Estimation



Using your knowledge of Rounding Numbers, you can develop a "feel" for the numbers involved in a calculation, and estimate the general range in which the answer lies, before actually carrying out the calculation itself. This skill obviously enables you to reduce errors in your work.

Addition

Example

Add

£ 3.90, £ 4.26, £ 13.29

Roughly speaking, this would be equal to

4 + 4 + 13 = 21

Compare this with the true answer of £ 21.45

So you can see that using this procedure (having a "feel" for the numbers, estimating the answer beforehand), we can at least eliminate totally wrong answers. If you had placed the decimal point in the wrong place in the one of the numbers, this would be picked up.

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Multiplication

Example

22.1 x 5.2

Roughly speaking, this would be equal to

22 x 5 = 110

Compare this with the correct value of 114.92

Example

27.18 x 28.39

Roughly speaking, this would be equal to

30 x 30 = 900

Compare this with the correct value of 771.6402, which is not too close to the estimation, but near enough to be confident that it could indeed be the correct answer.

If you are confident with decimal numbers, it might be possible to approximate to an appropriate non-whole number (i.e. a number with figures to the right of the decimal point.

Original Problem Approximated by
10.89 x 2.53 11 x 2.5 = 27.5

1.456 x 16.3

1.5 x 16 = 24
7.79 x 16.12 7.75 x 16 = 124

The above procedure, although useful, is limited to use in a small number of cases.

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Division

Example

\[ \frac{123}{5.8} \]

can be approximated by

\[ \frac{123}{6} \]

which is approximately

20 or 21

So we know the result of $\frac{123}{5.8}$ will be close to 20 or 21 (the real figure will be slightly higher). Therefore we know the approximate value of the answer before we have started, and this will help to exclude any totally-incorrect calculations.

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Fractions

Sometimes, a "difficult" fraction can be approximated by an "easier" fraction. For example, a fraction like

\[ \frac{17}{168} \]

could be estimated as

\[ \frac{1}{10} \]

Sometimes a decimal can be approximated by an fraction, making estimation easier (you might need to consult the fractions and/or decimal modules before reading this).

Examples

Decimal Approximated by
0.57 $\frac{1}{2}$    (0.50)

0.26

$\frac{1}{4}$    (0.25)
0.129 $\frac{1}{8}$    (0.125)

Examples of Calculation

Original Problem Approximated by
24.6 x 0.56 25 x $\left(\frac{1}{2}\right)$ = 12.5

42.3 x 0.12

42 x $\left(\frac{1}{10}\right)$ = 4.2
51.21 x 0.027 50 x $\left(\frac{1}{40}\right)$ = 1.25

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Expressions involving a mixture of Operations

Example

Even something as 'complicated' as

\[( 48.3 \times 34.29 )\div ( 9.73 \times 4.63 ) \]

can be approximated by

\[ \frac{( 50 \times 35 )}{( 10 \times 5 )} = 35 \]

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More complicated procedures

Example

What about

7.23 x 0.032

If you are fully happy with the decimal system, then

  • Using a knowledge of the decimal system, you could recognize that multiplication by 0.01 shifts the decimal point in 7.23 by two places to the left, i.e. 0.0723, approximately 0.07.
  • Since 0.032 is approximately 3 times 0.1, multiply 0.07 by 3 to give approximately 0.21.
  • (Compare this with the true value of 0.23136)

0.093 x 0.059
  • Recognize that multiplying 0.093 by 0.01 will shift the decimal point two places to the left, producing 0.00093
  • Since 0.059 is approximately 6 times the value of 0.01, multiply 0.0009 (0.00093 from above, but rounded for simplicity) by 6, to produce 0.0054
  • (Compare this with the real value of 0.005487)
  • You obviously need to have a good understanding of the decimal system to use this method. Sometimes it is easier to use the methods outlined above, under 'Fractions'.

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Past Test Questions

Chris buys 4 CDs at £13.99 each. He gives the cashier £60. Which method would give Chris the closest estimate to let him check his change?

  • A    60 - 13 x 4
  • B    60 - 4 - 13
  • C    60 - 4 + 14
  • D    60 - 4 x 14


Cathy buys 13 T-Shirts that cost £ 9.89 each.

She estimates the total cost in four ways. Each way gives a different result. Which of the estimates below is closest to the total cost ?

  • A    £ 130.00
  • B    £ 128.70
  • C    £ 127.60
  • D    £ 123.50


Jess has to quickly work out an estimate of

\[ \frac{784}{(39.2 \times 0.47)}\]
in her head.

Show a method which she could use.

Use it to give an estimate for the value of the calculation.

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