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Primary Maths · Year 5Chapter 4 · Multiplication
Year 5 · Long Multiplication · Lesson 4.2

Year 5: Multiplying in your head

Part of the Year 5 long multiplication chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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4.2

Multiplying in your head

Year 5
Multiply numbers mentally drawing upon known facts, Partitioning, doubling and halving, factor pairs, compensation, Meaning of the equals sign

Good mathematicians do not always write a calculation down. They choose a clever strategy. All of these strategies start from facts you already know.

Use a known fact. If you know 6 × 7 = 42, you also know 60 × 7 = 420, 6 × 70 = 420 and 60 × 70 = 4,200. The fact stays the same; the zeros move. Partition. Split a number into parts you can multiply easily: 27 × 4 = (20 × 4) + (7 × 4) = 80 + 28 = 108.

20807284
27 × 4 split into 20 × 4 and 7 × 4. (Not drawn to scale.)

Double and halve. If you double one number and halve the other, the answer stays the same: 16 × 35 = 8 × 70 = 4 × 140 = 560. Use factors. Multiplying by 12 is the same as multiplying by 3 and then by 4, because 12 = 3 × 4. Compensate. To multiply by 19, multiply by 20 (easy) and take away one lot: 34 × 19 = 34 × 20 − 34. Here is a menu of tricks.

To multiply by…Try…Example
10, 100, 1,000Move every digit 1, 2 or 3 places to the left.36 × 100 = 3,600
4Double, then double again.23 × 4: 46, then 92
5Multiply by 10, then halve.46 × 5: 460, then 230
9 or 99Multiply by 10 (or 100), then take away one lot.43 × 9 = 430 − 43
11 or 101Multiply by 10 (or 100), then add one lot.32 × 11 = 320 + 32
12Multiply by 3, then by 4 (or by 10, then add 2 lots).24 × 12: 72, then 288
15Multiply by 10, then add half of that.18 × 15 = 180 + 90
20Double, then multiply by 10.35 × 20: 70, then 700
25Multiply by 100, then halve twice.36 × 25: 3,600, 1,800, 900
19, 21, 29, 31Use 20 or 30, then take away or add.34 × 19 = 680 − 34

The equals sign. The equals sign (=) does not mean “now write the answer”. It means “is the same value as”. Both sides must balance, like a see-saw. We can have a calculation on both sides: 3 × 4 = 6 × 2 is true, because both sides are worth 12.

3 × 4126 × 212=
Both sides are worth 12, so the balance is level.

Key ideas

  • Use a known fact and put the zeros back: 6 × 7 = 42, so 60 × 70 = 4,200 (count all the zeros).
  • Partition into parts that are easy to multiply, then add the answers.
  • Double one number and halve the other: the answer stays the same.
  • Multiply by factors one at a time: × 12 is × 3 then × 4.
  • × 5 = × 10 then halve. × 25 = × 100 then halve twice. × 9 = × 10 then take away one lot. × 19 = × 20 then take away one lot.
  • The equals sign means “is the same value as”. Both sides must have the same value.
Worked example 4.2.1
Use a known fact to work out 60 × 70.

Key words. Place value tells you what each digit is worth. A place holder is a zero that keeps the other digits in the right columns. Multiplying by 10, 100 or 1,000 makes a number 10, 100 or 1,000 times bigger.

Method. Use the fact 6 × 7 = 42. Multiply the numbers without their zeros, then put all the zeros back.

  1. Find the multiplication fact you know. Take the zeros off both numbers. What is left is a times table fact.
    60 = 6 × 10  and  70 = 7 × 10
    The fact is 6 × 7 = 42
  2. Count the zeros you took off. Each zero means one × 10. Together they make 100.
    1 zero + 1 zero = 2 zeros, so × 100
  3. Multiply the fact by that power of ten. Move the digits of the fact to the left, filling the gaps with zeros.
    42 × 100 = 4,200
  4. Write the answer.
    60 × 70 = 4,200

Check. Check by dividing: 4,200 ÷ 70 = 60. ✓

Watch out. Count ALL the zeros from BOTH numbers, then put them after 42.

Worked example 4.2.2
Use a known fact to work out 300 × 80.

Key words. Place value tells you what each digit is worth. A place holder is a zero that keeps the other digits in the right columns. Multiplying by 10, 100 or 1,000 makes a number 10, 100 or 1,000 times bigger.

Method. Use the fact 3 × 8 = 24. Multiply the numbers without their zeros, then put all the zeros back.

  1. Find the multiplication fact you know. Take the zeros off both numbers. What is left is a times table fact.
    300 = 3 × 100  and  80 = 8 × 10
    The fact is 3 × 8 = 24
  2. Count the zeros you took off. Each zero means one × 10. Together they make 1,000.
    2 zeros + 1 zero = 3 zeros, so × 1,000
  3. Multiply the fact by that power of ten. Move the digits of the fact to the left, filling the gaps with zeros.
    24 × 1,000 = 24,000
  4. Write the answer.
    300 × 80 = 24,000

Check. Check by dividing: 24,000 ÷ 80 = 300. ✓

Watch out. Count ALL the zeros from BOTH numbers, then put them after 24.

Worked example 4.2.3
Work out 27 × 4 by partitioning.

Key words. To partition a number is to split it into parts, such as 47 = 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 one.

Method. Partition 27 into place-value parts. Multiply each part by 4, then add the answers.

  1. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
    27 = 20 + 7
  2. Multiply each part by 4. Use a times table fact, then put the zeros back.
    20 × 4 = 80
    7 × 4 = 28
  3. Add the answers. Add them one at a time, starting with the biggest.
    80 + 28 = 108
  4. Write the answer.
    27 × 4 = 108

Check. Estimate: 27 is about 30. 30 × 4 = 120. Our answer, 108, is close to 120. ✓

Watch out. Multiply every part. Do not forget a part, and do not lose the zeros: 40 × 6 is 240, not 24.

Worked example 4.2.4
Work out 208 × 5 by partitioning.

Key words. To partition a number is to split it into parts, such as 47 = 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 one.

Method. Partition 208 into place-value parts. Multiply each part by 5, then add the answers.

  1. Partition the bigger number. Split it into hundreds, tens and ones. Each part is easy to multiply.
    208 = 200 + 8
  2. Multiply each part by 5. Use a times table fact, then put the zeros back.
    200 × 5 = 1,000
    8 × 5 = 40
  3. Add the answers. Add them one at a time, starting with the biggest.
    1,000 + 40 = 1,040
  4. Write the answer.
    208 × 5 = 1,040

Check. Estimate: 208 is about 200. 200 × 5 = 1,000. Our answer, 1,040, is close to 1,000. ✓

Watch out. Multiply every part. Do not forget a part, and do not lose the zeros: 40 × 6 is 240, not 24.

Worked example 4.2.5
Work out 13 × 15 by partitioning the 15.

Key words. To partition a number is to split it into parts, such as 47 = 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 one.

Method. Partition 15 into tens and ones. Multiply 13 by each part, then add the answers.

  1. Partition 15. Split the second number into tens and ones. Multiplying by tens and by ones is easier.
    15 = 10 + 5
  2. Multiply by each part. Multiply by the tens first (move the digits), then by the ones.
    13 × 10 = 130
    13 × 5 = 10 × 5 + 3 × 5 = 50 + 15 = 65
  3. Add the answers.
    130 + 65 = 195
  4. Write the answer.
    13 × 15 = 195

Check. Estimate: 13 is about 10; 15 is about 20. 10 × 20 = 200. Our answer, 195, is close to 200. ✓

Watch out. You must multiply 13 by BOTH parts, not just by the tens.

Worked example 4.2.6
Work out 35 × 16 by doubling and halving.

Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To double is to multiply by 2. To halve is to divide by 2.

Method. Halve one number and double the other. The answer stays the same, but the sum gets easier.

  1. Look for an even number to halve. If you halve one number and double the other, the answer stays the same. This is because 2 × (1/2) = 1.
    35 × 16: halve 16, double 35
  2. Halve one number and double the other. Do this until the multiplication is easy.
    35 × 16 = (35 × 2) × (16 ÷ 2) = 70 × 8
    70 × 8 = (70 × 2) × (8 ÷ 2) = 140 × 4
  3. Work out the easy multiplication. Now use a fact you know or move the digits.
    140 × 4 = 100 × 4 + 40 × 4 = 400 + 160 = 560
  4. Write the answer.
    35 × 16 = 560

Check. Check with a different method: 35 × 16 = 10 × 35 + 6 × 35 = 350 + 210 = 560

Watch out. Halve ONE number and double the OTHER. If you halve both, the answer is four times too small.

Worked example 4.2.7
Work out 24 × 12 by multiplying by factors.

Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To double is to multiply by 2. To halve is to divide by 2.

Method. 12 is 3 × 4. Multiply 24 by each factor in turn.

  1. Split 12 into factors. A factor pair gives two easier multiplications that make the same answer.
    12 = 3 × 4
  2. Multiply by 3. Use the 3 times table and partition if you need to.
    24 × 3 = 20 × 3 + 4 × 3 = 60 + 12 = 72
  3. Multiply by 4. Use the 4 times table and partition if you need to.
    72 × 4 = 70 × 4 + 2 × 4 = 280 + 8 = 288
  4. Write the answer.
    24 × 12 = 288

Check. Estimate: 24 is about 20; 12 is about 10. 20 × 10 = 200. Our answer, 288, is a bit more than 200. ✓

Watch out. Multiply by each factor, one after the other. Do not add the factors: 12 is 3 × 4, not 3 + 4.

Worked example 4.2.8
Work out 46 × 5 in your head.

Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To double is to multiply by 2. To halve is to divide by 2.

Method. Multiplying by 5 is the same as multiplying by 10 and then halving, because 5 is half of 10.

  1. Multiply by 10. 5 is half of 10, so first multiply by 10: move the digits one place to the left.
    46 × 10 = 460
  2. Halve the answer. We multiplied by a number that was 2 times too big, so halve it.
    460 ÷ 2 = 230
  3. Write the answer.
    46 × 5 = 230

Check. Estimate: 46 is about 50. 50 × 5 = 250. Our answer, 230, is close to 250. ✓

Watch out. Remember to halve at the end. 10 is twice as big as 5.

Worked example 4.2.9
Work out 36 × 25 in your head.

Key words. A mental strategy is a way to work something out in your head by using facts you already know. A factor is a number that divides exactly into another number. To double is to multiply by 2. To halve is to divide by 2.

Method. Multiplying by 25 is the same as multiplying by 100 and then dividing by 4 (halving twice), because 25 is a quarter of 100.

  1. Multiply by 100. 25 is a quarter of 100, so first multiply by 100: move the digits two places to the left.
    36 × 100 = 3,600
  2. Halve the answer. To divide by 4, halve twice. First halve.
    3,600 ÷ 2 = 1,800
  3. Halve again. A quarter is half of a half.
    1,800 ÷ 2 = 900
  4. Write the answer.
    36 × 25 = 900

Check. Estimate: 36 is about 40; 25 is about 30. 40 × 30 = 1,200. Our answer, 900, is a bit less than 1,200. ✓

Watch out. Halve TWICE. Dividing by 4 means halving, then halving again.

Worked example 4.2.10
Work out 43 × 9 in your head.

Key words. To compensate is to make up for a change. Here we multiply by an easy number that is a little too big or too small, then correct the answer.

Method. 9 is close to 10. Multiply by 10, then take away 1 lot of 43.

  1. Spot an easy number close by. 9 is just 1 less than 10. Multiplying by 10 is easy.
    9 = 10 − 1
  2. Multiply by 10. Do the easy multiplication first.
    43 × 10 = 430
  3. Work out the extra 1 lot of 43. We multiplied by 10, but we wanted 9. That is 1 lot of 43 too many.
    43 × 1 = 43
  4. Correct the answer. Take away the extra lots.
    430 − 43 = 387
  5. Write the answer.
    43 × 9 = 387

Check. Another way: 43 × 9 = 40 × 9 + 3 × 9 = 360 + 27 = 387

Watch out. Take away 1 lot of 43, not just 1.

Worked example 4.2.11
Work out 34 × 19 in your head.

Key words. To compensate is to make up for a change. Here we multiply by an easy number that is a little too big or too small, then correct the answer.

Method. 19 is close to 20. Multiply by 20, then take away 1 lot of 34.

  1. Spot an easy number close by. 19 is just 1 less than 20. Multiplying by 20 is easy.
    19 = 20 − 1
  2. Multiply by 20. Do the easy multiplication first.
    34 × 2 = 68, then × 10: 68 × 10 = 680
  3. Work out the extra 1 lot of 34. We multiplied by 20, but we wanted 19. That is 1 lot of 34 too many.
    34 × 1 = 34
  4. Correct the answer. Take away the extra lots.
    680 − 34 = 646
  5. Write the answer.
    34 × 19 = 646

Check. Another way: 34 × 19 = 10 × 34 + 9 × 34 = 340 + 306 = 646

Watch out. Take away 1 lot of 34, not just 1.

Worked example 4.2.12
Find the missing number.   3 × 4 = 6 × ?
3 × 4126 × ?12=

Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. It does not mean “the answer comes next”.

Method. Work out the value of the side with no missing number. The other side must have the same value.

  1. Work out the left side. The equals sign means both sides are worth the same. Find the value of the side with no missing number.
    3 × 4 = 12
  2. Say what the right side must be. It has to balance the left side.
    6 × ? = 12
  3. Ask: how many 6s make 12? Count in steps, or use the inverse operation.
    12 ÷ 6 = 2
  4. Spot the pattern. If one number gets bigger, the other must get smaller by the same factor to keep the answer.
    6 is 2 times 3, so the missing number must be 2 times smaller than 4: 4 ÷ 2 = 2.
  5. Read the sentence back. Both sides must be equal.
    3 × 4 = 6 × 2 (both are 12)

Check. 6 × 2 = 12 and 3 × 4 = 12. ✓

Watch out. If you double one number you must halve the other to keep the same answer. The two sides need the same VALUE, not the same numbers.

Worked example 4.2.13
Find the missing number.   48 = 12 × ?

Key words. The equals sign (=) means “is the same value as”. The two sides of an equals sign balance, like the two sides of a see-saw. It does not mean “the answer comes next”.

Method. The two sides of an equals sign must have the same value. Work out the side you can, then find the missing number that makes the other side match.

  1. Work out the side with no missing number. The equals sign means both sides have the same value, so we need this value first.
    The left side is already a number: 48
  2. Say what the other side must be. The other side has to balance, so it must also be worth the same.
    12 × ? = 48
  3. Ask the question the missing number answers. Use the inverse (opposite) operation or a fact you know.
    What do I multiply 12 by to make 48?
    48 ÷ 12 = 4
  4. Write the missing number and read the sentence back. Both sides must be the same.
    4    (12 × 4 = 48)

Check. Put 4 in the box: 48 = 48 and 12 × 4 = 48. Both sides are equal. ✓

Watch out. The equals sign does not mean “now write the answer”. It means “is the same as”, so there can be a calculation on both sides.

Watch out

Do not stop at the first strategy that comes to mind: choose the one that makes the numbers easiest. And take care with the zeros. 60 × 70 is 4,200 (two zeros from the numbers, plus the zero in 42), not 420. When you double one number, you must halve the other. If you double both, the answer is four times bigger.

Exercise 4B  Multiplying in your head

Non-calculator
1
FUse a times table fact to work out each calculation.
(a)60 × 8 =
(b)700 × 4 =
(c)50 × 90 =
(d)300 × 70 =
2
FWork out each multiplication by partitioning.
(a)23 × 4 =
(b)38 × 6 =
(c)52 × 7 =
(d)206 × 5 =
3
FPartition the second number to work out each multiplication.
(a)12 × 14 =
(b)15 × 13 =
(c)22 × 12 =
(d)17 × 15 =
4
FDouble one number and halve the other to work out each calculation.
(a)25 × 16 =
(b)15 × 14 =
(c)45 × 18 =
(d)35 × 12 =
5
FMultiply by the factors to work out each calculation.
(a)17 × 4 =
(b)14 × 8 =
(c)23 × 12 =
(d)31 × 18 =
6
FWork out each multiplication in your head.
(a)64 × 5 =
(b)48 × 25 =
(c)36 × 50 =
(d)73 × 20 =
(e)26 × 15 =
7
FUse a nearby easy number to work out each multiplication.
(a)46 × 9 =
(b)52 × 11 =
(c)37 × 99 =
(d)24 × 19 =
(e)31 × 21 =
8
FWrite the missing number in each box. Both sides of the equals sign must have the same value.
(a)3 × 8 = 6 × ?
(b)5 × 8 = 10 × ?
(c)6 × 7 = 3 × ?
(d)4 × 15 = ? × 30
(e)5 × 6 = 20 + ?
(f)3 × 4 = ? − 8
9
FThe picture shows a balance. Find the missing number. 7 × 8 = 14 × ?
7 × 85614 × ??=
10
FChoose a clever method for each calculation and work it out.
(a)45 × 20 =
(b)6 × 98 =
(c)16 × 25 =
11
PSA baker puts 25 buns on each tray. She fills 36 trays. How many buns is that?
12
PSNoah says, “To multiply by 99, I multiply by 100 and take away one lot.” Use his method to work out 46 × 99.
13
RLeo works out 36 × 25 like this: “36 × 100 = 3,600, then halve it: 1,800.” What mistake has Leo made? What is the correct answer?
14
RAlways, sometimes or never?
(a)Multiplying a number by 5 gives the same answer as multiplying it by 10 and halving.
(b)If you double one number and halve the other, the answer to the multiplication gets bigger.
15
RProve it! Amir says, “15 × 18, 30 × 9 and 45 × 6 all have the same answer.” Prove that Amir is right.
16
RChallenge! Work out 4 × 17 × 25 in your head. Explain which order you used and why it is clever.

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