means.

Bring down the thousands digit to make 1. 1 is less than 25, so this digit of the answer is 0.
1 < 25, so 0 with remainder 1
Bring down the hundreds digit to make 10. 10 is less than 25, so this digit of the answer is 0.
10 < 25, so 0 with remainder 10
Bring down the tens digit to make 104. 25 goes into 104 4 times, and the remainder is what is left.
25 × 4 = 100, 104 - 100 = 4
Bring down the ones digit to make 46. 25 goes into 46 1 times, and the remainder is what is left.
25 × 1 = 25, 46 - 25 = 21
Write the answer.
42 classes

Check. 41 full classes hold 1025 pupils, and 21 pupils need one more class. ✓

Watch out. A remainder of 21 pupils still needs a class, so round up, not down.

2.5

Multiplying and dividing decimals, and the equals sign

Year 6
Multiply and divide whole numbers and decimals by 10, 100 and 1000 (Year 5-6), Multiply a number with up to two decimal places by a one-digit whole number, and use written division with up to two decimal places (Year 6), Solve problems, including the meaning of the equals sign (Year 5-6)

Video lesson: Multiplying and dividing decimals, and the equals signComing soon

Story. A roll of ribbon is 2.35 metres long. A craft club cuts 6 pieces of the same length, and the teacher wants to know the total length of ribbon used.

Rule

Multiplying by 10, 100 or 1000 moves every digit one, two or three places to the left. Dividing moves them to the right.

To multiply a decimal by a whole number, multiply as if there were no point, then put the point back so the answer has the same number of decimal places.

The equals sign means "the same as": both sides have the same value.

Key ideas

×10, ×100, ×1000 move digits to the left; ÷10, ÷100, ÷1000 move them to the right.
The answer to a decimal × whole number has the same number of decimal places as the decimal.
Write a decimal division to two places, adding zeros after the point if needed.
Both sides of an equals sign must have the same value.

Worked example 2.5.1

Work out 4.56 × 100.

Method. Multiply by 100: each digit moves two places to the left.

Move each digit two places left. The digit 4 moves to the hundreds, 5 to the tens, 6 to the ones.
4.56 → 456
Check the point. There are no decimal places left, so the answer is a whole number.
456
Write the answer. The answer is 456.
456
Write the answer.
456

Check. 456 ÷ 100 = 4.56. ✓

Watch out. Do not add zeros after the point for ×100 here: 4.56 × 100 is 456.

Worked example 2.5.2

Work out 3.45 × 6.

Method. Multiply as whole numbers, then put the decimal point back.

Ignore the decimal point for now. Multiply the digits 345 by 6, as whole numbers.
345 × 6 = 2070
Put the decimal point back. 3.45 has 2 decimal places, so the answer has 2 decimal places.
2070 → 20.70
Write the answer.
20.70

Check. 3.45 is about 3.5, and 3.5 × 6 = 21, close to 20.70. ✓

Watch out. 3.45 has two decimal places, so 20.70 has two decimal places too.

Worked example 2.5.3

Work out 27 ÷ 4 as a decimal.

Method. Divide the whole number, then continue with a decimal point and zeros.

Divide the whole number first. 4 goes into 27 6 times, with a remainder of 3.
27 ÷ 4 = 6 remainder 3
Write a decimal point, add a 0 and keep dividing. The remainder 3 becomes 30, and 4 goes into 30 7 times.
30 ÷ 4 = 7 remainder 2
Write a decimal point, add a 0 and keep dividing. The remainder 2 becomes 20, and 4 goes into 20 5 times.
20 ÷ 4 = 5 remainder 0
Write the answer.
6.75

Check. 4 × 6.75 = 27. ✓

Chapter summary

Skill	Method
Short multiplication	Multiply column by column from the right, carrying tens: 4,836 × 7 = 33,852.
Long multiplication	Multiply by each digit of the two-digit number, then add the partial products.
Short division	Divide one digit at a time from the left; the remainder is smaller than the divisor.
Long division	Divide, multiply, subtract, bring down; the remainder can be a whole number, a fraction or rounded.
Powers of 10	×10 moves digits left; ÷10 moves them right; ×100 moves two places; ×1000 moves three.
Decimal × whole number	Multiply as whole numbers, then keep the same number of decimal places.
Equals sign	Both sides of the equals sign must have the same value.

Can you…?

I can multiply a four-digit number by a one-digit number and by a two-digit number
I can divide by a one-digit number with short division and give the remainder
I can divide by a two-digit number with long division
I can say what a remainder means in a real problem, and round when needed
I can multiply and divide decimals by 10, 100 and 1000
I can use the equals sign to find a missing number

Key terms

partial product: part of a long multiplication, one for each digit of the second number
quotient: the answer to a division, before the remainder
remainder: what is left over after a division
divisor: the number you divide by
commutative: the order of the numbers does not change the answer, as in 3 × 8 = 8 × 3
carry: move a ten from one column to the next
equals sign: a sign that shows two things have the same value

Total 18 marks. Allow about 18 minutes.

← Chapter 1: Place value and the four operations Chapter 3: Fractions →

Multiplication and division

By the end of this chapter you will be able to

You should already know

Where this leads

Exercise P Do you remember?

Multiplying by a one-digit number

Rule

Key ideas

Exercise 2A Practice: multiplying by one digit

Multiplying by a two-digit number

Rule

Key ideas

Exercise 2B Practice: multiplying by a two-digit number

Short division and remainders

Rule

Key ideas

Exercise 2C Practice: short division and remainders

Long division by a two-digit number

Rule

Key ideas

Exercise 2D Practice: long division and remainders

Multiplying and dividing decimals, and the equals sign

Rule

Key ideas

Exercise 2E Practice: decimals, powers of 10 and the equals sign

Chapter summary

Can you…?

Key terms

Exercise 2Z Check-up

Exercise 2X Practice test questions