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Primary Maths · Year 5Chapter 2 · Adding and subtracting
Year 5 · Column Addition and Subtraction · Lesson 2.3

Year 5: Adding and subtracting in your head

Part of the Year 5 column addition and subtraction chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)

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2.3

Adding and subtracting in your head

Year 5
Add and subtract numbers mentally with increasingly large numbers

A written method always works, but it is not always the quickest. If the numbers are friendly, you can add and subtract in your head. Look at the numbers first, then pick a good strategy. Big numbers are not harder to use in your head: 340,000 + 270,000 is just 34 + 27 with some zeros.

1. Partitioning. Split (partition) a number into its parts and add or take away one part at a time, biggest first. For 36,400 + 25,800: 36,400 + 20,000 = 56,400, then + 5,000 = 61,400, then + 800 = 62,200. To subtract, keep the first number whole and take away the parts of the second number.

2. Bridging. Jump to the next round number (10, 100 or 1,000), then add or subtract what is left. For 5,970 + 2,400, add 30 to reach 6,000, then add the other 2,370.

5,9706,0008,370+30+2,370
Bridging through 1,000: 5,970 + 30 = 6,000, and 6,000 + 2,370 = 8,370.

3. Compensating. If a number is just below or above a round number (like 99, 999, 9,999 or 1,999), use the round number and then fix the answer. 36,450 + 9,999: add 10,000 (1 too many) to get 46,450, then take 1 away to get 46,449. To take away 1,999, take away 2,000 and put 1 back.

4. Using number facts. Number bonds to 100 give you bonds to 1,000, 10,000 and 100,000 for free. Because 75 + 25 = 100, we know that 7,500 + 2,500 = 10,000.

Number factBigger factEven biggerBiggest
75 + 25 = 100750 + 250 = 1,0007,500 + 2,500 = 10,00075,000 + 25,000 = 100,000
One number bond to 100 gives a whole family of facts. Each time, both numbers get 10 times bigger.

5. Multiples of 1,000 or 10,000. Count how many thousands (or ten thousands) there are, do the small calculation, and change back. 800,000 − 350,000 is 80 ten thousands − 35 ten thousands = 45 ten thousands = 450,000.

6. Near doubles. If two numbers are close, double the smaller and add the difference. 35,000 + 36,000: double 35,000 is 70,000, plus 1,000 is 71,000.

7. Counting up. To find the difference between two numbers that are close, start at the smaller number and count up to the bigger. For 100,000 − 96,250, count from 96,250 up to 97,000 (750), then on to 100,000 (3,000). The difference is 750 + 3,000 = 3,750.

96,25097,000100,000+750+3,000
Counting up from 96,250 to 100,000: 750 + 3,000 = 3,750.

Key words. To partition is to split a number into parts. To bridge is to jump to a round number. To compensate is to make up for a change you made. A number bond is a pair of numbers that make a total. A double is a number added to itself.

Key ideas

  • Partition: add or take away the parts of the second number one at a time, biggest first.
  • Bridge: jump to the next 10, 100 or 1,000, then do the rest.
  • Compensate: use a round number, then fix the answer (add 9,999 = add 10,000, take away 1).
  • Use number facts: 75 + 25 = 100, so 7,500 + 2,500 = 10,000.
  • Multiples of 1,000 or 10,000: count the units, calculate, then change back.
  • Near doubles: double the smaller number and add the difference.
  • Count up to find the difference when the numbers are close.
Worked example 2.3.1
Work out 36,400 + 25,800 in your head by partitioning.

Key words. To partition a number is to split it into parts, one part for each digit (for example 3,450 = 3,000 + 400 + 50).

Method. Keep the first number whole. Split (partition) the second number into its parts, then add the parts one at a time, biggest first.

  1. Partition the second number. Only the second number is split up.
    25,800 = 20,000 + 5,000 + 800
  2. Add the parts one at a time. Start with the biggest part. Each answer is the start of the next step.
    36,400 + 20,000 = 56,400
    56,400 + 5,000 = 61,400
    61,400 + 800 = 62,200
  3. Write the answer. The last number is the answer.
    36,400 + 25,800 = 62,200

Check. Use the opposite operation: 62,200 − 25,800 = 36,400. ✓

Watch out. Do not miss a part. Every digit of the second number that is not 0 gives one part to add.

Worked example 2.3.2
Work out 45,200 + 32,700 by splitting both numbers into parts.

Key words. To partition a number is to split it into parts, one part for each digit.

Method. Split (partition) both numbers into thousands, hundreds, tens and ones. Add the parts that match, then add the totals.

  1. Partition both numbers. Write each number as the value of its digits.
    45,200 = 40,000 + 5,000 + 200
    32,700 = 30,000 + 2,000 + 700
  2. Add the matching parts. Thousands with thousands, hundreds with hundreds, and so on.
    ten thousands: 40,000 + 30,000 = 70,000
    thousands: 5,000 + 2,000 = 7,000
    hundreds: 200 + 700 = 900
  3. Add the totals together. Start with the biggest and keep a running total.
    70,000 + 7,000 = 77,000
    77,000 + 900 = 77,900
  4. Write the answer.
    45,200 + 32,700 = 77,900

Check. Use the opposite operation: 77,900 − 32,700 = 45,200. ✓

Watch out. Do not forget a part. Every digit that is not 0 must be added.

Worked example 2.3.3
Work out 8,764 − 3,425 by partitioning the second number.

Key words. To partition a number is to split it into parts, one part for each digit.

Method. Keep the first number whole. Split (partition) the number you are taking away, then take the parts away one at a time, biggest first.

  1. Partition the number you take away. Only the second number is split up.
    3,425 = 3,000 + 400 + 20 + 5
  2. Take the parts away one at a time. Start with the biggest part. Use the new number as the start of the next step.
    8,764 − 3,000 = 5,764
    5,764 − 400 = 5,364
    5,364 − 20 = 5,344
    5,344 − 5 = 5,339
  3. Write the answer. The last number is the answer.
    8,764 − 3,425 = 5,339

Check. Use the opposite operation: 5,339 + 3,425 = 8,764. ✓

Watch out. Take the parts away from the first number, one after another. Do not take them away from each other.

Worked example 2.3.4
Work out 5,970 + 2,400 by bridging through 1,000.

Key words. To bridge through 1,000 is to use a jump that lands exactly on a multiple of 1,000, then jump again.

Method. Bridge through 1,000. Add enough to reach the next 1,000, then add what is left.

  1. How far is 5,970 from the next 1,000? 6,000 is the next multiple of 1,000.
    6,000 − 5,970 = 30
  2. Split 2,400 into 30 and 2,370. The first part takes us exactly to 6,000.
    2,400 = 30 + 2,370
  3. Add the first part to reach 6,000. This is the bridge.
    5,970 + 30 = 6,000
  4. Add the second part. This is easy because we are on a round number.
    6,000 + 2,370 = 8,370
  5. Show it on a number line.
    5,9706,0008,370+30+2,370

Check. Use the opposite operation: 8,370 − 2,400 = 5,970. ✓

Watch out. Make sure the two parts add back to 2,400: 30 + 2,370 = 2,400.

Worked example 2.3.5
Work out 7,300 − 2,450 by bridging through 1,000.

Method. Bridge through 1,000. Take away enough to land on the 1,000 below, then take away what is left.

  1. How far is 7,300 above the 1,000 below it? 7,000 is the multiple of 1,000 just below 7,300.
    7,300 − 7,000 = 300
  2. Split 2,450 into 300 and 2,150. The first part takes us exactly to 7,000.
    2,450 = 300 + 2,150
  3. Take away the first part to reach 7,000. This is the bridge.
    7,300 − 300 = 7,000
  4. Take away the second part. This is easy because we start from a round number.
    7,000 − 2,150 = 4,850
  5. Show it on a number line. We count back from the right.
    7,3007,0004,850−300−2,150

Check. Use the opposite operation: 4,850 + 2,450 = 7,300. ✓

Watch out. Make sure the two parts add back to 2,450: 300 + 2,150 = 2,450.

Worked example 2.3.6
Work out 36,450 + 9,999 by compensating.

Key words. To compensate is to make up for a change. You use an easier number, then correct the answer.

Method. Compensate: add the nearby round number 10,000 instead, then fix the answer.

  1. Spot a nearby round number. 10,000 is much easier to work with than 9,999.
    9,999 is 1 less than 10,000:  9,999 = 10,000 − 1
  2. Add 10,000 instead of 9,999. Round numbers are easy.
    36,450 + 10,000 = 46,450
  3. Fix the answer. We added 1 too many, so we must take 1 away.
    46,450 − 1 = 46,449

Check. Use the opposite operation: 46,449 − 9,999 = 36,450. ✓

Watch out. Do not forget the last step. You changed the number you add or take away, so the answer needs fixing.

Worked example 2.3.7
Work out 54,300 − 1,999 by compensating.

Key words. To compensate is to make up for a change. You use an easier number, then correct the answer.

Method. Compensate: take away the nearby round number 2,000 instead, then fix the answer.

  1. Spot a nearby round number. 2,000 is much easier to work with than 1,999.
    1,999 is 1 less than 2,000:  1,999 = 2,000 − 1
  2. Take away 2,000 instead of 1,999. Round numbers are easy.
    54,300 − 2,000 = 52,300
  3. Fix the answer. We took away 1 too many, so we must put 1 back.
    52,300 + 1 = 52,301

Check. Use the opposite operation: 52,301 + 1,999 = 54,300. ✓

Watch out. Do not forget the last step. You changed the number you add or take away, so the answer needs fixing.

Worked example 2.3.8
Work out 47,500 + 2,500 using a number fact.

Key words. A number bond is a pair of numbers that add up to a given total, such as 36 + 64 = 100.

Method. Look for a number fact you know. A pair of numbers that make 100 gives a pair that make 1,000, 10,000 or 100,000.

  1. Split 47,500 into a round part and the rest. The rest, 7,500, is the part that goes with 2,500.
    47,500 = 40,000 + 7,500
  2. Spot the number bond. Think of 7,500 and 2,500 as 75 and 25 lots of 100. You know that 75 + 25 = 100.
    75 + 25 = 100, so 7,500 + 2,500 = 10,000
  3. Put the round numbers together. 10,000 is easy to add to 40,000.
    40,000 + 10,000 = 50,000

Check. Use the opposite operation: 50,000 − 2,500 = 47,500. ✓

Watch out. The bond must make 100 (then 1,000, 10,000 and so on). Check: 75 + 25 = 100.

Worked example 2.3.9
Work out 10,000 − 3,600 using a number fact.

Method. Use a number bond to 100. Count how many lots of the small unit there are, then use the fact you know.

  1. Find the unit. 10,000 is 100 lots of 100.
    10,000 = 100 × 100
  2. Count the lots in 3,600. 3,600 is 36 lots of 100.
    3,600 = 36 × 100
  3. Use the bond to 100. What goes with 36 to make 100?
    36 + 64 = 100, so 100 − 36 = 64
  4. Change back to the real numbers. 64 lots of 100.
    64 × 100 = 6,400

Check. Use addition: 6,400 + 3,600 = 10,000. ✓

Watch out. Watch the zeros. 100 lots of 100 is 10,000, but 100 lots of 1,000 is 100,000.

Worked example 2.3.10
Work out 340,000 + 270,000 by counting ten thousands.

Method. Both numbers are multiples of 10,000. Count how many ten thousands each one is, do the small calculation, then change back.

  1. Count the ten thousands. 340,000 is 34 ten thousands. 270,000 is 27 ten thousands.
    340,000 = 34 × 10,000
    270,000 = 27 × 10,000
  2. Do the small calculation. It is just like adding or subtracting 34 and 27.
    34 + 27 = 61
    (34 + 20 = 54, 54 + 7 = 61)
  3. Change back to 10,000s. 61 ten thousands is 61 × 10,000.
    61 × 10,000 = 610,000

Check. Use the opposite operation: 610,000 − 270,000 = 340,000. ✓

Watch out. Count the zeros carefully. 10,000 has 4 zeros, so the answer needs them too.

Worked example 2.3.11
Work out 800,000 − 350,000 by counting ten thousands.

Method. Both numbers are multiples of 10,000. Count how many ten thousands each one is, do the small calculation, then change back.

  1. Count the ten thousands. 800,000 is 80 ten thousands. 350,000 is 35 ten thousands.
    800,000 = 80 × 10,000
    350,000 = 35 × 10,000
  2. Do the small calculation. It is just like adding or subtracting 80 and 35.
    80 − 35 = 45
    (80 − 30 = 50, 50 − 5 = 45)
  3. Change back to 10,000s. 45 ten thousands is 45 × 10,000.
    45 × 10,000 = 450,000

Check. Use the opposite operation: 450,000 + 350,000 = 800,000. ✓

Watch out. Count the zeros carefully. 10,000 has 4 zeros, so the answer needs them too.

Worked example 2.3.12
Work out 35,000 + 36,000 using a near double.

Key words. A double is a number added to itself. A near double is a sum of two numbers that are almost the same.

Method. Near doubles: if two numbers are close, double the smaller one, then add the small difference.

  1. Spot the near double. The numbers are very close together.
    36,000 is 1,000 more than 35,000, so 36,000 = 35,000 + 1,000
  2. Double 35,000. Double each part of the number, then add.
    double 30,000 = 60,000
    double 5,000 = 10,000
    60,000 + 10,000 = 70,000
  3. Add the extra 1,000. 36,000 is 1,000 more than 35,000, so the total is 1,000 more than the double.
    70,000 + 1,000 = 71,000

Check. Use the opposite operation: 71,000 − 36,000 = 35,000. ✓

Watch out. Double the SMALLER number and add the difference. Do not add it to the bigger number.

Worked example 2.3.13
Work out 100,000 − 96,250 by counting up.

Key words. The difference between two numbers is how far apart they are. Counting up is a good way to find a difference when the numbers are close.

Method. To find the difference between two numbers, start at the smaller number and count up to the bigger one. Add up the jumps.

  1. Start at the smaller number. The difference is how far we have to count up.
    Start: 96,250.   Finish: 100,000.
  2. Count up in easy jumps. Jump to a round number first, then add the rest.
    96,250 + 750 = 97,000
    97,000 + 3,000 = 100,000
    96,25097,000100,000+750+3,000
  3. Add up the jumps. The total of all the jumps is the difference.
    750 + 3,000 = 3,750

Check. Use addition: 96,250 + 3,750 = 100,000. ✓

Watch out. Add up the jumps. The answer is the size of the jumps, not the number you finish on.

Watch out

Compensating has two steps. People often do the first step (add 10,000) and then forget to fix the answer. 36,450 + 9,999 is not 46,450. Also be careful with the direction: when you take away too much, you must put some back. And with number facts, count the zeros: 75 + 25 = 100 but 75,000 + 25,000 = 100,000.

Exercise 2C  Mental strategies

Non-calculator
1
FAdd by partitioning. Keep the first number whole and add the parts of the second number one at a time.
(a)36,400 + 25,800
(b)4,675 + 3,230
(c)52,300 + 17,450
(d)240,500 + 135,200
2
FSubtract by partitioning the second number. Take the parts away one at a time.
(a)8,764 − 3,425
(b)54,300 − 21,800
(c)96,500 − 43,250
3
FAdd by bridging through the next 100 or 1,000.
(a)5,970 + 2,400
(b)7,980 + 300
(c)46,950 + 4,200
(d)2,990 + 3,500
4
FSubtract by bridging. Take away enough to land on a round number first.
(a)7,300 − 2,450
(b)6,020 − 450
(c)52,004 − 1,500
(d)30,070 − 350
5
FAdd by compensating. Use a nearby round number, then fix the answer.
(a)36,450 + 9,999
(b)4,586 + 99
(c)24,315 + 198
(d)7,348 + 1,999
6
FSubtract by compensating. Use a nearby round number, then fix the answer.
(a)54,300 − 1,999
(b)8,645 − 99
(c)6,480 − 298
(d)90,250 − 9,998
7
FUse a number fact to add. Think about which pair of numbers makes 100.
(a)47,500 + 2,500
(b)63,600 + 6,400
(c)8,250 + 750
(d)372,000 + 28,000
8
FSubtract from 1,000, 10,000 or 100,000. Use a number bond to 100.
(a)10,000 − 3,600
(b)100,000 − 35,000
(c)1,000 − 650
(d)100,000 − 82,000
9
FCount how many thousands or ten thousands there are, then work out each answer.
(a)340,000 + 270,000
(b)800,000 − 350,000
(c)47,000 + 38,000
(d)91,000 − 56,000
10
FUse a near double to add.
(a)35,000 + 36,000
(b)2,500 + 2,600
(c)48,000 + 49,000
(d)350,000 + 360,000
11
FFind each difference by counting up from the smaller number.
(a)100,000 − 96,250
(b)60,000 − 58,450
(c)30,002 − 29,985
(d)1,000,000 − 987,500
12
RMatch each calculation to the best strategy:   counting up,   compensating,   a near double,   a number fact. Then work out each answer.
60,000 − 59,994    7,348 + 999    45,000 + 45,200    8,500 + 1,500
13
PSA phone costs £399 and a tablet costs £249. How much do they cost together? Work it out in your head.
14
PSA school sells raffle tickets. There are 10,000 tickets and 6,400 have been sold. How many tickets are left? Work it out in your head.
15
RLeo works out 7,348 + 999. He thinks, “Add 1,000 to get 8,348, and that is my answer.” Spot Leo’s mistake and write the correct answer.
16
RProve it.   Show that 49,999 + 29,999 = 79,998 without using the column method.
17
RTwo numbers add up to 70,000. The difference between them is 12,000. What are the two numbers? Work it out in your head.
BiggerSmaller??12,000

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