Year 5: Estimating, rounding and checking
Part of the Year 5 column addition and subtraction chapter. Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Estimating, rounding and checking
Before you do a calculation, it helps to estimate the answer. An estimate is a good guess that is close to the real answer. To estimate, round the numbers, then do the easy calculation. In Chapter 1 you rounded to the nearest 10, 100, 1,000, 10,000 and 100,000. For 48,372 + 29,615, round to the nearest 1,000: 48,000 + 30,000 = 78,000. We write 48,372 + 29,615 ≈ 78,000. The sign ≈ means “is about equal to”.
An estimate tells you if your answer is sensible. If your exact answer is far from your estimate, you have made a mistake somewhere.
How accurate does it need to be? Rounding to a bigger number (like the nearest 10,000) is quicker but less accurate. Rounding to a smaller number (like the nearest 100) takes longer but is closer. The right level depends on the question. For “about how many fans came?” the nearest 1,000 is fine. For “do I have enough money?” you may need to be much more careful.
| Situation | Good level of accuracy |
|---|---|
| About how many fans were at the match? | nearest 1,000 |
| About how far is it from London to Glasgow in metres? | nearest 10,000 |
| Is my estimate close enough to check a column sum? | nearest 100 or 1,000 |
| Do I have exactly enough money for the bill? | exact (no rounding) |
Checking with the inverse. Addition and subtraction are inverse (opposite) operations. They undo each other. If 45,368 + 27,594 = 72,962, then 72,962 − 27,594 should take us back to 45,368. The numbers make a fact family: two additions and two subtractions.
The fact family: 45,368 + 27,594 = 72,962 27,594 + 45,368 = 72,962 72,962 − 27,594 = 45,368 72,962 − 45,368 = 27,594.
Missing numbers. To find a missing number in a sum like ? + 27,594 = 72,962, think of a bar model. The whole is 72,962. One part is 27,594. The other part is the missing number, so take the part away from the whole.
Key words. An estimate is a close answer. Inverse operations are opposites. Sensible means that the answer is the right sort of size. Accuracy means how close an answer is to the exact answer.
Key ideas
- To estimate, round each number (to the nearest 10, 100, 1,000, 10,000 or 100,000), then do the easy calculation.
- ≈ means “is about equal to”.
- Choose the level of accuracy to fit the question. If the estimate is close to a limit, use a closer estimate or the exact answer.
- An exact answer should be close to the estimate. If not, check your working.
- Addition and subtraction are inverses: if a + b = c, then c − b = a.
- Use a bar model to find a missing number: find a part by subtracting, find a whole by adding.
Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.
Method. Round each number to the nearest 1,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.
- Round 48,372 to the nearest 1,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.48,372 ≈ 48,000 (the hundreds digit is 3, so round down)
- Round 29,615 to the nearest 1,000. Do the same for the next number.29,615 ≈ 30,000 (the hundreds digit is 6, so round up)
- Do the easy calculation. Rounded numbers are easy to add or subtract.48,000 + 30,000 = 78,000
Check. The exact answer is 77,987. It is close to the estimate of 78,000. ✓
Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).
Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.
Method. Round each number to the nearest 10,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.
- Round 652,480 to the nearest 10,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.652,480 ≈ 650,000 (the thousands digit is 2, so round down)
- Round 187,950 to the nearest 10,000. Do the same for the next number.187,950 ≈ 190,000 (the thousands digit is 7, so round up)
- Do the easy calculation. Rounded numbers are easy to add or subtract.650,000 − 190,000 = 460,000
Check. The exact answer is 464,530. It is close to the estimate of 460,000. ✓
Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).
Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.
Method. Round each number to the nearest 1,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.
- Round 24,380 to the nearest 1,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.24,380 ≈ 24,000 (the hundreds digit is 3, so round down)
- Round 8,720 to the nearest 1,000. Do the same for the next number.8,720 ≈ 9,000 (the hundreds digit is 7, so round up)
- Round 15,950 to the nearest 1,000. Do the same for the next number.15,950 ≈ 16,000 (the hundreds digit is 9, so round up)
- Do the easy calculation. Rounded numbers are easy to add or subtract.24,000 + 9,000 + 16,000 = 49,000
Check. The exact answer is 49,050. It is close to the estimate of 49,000. ✓
Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).
Key words. To round a number is to change it to a nearby number that is easier to use. ≈ means “is about equal to”.
Method. Look at the digit to the right of the one you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.
- Find the two multiples of 10,000 that 275,490 is between. Rounding to the nearest 10,000 means choosing the closest multiple of 10,000.270,000 < 275,490 < 280,000
- Look at the thousands digit (the digit just to the right of the ten thousands column). This digit tells us if we are past halfway.The thousands digit is 5.
- Decide: round up or round down? 5 or more: round up. 4 or less: round down.5 is 5 or more, so round up.
- Write the rounded number. Every digit to the right of the ten thousands column becomes a zero.275,490 ≈ 280,000 (to the nearest 10,000)
Check. Distances: 275,490 − 270,000 = 5,490 and 280,000 − 275,490 = 4,510. 280,000 is the nearer one. ✓
Watch out. Only the digit just to the right decides. Do not look at digits further along.
Method. Estimate first, but be careful: if the estimate is close to the amount we have, we must work out the exact answer.
- Round each price to the nearest £1,000. This is quick, but it may not be accurate enough.Tables: £2,540 ≈ £3,000
Chairs: £3,560 ≈ £4,000
Whiteboard: £1,390 ≈ £1,000
£3,000 + £4,000 + £1,000 = £8,000 - Compare with Sofia’s £7,500. £8,000 is more than £7,500, so this estimate says there is not enough. But the estimate is £500 over, which is a lot of rounding error.£8,000 > £7,500
- Round to the nearest £100 for a closer estimate. Rounding to a smaller place value gives a more accurate answer.Tables: £2,540 ≈ £2,500
Chairs: £3,560 ≈ £3,600
Whiteboard: £1,390 ≈ £1,400
£2,500 + £3,600 + £1,400 = £7,500 - Work out the exact total. The estimate is very close to £7,500, so only the exact answer can decide.£2,540 + £3,560 + £1,390 = £7,490
Th H T O 2 5 4 0 + 3 5 6 0 + 1 3 9 0 7 4 9 0 1 1 - Decide. £7,490 is less than £7,500.Sofia has enough. The total is £7,490, so £10 is left over.
Check. £7,490 + £10 = £7,500. ✓
Watch out. A rough estimate cannot answer a question when the numbers are close. Use a more accurate estimate, or the exact answer.
Method. Look at what happened to each number when it was rounded. Did it go up or down? Then think about what that does to the answer.
- Compare each number with its rounded number. If a number was rounded up, the rounded number is bigger than the real one.36,640 was rounded up to 37,000 (360 more). This makes the answer bigger.
28,580 was rounded up to 29,000 (420 more). This makes the answer bigger. - Decide what that means for the estimate. Both changes push the answer the same way, so we can be sure.The estimate (66,000) is more than the exact answer, so the exact answer is less than 66,000.
- Check with the exact answer. 36,640 + 28,580 = 65,220
TTh Th H T O 3 6 6 4 0 + 2 8 5 8 0 6 5 2 2 0 1 1 1
Check. 65,220 is less than 66,000. ✓
Watch out. When you add, rounding up makes the answer too big. When you subtract, rounding the first number up makes the answer too big, but rounding the second number up makes it too small.
Method. Estimate the answer by rounding. If Jack’s answer is nowhere near the estimate, it cannot be right.
- Round the numbers. Round to the nearest 1,000 so the calculation is easy.64,205 ≈ 64,000 (the hundreds digit is 2, so round down)
38,790 ≈ 39,000 (the hundreds digit is 7, so round up) - Work out the estimate. 64,000 − 39,000 = 25,000
- Compare with Jack’s answer. Jack got 34,595. The estimate says the answer should be close to 25,000.34,595 is a long way from 25,000, so it is not sensible.
- Find the correct answer. Use the column method carefully.
TTh Th H T O 5 613 411 210 05 − 3 8 7 9 0 2 5 4 1 5
64,205 − 38,790 = 25,415
Check. The correct answer 25,415 is close to the estimate 25,000. ✓
Watch out. Always ask: is my answer about the right size? An estimate takes a few seconds and can catch big mistakes.
Key words. Inverse operations are opposites. Addition is the inverse of subtraction, and subtraction is the inverse of addition.
Method. Use the inverse (opposite) operation to check. Addition and subtraction undo each other. If you go back, you should reach the number you started with.
- Write Sam’s answer as an inverse calculation. The opposite operation should take us back to the first number.45,368 + 27,594 = 72,962 so 72,962 − 27,594 should be 45,368
- Work it out. Use the column method.72,962 − 27,594 = 45,368
TTh Th H T O 6 712 28 915 612 2− 2 7 5 9 4 4 5 3 6 8 - Compare. We should get 45,368.We got 45,368. 45,368 = 45,368, so Sam is correct.
Check. The correct answer is 72,962. Sam was right. ✓
Watch out. The inverse of + is −. The inverse of − is +. If the check does not work, one of the calculations has a mistake.
Key words. Inverse operations are opposites. Addition is the inverse of subtraction, and subtraction is the inverse of addition.
Method. Use the inverse (opposite) operation to check. Addition and subtraction undo each other. If you go back, you should reach the number you started with.
- Write Priya’s answer as an inverse calculation. The opposite operation should take us back to the first number.81,305 − 36,749 = 44,666 so 44,666 + 36,749 should be 81,305
- Work it out. Use the column method.44,666 + 36,749 = 81,415
TTh Th H T O 4 4 6 6 6 + 3 6 7 4 9 8 1 4 1 5 1 1 1 1 - Compare. We should get 81,305.We got 81,415. 81,415 is not 81,305, so Priya is wrong.
- Find the correct answer. Do the original calculation again.81,305 − 36,749 = 44,556
TTh Th H T O 7 810 112 39 015 5− 3 6 7 4 9 4 4 5 5 6
Check. The correct answer is 44,556. 44,556 + 36,749 = 81,305. ✓
Watch out. The inverse of + is −. The inverse of − is +. If the check does not work, one of the calculations has a mistake.
Method. Draw a bar model, then use the inverse (opposite) operation to find the missing number.
- Look at the number sentence. The box stands for the number we want to find.? + 27,594 = 72,962
- Draw a bar model. The whole is the total, and we know one part. The other part is the missing number.
- Choose the operation. To find a missing part, take the part we know away from the whole.72,962 − 27,594
- Work it out. Use the column method if the numbers are big.72,962 − 27,594 = 45,368
TTh Th H T O 6 712 28 915 612 2− 2 7 5 9 4 4 5 3 6 8 - Check in the number sentence. Put the answer back in the box.45,368 + 27,594 = 72,962 ✓
Check. Read the whole number sentence with your answer in the box. It must be true. ✓
Watch out. The missing number is not always the last number. Think about which number is the whole, then decide whether to add or subtract.
Method. Draw a bar model, then use the inverse (opposite) operation to find the missing number.
- Look at the number sentence. The box stands for the number we want to find.500,000 − ? = 284,615
- Draw a bar model. The whole is the first number. The missing number is one of the parts.
- Choose the operation. To find a missing part, take the part we know away from the whole.500,000 − 284,615
- Work it out. Use the column method if the numbers are big.500,000 − 284,615 = 215,385
HTh TTh Th H T O 4 59 09 09 09 010 0− 2 8 4 6 1 5 2 1 5 3 8 5 - Check in the number sentence. Put the answer back in the box.500,000 − 215,385 = 284,615 ✓
Check. Read the whole number sentence with your answer in the box. It must be true. ✓
Watch out. The missing number is not always the last number. Think about which number is the whole, then decide whether to add or subtract.
Watch out
An estimate is not the exact answer, so do not expect it to match exactly. It should only be close. When you check with the inverse, use the opposite sign: check an addition by subtracting, and check a subtraction by adding. And when a total is close to a limit (such as a budget), rounding a lot can give the wrong decision.Exercise 2D Estimating, inverse operations and missing numbers
Non-calculatorKey words. To round a number is to change it to a nearby number that is easier to use. ≈ means “is about equal to”.
Method. Look at the digit to the right of the one you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.
- Find the two multiples of 1,000 that 46,382 is between. Rounding to the nearest 1,000 means choosing the closest multiple of 1,000.46,000 < 46,382 < 47,000
- Look at the hundreds digit (the digit just to the right of the thousands column). This digit tells us if we are past halfway.The hundreds digit is 3.
- Decide: round up or round down? 5 or more: round up. 4 or less: round down.3 is 4 or less, so round down.
- Write the rounded number. Every digit to the right of the thousands column becomes a zero.46,382 ≈ 46,000 (to the nearest 1,000)
Check. Distances: 46,382 − 46,000 = 382 and 47,000 − 46,382 = 618. 46,000 is the nearer one. ✓
Watch out. Only the digit just to the right decides. Do not look at digits further along.
Key words. To round a number is to change it to a nearby number that is easier to use. ≈ means “is about equal to”.
Method. Look at the digit to the right of the one you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.
- Find the two multiples of 10,000 that 275,490 is between. Rounding to the nearest 10,000 means choosing the closest multiple of 10,000.270,000 < 275,490 < 280,000
- Look at the thousands digit (the digit just to the right of the ten thousands column). This digit tells us if we are past halfway.The thousands digit is 5.
- Decide: round up or round down? 5 or more: round up. 4 or less: round down.5 is 5 or more, so round up.
- Write the rounded number. Every digit to the right of the ten thousands column becomes a zero.275,490 ≈ 280,000 (to the nearest 10,000)
Check. Distances: 275,490 − 270,000 = 5,490 and 280,000 − 275,490 = 4,510. 280,000 is the nearer one. ✓
Watch out. Only the digit just to the right decides. Do not look at digits further along.
Key words. To round a number is to change it to a nearby number that is easier to use. ≈ means “is about equal to”.
Method. Look at the digit to the right of the one you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.
- Find the two multiples of 100,000 that 638,251 is between. Rounding to the nearest 100,000 means choosing the closest multiple of 100,000.600,000 < 638,251 < 700,000
- Look at the ten thousands digit (the digit just to the right of the hundred thousands column). This digit tells us if we are past halfway.The ten thousands digit is 3.
- Decide: round up or round down? 5 or more: round up. 4 or less: round down.3 is 4 or less, so round down.
- Write the rounded number. Every digit to the right of the hundred thousands column becomes a zero.638,251 ≈ 600,000 (to the nearest 100,000)
Check. Distances: 638,251 − 600,000 = 38,251 and 700,000 − 638,251 = 61,749. 600,000 is the nearer one. ✓
Watch out. Only the digit just to the right decides. Do not look at digits further along.
Key words. To round a number is to change it to a nearby number that is easier to use. ≈ means “is about equal to”.
Method. Look at the digit to the right of the one you are rounding to. If it is 5 or more, round up. If it is 4 or less, round down.
- Find the two multiples of 100 that 84,950 is between. Rounding to the nearest 100 means choosing the closest multiple of 100.84,900 < 84,950 < 85,000
- Look at the tens digit (the digit just to the right of the hundreds column). This digit tells us if we are past halfway.The tens digit is 5.
- Decide: round up or round down? 5 or more: round up. 4 or less: round down.5 is 5 or more, so round up.
- Write the rounded number. Every digit to the right of the hundreds column becomes a zero.84,950 ≈ 85,000 (to the nearest 100)
Check. Distances: 84,950 − 84,900 = 50 and 85,000 − 84,950 = 50. 85,000 is the nearer one. ✓
Watch out. Only look at the digit just to the right. Halfway (like 6,500 to the nearest 1,000) always rounds up.
Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.
Method. Round each number to the nearest 1,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.
- Round 48,372 to the nearest 1,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.48,372 ≈ 48,000 (the hundreds digit is 3, so round down)
- Round 29,615 to the nearest 1,000. Do the same for the next number.29,615 ≈ 30,000 (the hundreds digit is 6, so round up)
- Do the easy calculation. Rounded numbers are easy to add or subtract.48,000 + 30,000 = 78,000
Check. The exact answer is 77,987. It is close to the estimate of 78,000. ✓
Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).
Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.
Method. Round each number to the nearest 1,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.
- Round 63,540 to the nearest 1,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.63,540 ≈ 64,000 (the hundreds digit is 5, so round up)
- Round 21,480 to the nearest 1,000. Do the same for the next number.21,480 ≈ 21,000 (the hundreds digit is 4, so round down)
- Do the easy calculation. Rounded numbers are easy to add or subtract.64,000 − 21,000 = 43,000
Check. The exact answer is 42,060. It is close to the estimate of 43,000. ✓
Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).
Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.
Method. Round each number to the nearest 1,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.
- Round 71,906 to the nearest 1,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.71,906 ≈ 72,000 (the hundreds digit is 9, so round up)
- Round 38,095 to the nearest 1,000. Do the same for the next number.38,095 ≈ 38,000 (the hundreds digit is 0, so round down)
- Do the easy calculation. Rounded numbers are easy to add or subtract.72,000 + 38,000 = 110,000
Check. The exact answer is 110,001. It is close to the estimate of 110,000. ✓
Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).
Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.
Method. Round each number to the nearest 1,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.
- Round 95,210 to the nearest 1,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.95,210 ≈ 95,000 (the hundreds digit is 2, so round down)
- Round 47,860 to the nearest 1,000. Do the same for the next number.47,860 ≈ 48,000 (the hundreds digit is 8, so round up)
- Do the easy calculation. Rounded numbers are easy to add or subtract.95,000 − 48,000 = 47,000
Check. The exact answer is 47,350. It is close to the estimate of 47,000. ✓
Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).
Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.
Method. Round each number to the nearest 10,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.
- Round 652,480 to the nearest 10,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.652,480 ≈ 650,000 (the thousands digit is 2, so round down)
- Round 187,950 to the nearest 10,000. Do the same for the next number.187,950 ≈ 190,000 (the thousands digit is 7, so round up)
- Do the easy calculation. Rounded numbers are easy to add or subtract.650,000 − 190,000 = 460,000
Check. The exact answer is 464,530. It is close to the estimate of 460,000. ✓
Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).
Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.
Method. Round each number to the nearest 10,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.
- Round 394,200 to the nearest 10,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.394,200 ≈ 390,000 (the thousands digit is 4, so round down)
- Round 211,800 to the nearest 10,000. Do the same for the next number.211,800 ≈ 210,000 (the thousands digit is 1, so round down)
- Do the easy calculation. Rounded numbers are easy to add or subtract.390,000 + 210,000 = 600,000
Check. The exact answer is 606,000. It is close to the estimate of 600,000. ✓
Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).
Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.
Method. Round each number to the nearest 10,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.
- Round 870,450 to the nearest 10,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.870,450 ≈ 870,000 (the thousands digit is 0, so round down)
- Round 215,600 to the nearest 10,000. Do the same for the next number.215,600 ≈ 220,000 (the thousands digit is 5, so round up)
- Do the easy calculation. Rounded numbers are easy to add or subtract.870,000 − 220,000 = 650,000
Check. The exact answer is 654,850. It is close to the estimate of 650,000. ✓
Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).
Method. Round the numbers, work out an easy estimate, then find the choice that is closest.
- Round the numbers to the nearest 1,000. Rounded numbers are quick to add or subtract.61,840 ≈ 62,000 (the hundreds digit is 8, so round up)
28,260 ≈ 28,000 (the hundreds digit is 2, so round down) - Work out the estimate. 62,000 + 28,000 = 90,000
- Choose the closest answer. The choices are very different sizes, so only one can be close.A: 9,000
B: 90,000 (close to the estimate)
C: 900,000
D: 19,000
Check. The exact answer is 90,100, which is close to 90,000. ✓
Watch out. You do not need the exact answer. Round, work out the easy calculation, and look for the closest choice.
Method. Round the numbers, work out an easy estimate, then find the choice that is closest.
- Round the numbers to the nearest 10,000. Rounded numbers are quick to add or subtract.803,450 ≈ 800,000 (the thousands digit is 3, so round down)
297,600 ≈ 300,000 (the thousands digit is 7, so round up) - Work out the estimate. 800,000 − 300,000 = 500,000
- Choose the closest answer. The choices are very different sizes, so only one can be close.A: 50,000
B: 500,000 (close to the estimate)
C: 5,000,000
D: 1,100,000
Check. The exact answer is 505,850, which is close to 500,000. ✓
Watch out. You do not need the exact answer. Round, work out the easy calculation, and look for the closest choice.
Method. The inverse of addition is subtraction. Start with the answer and take away one of the numbers you added.
- Write the calculation and its answer. The answer is the whole.42,080 + 15,365 = 57,445
- Take one part away from the whole. This undoes the addition and takes us back to the other part.
TTh Th H T O 5 7 3 414 45 − 1 5 3 6 5 4 2 0 8 0 - Write the inverse calculation. 57,445 − 15,365 = 42,080 (or 57,445 − 42,080 = 15,365)
Check. It works because 42,080 is what is left when you take 15,365 from 57,445. ✓
Watch out. The inverse of + is −. Start with the biggest number, the answer.
Method. The inverse of subtraction is addition. Add the number you took away back on to the answer.
- Write the calculation and its answer. The number we started with is the whole.90,020 − 36,475 = 53,545
- Add the part back on. This undoes the subtraction and takes us back to the start.
TTh Th H T O 5 3 5 4 5 + 3 6 4 7 5 9 0 0 2 0 1 1 1 1
53,545 + 36,475 = 90,020 - Write the inverse calculation. 53,545 + 36,475 = 90,020
Check. Check: 90,020 − 53,545 = 36,475, the number we took away. ✓
Watch out. The inverse of − is +. The answer to the subtraction plus the number taken away makes the number you started with.
Method. The inverse of addition is subtraction. Start with the answer and take away one of the numbers you added.
- Write the calculation and its answer. The answer is the whole.275,500 + 386,200 = 661,700
- Take one part away from the whole. This undoes the addition and takes us back to the other part.
HTh TTh Th H T O 5 615 611 17 0 0 − 3 8 6 2 0 0 2 7 5 5 0 0 - Write the inverse calculation. 661,700 − 386,200 = 275,500 (or 661,700 − 275,500 = 386,200)
Check. It works because 275,500 is what is left when you take 386,200 from 661,700. ✓
Watch out. The inverse of + is −. Start with the biggest number, the answer.
Method. The inverse of subtraction is addition. Add the number you took away back on to the answer.
- Write the calculation and its answer. The number we started with is the whole.600,000 − 124,500 = 475,500
- Add the part back on. This undoes the subtraction and takes us back to the start.
HTh TTh Th H T O 4 7 5 5 0 0 + 1 2 4 5 0 0 6 0 0 0 0 0 1 1 1
475,500 + 124,500 = 600,000 - Write the inverse calculation. 475,500 + 124,500 = 600,000
Check. Check: 600,000 − 475,500 = 124,500, the number we took away. ✓
Watch out. The inverse of − is +. The answer to the subtraction plus the number taken away makes the number you started with.
Key words. Inverse operations are opposites. Addition is the inverse of subtraction, and subtraction is the inverse of addition.
Method. Use the inverse (opposite) operation to check. Addition and subtraction undo each other. If you go back, you should reach the number you started with.
- Write Aisha’s answer as an inverse calculation. The opposite operation should take us back to the first number.36,404 + 27,918 = 64,322 so 64,322 − 27,918 should be 36,404
- Work it out. Use the column method.64,322 − 27,918 = 36,404
TTh Th H T O 5 613 413 31 212 2− 2 7 9 1 8 3 6 4 0 4 - Compare. We should get 36,404.We got 36,404. 36,404 = 36,404, so Aisha is correct.
Check. The correct answer is 64,322. Aisha was right. ✓
Watch out. The inverse of + is −. The inverse of − is +. If the check does not work, one of the calculations has a mistake.
Key words. Inverse operations are opposites. Addition is the inverse of subtraction, and subtraction is the inverse of addition.
Method. Use the inverse (opposite) operation to check. Addition and subtraction undo each other. If you go back, you should reach the number you started with.
- Write Aisha’s answer as an inverse calculation. The opposite operation should take us back to the first number.70,005 − 28,864 = 41,241 so 41,241 + 28,864 should be 70,005
- Work it out. Use the column method.41,241 + 28,864 = 70,105
TTh Th H T O 4 1 2 4 1 + 2 8 8 6 4 7 0 1 0 5 1 1 1 - Compare. We should get 70,005.We got 70,105. 70,105 is not 70,005, so Aisha is wrong.
- Find the correct answer. Do the original calculation again.70,005 − 28,864 = 41,141
TTh Th H T O 6 79 09 010 05 − 2 8 8 6 4 4 1 1 4 1
Check. The correct answer is 41,141. 41,141 + 28,864 = 70,005. ✓
Watch out. The inverse of + is −. The inverse of − is +. If the check does not work, one of the calculations has a mistake.
Key words. Inverse operations are opposites. Addition is the inverse of subtraction, and subtraction is the inverse of addition.
Method. Use the inverse (opposite) operation to check. Addition and subtraction undo each other. If you go back, you should reach the number you started with.
- Write Aisha’s answer as an inverse calculation. The opposite operation should take us back to the first number.51,904 + 38,444 = 90,348 so 90,348 − 38,444 should be 51,904
- Work it out. Use the column method.90,348 − 38,444 = 51,904
TTh Th H T O 8 99 013 34 8 − 3 8 4 4 4 5 1 9 0 4 - Compare. We should get 51,904.We got 51,904. 51,904 = 51,904, so Aisha is correct.
Check. The correct answer is 90,348. Aisha was right. ✓
Watch out. The inverse of + is −. The inverse of − is +. If the check does not work, one of the calculations has a mistake.
Key words. Inverse operations are opposites. Addition is the inverse of subtraction, and subtraction is the inverse of addition.
Method. Use the inverse (opposite) operation to check. Addition and subtraction undo each other. If you go back, you should reach the number you started with.
- Write Aisha’s answer as an inverse calculation. The opposite operation should take us back to the first number.84,300 − 45,688 = 38,712 so 38,712 + 45,688 should be 84,300
- Work it out. Use the column method.38,712 + 45,688 = 84,400
TTh Th H T O 3 8 7 1 2 + 4 5 6 8 8 8 4 4 0 0 1 1 1 1 - Compare. We should get 84,300.We got 84,400. 84,400 is not 84,300, so Aisha is wrong.
- Find the correct answer. Do the original calculation again.84,300 − 45,688 = 38,612
TTh Th H T O 7 813 412 39 010 0− 4 5 6 8 8 3 8 6 1 2
Check. The correct answer is 38,612. 38,612 + 45,688 = 84,300. ✓
Watch out. The inverse of + is −. The inverse of − is +. If the check does not work, one of the calculations has a mistake.
Method. Draw a bar model, then use the inverse (opposite) operation to find the missing number.
- Look at the number sentence. The box stands for the number we want to find.? + 25,000 = 60,000
- Draw a bar model. The whole is the total, and we know one part. The other part is the missing number.
- Choose the operation. To find a missing part, take the part we know away from the whole.60,000 − 25,000
- Work it out. Use the column method if the numbers are big.60,000 − 25,000 = 35,000
TTh Th H T O 5 610 00 0 0 − 2 5 0 0 0 3 5 0 0 0 - Check in the number sentence. Put the answer back in the box.35,000 + 25,000 = 60,000 ✓
Check. Read the whole number sentence with your answer in the box. It must be true. ✓
Watch out. The missing number is not always the last number. Think about which number is the whole, then decide whether to add or subtract.
Method. Draw a bar model, then use the inverse (opposite) operation to find the missing number.
- Look at the number sentence. The box stands for the number we want to find.18,500 + ? = 40,000
- Draw a bar model. The whole is the total, and we know one part. The other part is the missing number.
- Choose the operation. To find a missing part, take the part we know away from the whole.40,000 − 18,500
- Work it out. Use the column method if the numbers are big.40,000 − 18,500 = 21,500
TTh Th H T O 3 49 010 00 0 − 1 8 5 0 0 2 1 5 0 0 - Check in the number sentence. Put the answer back in the box.18,500 + 21,500 = 40,000 ✓
Check. Read the whole number sentence with your answer in the box. It must be true. ✓
Watch out. The missing number is not always the last number. Think about which number is the whole, then decide whether to add or subtract.
Method. Draw a bar model, then use the inverse (opposite) operation to find the missing number.
- Look at the number sentence. The box stands for the number we want to find.? − 12,750 = 21,400
- Draw a bar model. The missing number is the whole. It is made of the two parts we know.
- Choose the operation. To find the whole, add the two parts together.21,400 + 12,750
- Work it out. Use the column method if the numbers are big.21,400 + 12,750 = 34,150
TTh Th H T O 2 1 4 0 0 + 1 2 7 5 0 3 4 1 5 0 1 - Check in the number sentence. Put the answer back in the box.34,150 − 12,750 = 21,400 ✓
Check. Read the whole number sentence with your answer in the box. It must be true. ✓
Watch out. The missing number is not always the last number. Think about which number is the whole, then decide whether to add or subtract.
Method. Draw a bar model, then use the inverse (opposite) operation to find the missing number.
- Look at the number sentence. The box stands for the number we want to find.90,000 − ? = 67,850
- Draw a bar model. The whole is the first number. The missing number is one of the parts.
- Choose the operation. To find a missing part, take the part we know away from the whole.90,000 − 67,850
- Work it out. Use the column method if the numbers are big.90,000 − 67,850 = 22,150
TTh Th H T O 8 99 09 010 00 − 6 7 8 5 0 2 2 1 5 0 - Check in the number sentence. Put the answer back in the box.90,000 − 22,150 = 67,850 ✓
Check. Read the whole number sentence with your answer in the box. It must be true. ✓
Watch out. The missing number is not always the last number. Think about which number is the whole, then decide whether to add or subtract.
Method. Draw a bar model, then use the inverse (opposite) operation to find the missing number.
- Look at the number sentence. The box stands for the number we want to find.345,000 + ? = 800,000
- Draw a bar model. The whole is the total, and we know one part. The other part is the missing number.
- Choose the operation. To find a missing part, take the part we know away from the whole.800,000 − 345,000
- Work it out. Use the column method if the numbers are big.800,000 − 345,000 = 455,000
HTh TTh Th H T O 7 89 010 00 0 0 − 3 4 5 0 0 0 4 5 5 0 0 0 - Check in the number sentence. Put the answer back in the box.345,000 + 455,000 = 800,000 ✓
Check. Read the whole number sentence with your answer in the box. It must be true. ✓
Watch out. The missing number is not always the last number. Think about which number is the whole, then decide whether to add or subtract.
Key words. To estimate is to find an answer that is close, usually by rounding the numbers first. ≈ means “is about equal to”.
Method. Round each number to the nearest 1,000, then work out the easy calculation. The answer is only an estimate (a good guess), so it will not be exact.
- Round 48,372 to the nearest 1,000. Look at the digit just to the right. 5 or more: round up. 4 or less: round down.48,372 ≈ 48,000 (the hundreds digit is 3, so round down)
- Round 29,615 to the nearest 1,000. Do the same for the next number.29,615 ≈ 30,000 (the hundreds digit is 6, so round up)
- Do the easy calculation. Rounded numbers are easy to add or subtract.48,000 + 30,000 = 78,000
Check. The exact answer is 77,987. It is close to the estimate of 78,000. ✓
Watch out. An estimate is not the exact answer. Use the sign ≈ (about equal to).
Key words. Column addition is a written method. You write the numbers one under the other, with ones under ones, tens under tens and so on, and add one column at a time. When a column adds up to 10 or more, you carry to the next column. The sum or total is the answer to an addition.
Method. Write the numbers in columns with the same place values lined up. Add the ones first, then the tens, hundreds and so on. If a column makes 10 or more, carry to the next column.
- Line up the digits in columns. Ones go under ones, tens under tens, hundreds under hundreds and so on, so every digit is added to digits of the same value.
TTh Th H T O 4 8 3 7 2 + 2 9 6 1 5 - Add the ones. Always start with the ones, on the right.2 + 5 = 7
7 is less than 10, so write 7 in the ones column. There is nothing to carry. - Add the tens. Now the next column to the left.7 + 1 = 8
8 is less than 10, so write 8 in the tens column. There is nothing to carry. - Add the hundreds. Now the next column to the left.3 + 6 = 9
9 is less than 10, so write 9 in the hundreds column. There is nothing to carry. - Add the thousands. Now the next column to the left.8 + 9 = 17
17 thousands is 1 ten thousand and 7 thousands. Write 7 in the thousands column and carry 1 to the ten thousands column. - Add the ten thousands. Now the next column to the left. Add the 1 we carried as well.4 + 2 + 1 = 7 (the red digit is the carry)
7 is less than 10, so write 7 in the ten thousands column. There is nothing to carry. - Read the answer. Check that the answer has a sensible number of digits, then write it.
TTh Th H T O 4 8 3 7 2 + 2 9 6 1 5 7 7 9 8 7 1
48,372 + 29,615 = 77,987
Check. Use the opposite (inverse) operation: 77,987 − 29,615 = 48,372. ✓
Estimate: 48,000 + 30,000 = 78,000 (rounding to the nearest 1,000). It is close to the exact answer. ✓
Watch out. Do not forget the carried digits. Write each one small under the answer line, and add it in the next column.
Method. Round both numbers to the nearest 1,000 and add. Then find the exact total to be sure.
- Round both numbers to the nearest 1,000. The hundreds digit tells us whether to round up or down.31,480 ≈ 31,000 (the hundreds digit is 4, so round down)
30,270 ≈ 30,000 (the hundreds digit is 2, so round down) - Add the rounded numbers. This is the estimate.31,000 + 30,000 = 61,000
- Work out the exact total. The estimate is close to 60,000, so we need the exact total to be sure.31,480 + 30,270 = 61,750
TTh Th H T O 3 1 4 8 0 + 3 0 2 7 0 6 1 7 5 0 1 - Answer. 61,750 is more than 60,000.Anil is right: 61,750 tickets were sold, which is 1,750 more than the 60,000 seats.
Check. 61,750 − 30,270 = 31,480. ✓
Watch out. An estimate cannot always decide a question. If the estimate is very close to the limit, work out the exact answer too.
Method. Estimate first, but be careful: if the estimate is close to the amount we have, we must work out the exact answer.
- Round each price to the nearest £1,000. This is quick, but it may not be accurate enough.Sofa: £3,540 ≈ £4,000
Table: £2,560 ≈ £3,000
Bed: £2,880 ≈ £3,000
£4,000 + £3,000 + £3,000 = £10,000 - Compare with Mr Khan’s £9,000. £10,000 is more than £9,000, so this estimate says there is not enough. But the estimate is £1,000 over, which is a lot of rounding error.£10,000 > £9,000
- Round to the nearest £100 for a closer estimate. Rounding to a smaller place value gives a more accurate answer.Sofa: £3,540 ≈ £3,500
Table: £2,560 ≈ £2,600
Bed: £2,880 ≈ £2,900
£3,500 + £2,600 + £2,900 = £9,000 - Work out the exact total. The estimate is very close to £9,000, so only the exact answer can decide.£3,540 + £2,560 + £2,880 = £8,980
Th H T O 3 5 4 0 + 2 5 6 0 + 2 8 8 0 8 9 8 0 1 1 - Decide. £8,980 is less than £9,000.Mr Khan has enough. The total is £8,980, so £20 is left over.
Check. £8,980 + £20 = £9,000. ✓
Watch out. A rough estimate cannot answer a question when the numbers are close. Use a more accurate estimate, or the exact answer.
Method. Estimate the answer by rounding. If Mia’s answer is nowhere near the estimate, it cannot be right.
- Round the numbers. Round to the nearest 1,000 so the calculation is easy.37,859 ≈ 38,000 (the hundreds digit is 8, so round up)
26,478 ≈ 26,000 (the hundreds digit is 4, so round down) - Work out the estimate. 38,000 + 26,000 = 64,000
- Compare with Mia’s answer. Mia got 53,227. The estimate says the answer should be close to 64,000.53,227 is a long way from 64,000, so it is not sensible.
- Find the correct answer. Use the column method carefully.
TTh Th H T O 3 7 8 5 9 + 2 6 4 7 8 6 4 3 3 7 1 1 1 1
37,859 + 26,478 = 64,337
Check. The correct answer 64,337 is close to the estimate 64,000. ✓
Watch out. Always ask: is my answer about the right size? An estimate takes a few seconds and can catch big mistakes.
Method. Look at what happened to each number when it was rounded. Did it go up or down? Then think about what that does to the answer.
- Compare each number with its rounded number. If a number was rounded up, the rounded number is bigger than the real one.36,640 was rounded up to 37,000 (360 more). This makes the answer bigger.
28,580 was rounded up to 29,000 (420 more). This makes the answer bigger. - Decide what that means for the estimate. Both changes push the answer the same way, so we can be sure.The estimate (66,000) is more than the exact answer, so the exact answer is less than 66,000.
- Check with the exact answer. 36,640 + 28,580 = 65,220
TTh Th H T O 3 6 6 4 0 + 2 8 5 8 0 6 5 2 2 0 1 1 1
Check. 65,220 is less than 66,000. ✓
Watch out. When you add, rounding up makes the answer too big. When you subtract, rounding the first number up makes the answer too big, but rounding the second number up makes it too small.
Method. Try some examples. Look for one where the estimate is bigger and one where it is smaller.
- Try 8,260 − 3,840. 8,260 rounds down and 3,840 rounds up. Both make the answer smaller.8,000 − 4,000 = 4,000, but the exact answer is 4,420. The estimate is smaller.
- Try 8,740 − 3,260. 8,740 rounds up and 3,260 rounds down. Both make the answer bigger.9,000 − 3,000 = 6,000, but the exact answer is 5,480. The estimate is bigger.
- Decide. We found one example of each.Sometimes
Check. It can also be exactly right (for example 9,000 − 3,000 = 6,000 where nothing needs rounding). ✓
Watch out. “Sometimes” needs two examples: one that works and one that does not.
Method. Think about what the number is used for. Is a rough answer enough?
- Read the situation. What does the person need to know?A newspaper says how many people were at a football match (48,372).
- Choose the level of accuracy. We only need a rough idea of the crowd size.the nearest 1,000 (about 48,000)
Check. A good choice gives an answer that is close enough for the job. ✓
Watch out. Rounding more makes the answer quicker but less accurate. Do not round when the exact amount matters.
Method. Think about what the number is used for. Is a rough answer enough?
- Read the situation. What does the person need to know?A shopkeeper checks that three prices add to exactly the amount on the till.
- Choose the level of accuracy. The till must be exactly right.no rounding (exact) (the exact total)
Check. A good choice gives an answer that is close enough for the job. ✓
Watch out. Rounding more makes the answer quicker but less accurate. Do not round when the exact amount matters.
Method. Think about what the number is used for. Is a rough answer enough?
- Read the situation. What does the person need to know?A pilot says how far the plane flew (5,632 km).
- Choose the level of accuracy. A journey this long does not need to be exact to the last kilometre.the nearest 100 (about 5,600 km)
Check. A good choice gives an answer that is close enough for the job. ✓
Watch out. Rounding more makes the answer quicker but less accurate. Do not round when the exact amount matters.
Method. Think about what the number is used for. Is a rough answer enough?
- Read the situation. What does the person need to know?Aisha checks she has enough for a £48 bill. She has £47.
- Choose the level of accuracy. Rounding £47 and £48 to the nearest £10 would give the same number and hide the problem.no rounding (exact) (the exact amounts)
Check. A good choice gives an answer that is close enough for the job. ✓
Watch out. Rounding more makes the answer quicker but less accurate. Do not round when the exact amount matters.