**ESTIMATION AND ****APPROXIMATION**

**ROUNDING OFF**

To ’round off’ or ‘approximate’ a number to a desired degree of accuracy, we round the number *up* if the next digit is 5 or more or we round the number *down* if the next digit is less than 5. We represent approximately as** ‘~’**

**Examples**

73 is close to 70 if approximating to or rounding in “tens”. So

73 ~ 70 (Read as 73 is approximately equal to 70)

86 is close to 90 when rounded off to the nearest “tens”, or approximate to the nearest “tens”

86 ~ 90

650 ~ 700 when rounded off to the nearest hundreds

26432 rounded off to

nearest 100 is 264 / 32 ~ 26400

nearest 10,000 is 2 / 6432 ~ 30000

**Strategy to round off numbers**

Put a line where you want to round off. In the above example 3 to round off 650, put a big line after 6, because 6 is in the hundreds place, and you want to round to the nearest hundred.

6/50

The digit before the big line (6 in this case) will go up by 1, and the rest of the digits after the line will become 0, since the number after the line is 5. So the answer is 700.

**Estimation**

Sometimes, we may have to guess or estimate numbers as close as possible to their real values. This process of guessing the numbers sensibly is called estimation. Generally you can use estimation to work out whether your calculation or the answer you are working towards is right or wrong.

We use a simple strategy to estimate a number for calculations – we use the first digit position for rounding off. This first digit is also called the leading figure.

To estimate the answer of a calculation using leading figure estimation:

round each number to its nearest leading figure

evaluate the resulting expression

Let’s look at some examples to demonstrate this strategy.

Note that the sign **~**, means approximation.

234 + 57 ~ 200 + 60 = 260

Here 234 is close to 200, and 57 is closer to 60 – the leading figures here are 2 and 5 respectively. So the answer to this equation should be close to 260.

The correct answer for 234 + 57 = 291, which is a bit further away from the estimated answer of 260.

**So how do we get a better estimation?**

We need to change the leading figure for 234 – earlier, we considered 2 to be the leading figure, but now we should consider 3 to be the leading figure. So

234 + 57 = 230 + 60 = 290, which is very close to 291 (the correct answer).

287 x 34

287 ~ 300, and 34 ~ 30 (leading figures being 2 and 3 respectively)

So 287 x 34 ~ 300 x 30 = 9000

The correct answer of 287 x 34 is 9758.

If 523 people pass through the gate of a stadium every two minutes, estimate how many people attending the game if the gates were open for only an hour.

523 people in 2 minutes ~ 500 people in 2 minutes

In 60 minutes, there are 30 groups of 2 minutes. So

523 x 30 ~ 500 x 30 = 15000 people

Round off each number to one significant figure, then approximate the answers.

**Example**

a. 47 + 31

b. 291 + 603

**Solution**

a. 47 + 31 ~ 50 + 30 = 80

b. 291 + 603 ~ 300 + 600 = 900

**Exercises**

Round off each number to one significant figure. Then approximate each answer

1. 23 + 19

a. 40

2. 24 X 37

a. 940 b. 800 c. 700 d. 650

3. 73 – 48

a. 24 b. 34 c. 20 d. 30

Round off each number to the nearest whole number. Then find the approximate answer.

4. 6.2 + 3.7

a. 9 b. 6 c. 7 d. 10

5. 3.4 X 5.8

a. 18 b. 24 c. 19 d. 20

**Answers**

1. A 2. B 3. C 4. D 5. A

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