GCSE Maths: Estimation and checking answers (Higher)
Part of the GCSE Maths decimals, rounding and estimation chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Estimation and checking answers
An estimate is an approximate answer found using simple numbers. We use estimates to check that an answer is sensible, and to get a rough idea of the size of a calculation when an exact answer is not needed. This is very useful when you use a calculator: it is easy to press a wrong key or to miss a decimal point, and an estimate will catch it.
How to estimate
- Round every number in the calculation to 1 significant figure.
- Write the calculation again with the rounded numbers. Keep the same operations.
- Work it out (without a calculator), remembering the order of operations: brackets first, then multiply and divide, then add and subtract.
Rounding to 1 significant figure leaves one non-zero digit, so the numbers are easy to work with. For example 4.8 × 31 becomes 5 × 30 = 150. 596 ÷ 2.9 becomes 600 ÷ 3 = 200. The estimate does not have to be exact, it only has to be close. A good estimate is usually within a few percent of the exact answer.
Be careful with these. Never round a number to 0 (0.52 becomes 0.5, not 1 or 0). Round each number separately, before you do the calculation. Do not round the answer at the end: the estimate is what you get from the rounded numbers.
Using an estimate to check an answer
Work out the estimate. If the answer from the calculator is close to the estimate, it is probably correct. If it is about 10 or 100 times too big or too small, the decimal point is probably in the wrong place.
Is the estimate too big or too small?
You can often say whether an estimate is an over-estimate (bigger than the exact value) or an under-estimate (smaller). Think about each number: which way was it rounded, and how does that number change the answer? If you multiply by a number that was rounded up, the answer is pushed up. But if you divide by a number that was rounded up, the answer is pushed down. If the roundings push in different directions you cannot tell.
Key ideas
- Round every number to 1 significant figure, then do the easy calculation.
- Follow BIDMAS: brackets, then × and ÷, then + and −.
- Never round a number to 0.
- If the exact answer is far from the estimate (about 10 times too big or small), check the decimal point.
- Rounding a number up makes the answer bigger if it is multiplied or added, but smaller if it is divided by or taken away.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.4.8 → 5 (the next digit is 8, so round up)
31 → 30 (the next digit is 1, so round down) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.5 × 30
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.5 × 30
= 150 - State the estimate. The estimate is 150.
Check. A calculator gives 149 (3 s.f.) for the exact calculation. That is close to the estimate of 150. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.6.2 → 6 (the next digit is 2, so round down)
3.9 → 4 (the next digit is 9, so round up)
5.1 → 5 (the next digit is 1, so round down)
1.2 → 1 (the next digit is 2, so round down) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.6 + 45 − 1
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.6 + 45 − 1
= 104
= 2.5 - State the estimate. The estimate is 2.5.
Check. A calculator gives 2.59 (3 s.f.) for the exact calculation. That is close to the estimate of 2.5. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.596 → 600 (the next digit is 9, so round up)
2.9 → 3 (the next digit is 9, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.600 ÷ 3
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.600 ÷ 3
= 200 - State the estimate. The estimate is 200.
Check. A calculator gives 206 (3 s.f.) for the exact calculation. That is close to the estimate of 200. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.0.42 → 0.4 (the next digit is 2, so round down)
19.6 → 20 (the next digit is 9, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.0.4 × 20
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.0.4 × 20
= 8 - State the estimate. The estimate is 8.
Check. A calculator gives 8.23 (3 s.f.) for the exact calculation. That is close to the estimate of 8. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Method. Estimate the answer with numbers rounded to 1 significant figure and compare it with the answer given.
- Estimate the answer by rounding each number to 1 significant figure. The estimate tells us roughly how big the answer should be.5 × 30 = 150
- Compare the estimate with the answer given. If the answer is close to the estimate, it is probably right. If it is about 10 times too big or too small, the decimal point is probably in the wrong place.Answer given: 1488. Estimate: 150.
- Decide. The two numbers are not close.1488 is far from 150, so the answer is not sensible (it is about 10 times the estimate).
Check. The exact answer is 149 (3 s.f.), so the answer given was wrong. ✓
Watch out. An estimate only checks the size of the answer. If it matches the estimate, the answer could still have a small mistake.
Method. Work out what each rounding does to the answer. If they all push the answer the same way, you know whether the estimate is too big or too small.
- Round to 1 significant figure and work out the estimate. This is the normal estimating method.5 × 4 ÷ 0.4 = 50
- Look at what each rounding does to the answer. Rounding a number up or down changes the answer. Whether the answer gets bigger or smaller depends on where the number is in the calculation: a number being added or multiplied pushes the answer the same way; a number we divide by pushes it the opposite way.4.8 is rounded up to 5. A bigger number here makes the answer bigger, so this makes the estimate bigger than the true answer.
3.9 is rounded up to 4. A bigger number here makes the answer bigger, so this makes the estimate bigger than the true answer.
0.44 is rounded down to 0.4. A bigger number here makes the answer smaller, so this makes the estimate bigger than the true answer. - Decide. Every rounding pushes the estimate in the same direction, so we can be sure.The estimate is an over-estimate: it is bigger than the exact value.
Check. The exact value is 42.5 (3 s.f.) and the estimate is 50; the estimate is bigger. ✓
Watch out. When you divide by a number that was rounded up, the estimate gets smaller. Think about each number separately.
Watch out
When you estimate, round each number to 1 significant figure first. Do not round the final answer, and do not round a small decimal to 0: 0.52 → 0.5, not 1. And remember: 1 significant figure for 0.0472 is 0.05, not 0.Exercise 4H Estimation and checking answers
Non-calculatorKey words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.4.8 → 5 (the next digit is 8, so round up)
6.1 → 6 (the next digit is 1, so round down) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.5 × 6
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.5 × 6
= 30 - State the estimate. The estimate is 30.
Check. A calculator gives 29.3 (3 s.f.) for the exact calculation. That is close to the estimate of 30. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.9.7 → 10 (the next digit is 7, so round up)
3.2 → 3 (the next digit is 2, so round down) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.10 × 3
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.10 × 3
= 30 - State the estimate. The estimate is 30.
Check. A calculator gives 31 (3 s.f.) for the exact calculation. That is close to the estimate of 30. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.31 → 30 (the next digit is 1, so round down)
1.9 → 2 (the next digit is 9, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.30 × 2
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.30 × 2
= 60 - State the estimate. The estimate is 60.
Check. A calculator gives 58.9 (3 s.f.) for the exact calculation. That is close to the estimate of 60. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.0.82 → 0.8 (the next digit is 2, so round down)
48 → 50 (the next digit is 8, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.0.8 × 50
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.0.8 × 50
= 40 - State the estimate. The estimate is 40.
Check. A calculator gives 39.4 (3 s.f.) for the exact calculation. That is close to the estimate of 40. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.72 → 70 (the next digit is 2, so round down)
0.48 → 0.5 (the next digit is 8, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.70 × 0.5
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.70 × 0.5
= 35 - State the estimate. The estimate is 35.
Check. A calculator gives 34.6 (3 s.f.) for the exact calculation. That is close to the estimate of 35. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.6.3 → 6 (the next digit is 3, so round down)
8.9 → 9 (the next digit is 9, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.6 × 9
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.6 × 9
= 54 - State the estimate. The estimate is 54.
Check. A calculator gives 56.1 (3 s.f.) for the exact calculation. That is close to the estimate of 54. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.596 → 600 (the next digit is 9, so round up)
2.9 → 3 (the next digit is 9, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.600 ÷ 3
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.600 ÷ 3
= 200 - State the estimate. The estimate is 200.
Check. A calculator gives 206 (3 s.f.) for the exact calculation. That is close to the estimate of 200. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.81.3 → 80 (the next digit is 1, so round down)
3.9 → 4 (the next digit is 9, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.80 ÷ 4
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.80 ÷ 4
= 20 - State the estimate. The estimate is 20.
Check. A calculator gives 20.8 (3 s.f.) for the exact calculation. That is close to the estimate of 20. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.4.9 → 5 (the next digit is 9, so round up)
0.51 → 0.5 (the next digit is 1, so round down) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.5 ÷ 0.5
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.5 ÷ 0.5
= 10 - State the estimate. The estimate is 10.
Check. A calculator gives 9.61 (3 s.f.) for the exact calculation. That is close to the estimate of 10. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.1920 → 2000 (the next digit is 9, so round up)
39 → 40 (the next digit is 9, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.2000 ÷ 40
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.2000 ÷ 40
= 50 - State the estimate. The estimate is 50.
Check. A calculator gives 49.2 (3 s.f.) for the exact calculation. That is close to the estimate of 50. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.6.4 → 6 (the next digit is 4, so round down)
0.31 → 0.3 (the next digit is 1, so round down) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.6 ÷ 0.3
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.6 ÷ 0.3
= 20 - State the estimate. The estimate is 20.
Check. A calculator gives 20.6 (3 s.f.) for the exact calculation. That is close to the estimate of 20. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.3.8 → 4 (the next digit is 8, so round up)
6.1 → 6 (the next digit is 1, so round down)
2.9 → 3 (the next digit is 9, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.(4 + 6) × 3
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.(4 + 6) × 3
= 10 × 3
= 30 - State the estimate. The estimate is 30.
Check. A calculator gives 28.7 (3 s.f.) for the exact calculation. That is close to the estimate of 30. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.4.9 → 5 (the next digit is 9, so round up)
21 → 20 (the next digit is 1, so round down)
0.48 → 0.5 (the next digit is 8, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.5 × 200.5
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.5 × 200.5
= 1000.5
= 200 - State the estimate. The estimate is 200.
Check. A calculator gives 214 (3 s.f.) for the exact calculation. That is close to the estimate of 200. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.7.9 → 8 (the next digit is 9, so round up)
2.1 → 2 (the next digit is 1, so round down)
0.52 → 0.5 (the next digit is 2, so round down) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.8 − 20.5
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.8 − 20.5
= 60.5
= 12 - State the estimate. The estimate is 12.
Check. A calculator gives 11.2 (3 s.f.) for the exact calculation. That is close to the estimate of 12. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.48.2 → 50 (the next digit is 8, so round up)
19.7 → 20 (the next digit is 9, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.50 − 20
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.50 − 20
= 30 - State the estimate. The estimate is 30.
Check. A calculator gives 28.5 (3 s.f.) for the exact calculation. That is close to the estimate of 30. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.38 → 40 (the next digit is 8, so round up)
4.95 → 5 (the next digit is 9, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.40 × 5
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.40 × 5
= 200 - State the estimate. The estimate is 200.
Check. A calculator gives 188 (3 s.f.) for the exact calculation. That is close to the estimate of 200. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.312 → 300 (the next digit is 1, so round down)
19.5 → 20 (the next digit is 9, so round up) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.300 × 20
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.300 × 20
= 6000 - State the estimate. The estimate is 6000.
Check. A calculator gives 6080 (3 s.f.) for the exact calculation. That is close to the estimate of 6000. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.6.2 → 6 (the next digit is 2, so round down)
487 → 500 (the next digit is 8, so round up)
100 is already written with 1 significant figure, so it stays 100. - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.6 × 500100
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.6 × 500100
= 3000100
= 30 - State the estimate. The estimate is 30.
Check. A calculator gives 30.2 (3 s.f.) for the exact calculation. That is close to the estimate of 30. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Method. Round each price up to the next whole pound. If the total of the rounded-up prices is within the money Jo has, she definitely has enough.
- Round each price up. Rounding up means the total of the rounded prices is bigger than the real total, so it is a safe check.£14.75 → £15
£22.99 → £23
£8.49 → £9 - Add the first two rounded prices. Line up the decimal points, fill the gaps with zeros, and add column by column from the right.15 + 23 -- 38
- Add the third. Line up the decimal points, fill the gaps with zeros, and add column by column from the right.38 + 9 -- 47
- Compare with the money. The rounded-up total is 47, which is less than 50.£47 < £50
- Write the answer. The real total is less than £47, which is less than £50.Yes, Jo has enough (the real total is £46.23).
Check. 14.75 + 22.99 + 8.49 = 46.23, which is less than 50. ✓
Watch out. When you want to be sure there is enough, round up, not to the nearest.
Method. Estimate the answer with numbers rounded to 1 significant figure and compare it with the answer given.
- Estimate the answer by rounding each number to 1 significant figure. The estimate tells us roughly how big the answer should be.5 × 30 = 150
- Compare the estimate with the answer given. If the answer is close to the estimate, it is probably right. If it is about 10 times too big or too small, the decimal point is probably in the wrong place.Answer given: 1488. Estimate: 150.
- Decide. The two numbers are not close.1488 is far from 150, so the answer is not sensible (it is about 10 times the estimate).
Check. The exact answer is 149 (3 s.f.), so the answer given was wrong. ✓
Watch out. An estimate only checks the size of the answer. If it matches the estimate, the answer could still have a small mistake.
Method. Estimate the answer with numbers rounded to 1 significant figure and compare it with the answer given.
- Estimate the answer by rounding each number to 1 significant figure. The estimate tells us roughly how big the answer should be.600 ÷ 3 = 200
- Compare the estimate with the answer given. If the answer is close to the estimate, it is probably right. If it is about 10 times too big or too small, the decimal point is probably in the wrong place.Answer given: 20.55. Estimate: 200.
- Decide. The two numbers are not close.20.55 is far from 200, so the answer is not sensible (it is about 0.1 times the estimate).
Check. The exact answer is 206 (3 s.f.), so the answer given was wrong. ✓
Watch out. An estimate only checks the size of the answer. If it matches the estimate, the answer could still have a small mistake.
Method. Estimate the answer with numbers rounded to 1 significant figure and compare it with the answer given.
- Estimate the answer by rounding each number to 1 significant figure. The estimate tells us roughly how big the answer should be.7 × 0.5 = 3.5
- Compare the estimate with the answer given. If the answer is close to the estimate, it is probably right. If it is about 10 times too big or too small, the decimal point is probably in the wrong place.Answer given: 3.456. Estimate: 3.5.
- Decide. The two numbers are close.3.456 is close to 3.5, so the answer is sensible.
Check. The exact answer is 3.46 (3 s.f.), so the answer given was close to it. ✓
Watch out. An estimate only checks the size of the answer. If it matches the estimate, the answer could still have a small mistake.
Method. Estimate the answer with numbers rounded to 1 significant figure and compare it with the answer given.
- Estimate the answer by rounding each number to 1 significant figure. The estimate tells us roughly how big the answer should be.80 ÷ 4 = 20
- Compare the estimate with the answer given. If the answer is close to the estimate, it is probably right. If it is about 10 times too big or too small, the decimal point is probably in the wrong place.Answer given: 20.846. Estimate: 20.
- Decide. The two numbers are close.20.846 is close to 20, so the answer is sensible.
Check. The exact answer is 20.8 (3 s.f.), so the answer given was close to it. ✓
Watch out. An estimate only checks the size of the answer. If it matches the estimate, the answer could still have a small mistake.
Method. Work out what each rounding does to the answer. If they all push the answer the same way, you know whether the estimate is too big or too small.
- Round to 1 significant figure and work out the estimate. This is the normal estimating method.5 × 4 ÷ 0.4 = 50
- Look at what each rounding does to the answer. Rounding a number up or down changes the answer. Whether the answer gets bigger or smaller depends on where the number is in the calculation: a number being added or multiplied pushes the answer the same way; a number we divide by pushes it the opposite way.4.8 is rounded up to 5. A bigger number here makes the answer bigger, so this makes the estimate bigger than the true answer.
3.9 is rounded up to 4. A bigger number here makes the answer bigger, so this makes the estimate bigger than the true answer.
0.44 is rounded down to 0.4. A bigger number here makes the answer smaller, so this makes the estimate bigger than the true answer. - Decide. Every rounding pushes the estimate in the same direction, so we can be sure.The estimate is an over-estimate: it is bigger than the exact value.
Check. The exact value is 42.5 (3 s.f.) and the estimate is 50; the estimate is bigger. ✓
Watch out. When you divide by a number that was rounded up, the estimate gets smaller. Think about each number separately.
Method. Work out what each rounding does to the answer. If they all push the answer the same way, you know whether the estimate is too big or too small.
- Round to 1 significant figure and work out the estimate. This is the normal estimating method.7 × 9 = 63
- Look at what each rounding does to the answer. Rounding a number up or down changes the answer. Whether the answer gets bigger or smaller depends on where the number is in the calculation: a number being added or multiplied pushes the answer the same way; a number we divide by pushes it the opposite way.6.7 is rounded up to 7. A bigger number here makes the answer bigger, so this makes the estimate bigger than the true answer.
8.6 is rounded up to 9. A bigger number here makes the answer bigger, so this makes the estimate bigger than the true answer. - Decide. Every rounding pushes the estimate in the same direction, so we can be sure.The estimate is an over-estimate: it is bigger than the exact value.
Check. The exact value is 57.6 (3 s.f.) and the estimate is 63; the estimate is bigger. ✓
Watch out. When you divide by a number that was rounded up, the estimate gets smaller. Think about each number separately.
Method. Work out what each rounding does to the answer. If they all push the answer the same way, you know whether the estimate is too big or too small.
- Round to 1 significant figure and work out the estimate. This is the normal estimating method.30 ÷ 0.5 = 60
- Look at what each rounding does to the answer. Rounding a number up or down changes the answer. Whether the answer gets bigger or smaller depends on where the number is in the calculation: a number being added or multiplied pushes the answer the same way; a number we divide by pushes it the opposite way.29 is rounded up to 30. A bigger number here makes the answer bigger, so this makes the estimate bigger than the true answer.
0.52 is rounded down to 0.5. A bigger number here makes the answer smaller, so this makes the estimate bigger than the true answer. - Decide. Every rounding pushes the estimate in the same direction, so we can be sure.The estimate is an over-estimate: it is bigger than the exact value.
Check. The exact value is 55.8 (3 s.f.) and the estimate is 60; the estimate is bigger. ✓
Watch out. When you divide by a number that was rounded up, the estimate gets smaller. Think about each number separately.
Method. Work out what each rounding does to the answer. If they all push the answer the same way, you know whether the estimate is too big or too small.
- Round to 1 significant figure and work out the estimate. This is the normal estimating method.4 × 20 = 80
- Look at what each rounding does to the answer. Rounding a number up or down changes the answer. Whether the answer gets bigger or smaller depends on where the number is in the calculation: a number being added or multiplied pushes the answer the same way; a number we divide by pushes it the opposite way.4.2 is rounded down to 4. A bigger number here makes the answer bigger, so this makes the estimate smaller than the true answer.
21 is rounded down to 20. A bigger number here makes the answer bigger, so this makes the estimate smaller than the true answer. - Decide. Every rounding pushes the estimate in the same direction, so we can be sure.The estimate is an under-estimate: it is smaller than the exact value.
Check. The exact value is 88.2 (3 s.f.) and the estimate is 80; the estimate is smaller. ✓
Watch out. When you divide by a number that was rounded up, the estimate gets smaller. Think about each number separately.
Method. Check the rounding of 0.52 and count the significant figures properly.
- Find the first significant figure of 0.52. The first non-zero digit is 5, in the tenths column.0.52
- Look at the decider. The next digit is 2, so round down. Zoe rounded to the nearest whole number instead of to 1 significant figure.0.52 → 0.5
- Round 48 and multiply. 48 → 50. Then multiply.0.5 × 50 = 25
Check. 0.52 × 48 = 24.96, close to 25. ✓
Watch out. 1 significant figure means one non-zero digit, wherever the decimal point is.
Method. Estimate by rounding each number to 1 significant figure, then compare with the three choices.
- Round each number to 1 s.f. 31.2 → 30 and 0.42 → 0.4.30 × 0.4
- Work out the estimate. 30 × 4 = 120, so 30 × 0.4 = 12.30 × 0.4 = 12
- Choose. 12 is one of the three choices.12
Check. The exact value is 13.104, which is closest to 12. ✓
Watch out. Multiplying by a number smaller than 1 makes the answer smaller than the other number.
Key words. An estimate is an approximate answer found with simple numbers. A significant figure is a digit that counts, starting from the first non-zero digit.
Method. Estimate by rounding every number to 1 significant figure, then do the easy calculation.
- Round every number to 1 significant figure. An estimate uses easy numbers. 1 significant figure leaves just one non-zero digit followed by zeros, which is easy to calculate with.9.8 → 10 (the next digit is 8, so round up)
41 → 40 (the next digit is 1, so round down) - Write the calculation again with the rounded numbers. Replace each number by its rounded value. Keep the same operations in the same places.√(10 × 40)
- Work it out, remembering the order of operations (brackets first, then multiply and divide, then add and subtract). These numbers are simple enough to do without a calculator.√(10 × 40)
= √400
= 20 - State the estimate. The estimate is 20.
Check. A calculator gives 20 (3 s.f.) for the exact calculation. That is close to the estimate of 20. ✓
Watch out. Round each number to 1 significant figure first, then calculate. Do not round the answer. And never round a number to 0.
Method. Round each number to 1 significant figure, calculate, and compare the estimate with 2.64.
- Round to 1 s.f. 0.0081 → 0.008, 62 → 60, 0.019 → 0.02.0.008, 60 and 0.02
- Calculate. 0.008 × 60 = 0.48 and 0.48 ÷ 0.02 = 24.0.008 × 600.02 = 0.480.02 = 24
- Compare. The estimate is 24, which is nearly 10 times bigger than 2.64.estimate = 24, but 2.64 is about 10 times smaller
- Explain. The exact answer is 26.43 (to 4 s.f.). 2.64 has the decimal point in the wrong place.
Check. 0.0081 × 62 = 0.5022 and 0.5022 ÷ 0.019 = 26.43. ✓
Watch out. An estimate that is about 10 times different from the answer shows the decimal point is wrong.