Estimation is a way of speeding up calculation. Usually, the resulting number is less accurate and less precise than the actual value, although this is not always the case, as we'll see below.

## Reformulation strategiesEdit

'Reformulation' means changing the numbers of a sum to make it easier to calculate. This results in a less accurate answer.

### ApproximationEdit

We can get the approximate values of numbers before evaluating the sum. We usually round up, down or off first. We can round the number to the same number of places or the same number of significant figures, depending on the situation. Here are some examples:

Example 1 | |
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Question | Estimate (a) 123 + 34.5678 - 62.4 and (b) 942.5 ÷ 4.1 by rounding up, down and off the numbers to a suitable number of places and/or significant figures. |

Solution | (a)We will round the numbers to the nearest integer. Rounding up: 123 + 34.5678 - 62.4 ≈ 123 + 35 - 63 = 95 Rounding down: 123 + 34.5678 - 62.4 ≈ 123 + 34 - 62 = 93 Rounding off: Rounding down: 123 + 34.5678 - 62.4 ≈ 123 + 34 - 62 = 93 (b) We will round the numbers to one significant figure. Rounding up: 942.5 ÷ 4.1 ≈ 1000 ÷ 5 = 200 Rounding down: 942.5 ÷ 4.1 ≈ 900 ÷ 4 = 225 Rounding off: 942.5 ÷ 4.1 ≈ 900 ÷ 4 = 225 |

Note that in (b) in the above example, it is inappropriate to round the second sum to the nearest integer (since that would make it difficult to calculate) or the nearest tenth or hundredth (since that would result in the divisor being zero and an undefined answer.)

Sometimes, it is enough to round off only one of the numbers. This is particularly common in subtraction:

Example 2 | |
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Question | Estimate (a) 124436 - 6132 and (b) 1200 ÷ 5.03 by rounding off one number. |

Solution | (a)124436 - 6132 ≈ 124436 - 6000 = 118436 (b)1200 ÷ 5.03 ≈ 1200 ÷ 5 = 240 |

When we estimate an addition or multiplication problem, we can find the range of the answer (between the rounded-up value and the rounded-down value.)

Example 3 | |
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Question | In the following, if possible, find the range of the result by rounding up and down to the nearest hundred. a) 3456 + 9123 b) 468.73 - 3546.7 c) 2345 × 132 |

Solution | a) Result after rounding up: 3456 + 9123 ≈ 3500 + 9200 = 12700 Result after rounding down: 3456 + 9123 ≈ 3400 + 9100 = 12500 ∴The result lies between 12500 and 12700. b) It is impossible. c) Result after rounding up: 2345 × 132 ≈ 2300 × 200 = 460,000 Result after rounding down: 2345 × 132 ≈ 2300 × 100 = 230,000 ∴The result lies between 230,000 and 460,000. |

### Compatible numbersEdit

Sometimes, we can tinker with the numbers so that they 'agree' with each other. There are several ways to do this. Look at these examples:

Example 4 | |
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Question | a) Estimate 8200 ÷ 9 with compatible numbers. b) Estimate 2556 - 4546 with compatible numbers. c) Estimate 1239 + 1772 with compatible numbers. |

Solution | a) 8200 ÷ 9 ≈ 8100 ÷ 9 = 900 b) 2556 - 4546 ≈ 2556 - 4556 = -2000 Alternative method: 2556 - 4546 ≈ 2546 - 4546 = -2000 c) 1239 + 1772 ≈ 1230 + 1770 = 3000 |

In a), we changed 8200 to 8100, which is divisible by nine.

In b), we changed the tens place so that the hundred, tens and units places of the two numbers all match. That way, we can obtain the estimated value of -2000 immediately.

In c), we changed the two numbers so that they add up to exactly 3000.

## Translation strategyEdit

### Order of operationsEdit

### ClustersEdit

Clustering is a strategy that involves both reformulation and translation.

## Compensation strategyEdit

## Vocabulary tableEdit

- Reformulation
- Cluster
- Compatible number
- Translation
- Compensation