Unit 14 Section 2 : Estimation

Estimation is important because it allows you to check that the answers you obtain on a calculator are reasonable. It is easy to make a simple mistake when using a calculator or working out a problem by hand, so an estimation of the expected answer is a useful check.

One of the simplest approaches to estimation is to round all the numbers involved to 1 significant figure.

Example 1

(a)

Estimate the value of 4.73 × 18.4.

4.73 × 18.4 ≈ 5 × 20 = 100
(b)

Use a calculator to find 4.73 × 18.4.

4.73 × 18.4 = 87.032
(c)

Compare the estimate with the exact value.

Because in (a) we rounded both numbers up, the estimate is slightly bigger than the actual value, but it does give us an idea of the size of the answer.

Example 2

The correct answer to 14.1 × 18.3 is listed below, along with 3 incorrect answers.

25.803
258.03
2580.3
25803

Use estimation to decide which is the correct answer.

14.1 × 18.3 ≈ 10 × 20 = 200
So the correct answer must be 258.03

Although we often round to 1 significant figure, we can sometimes produce better estimates by using other values that are still easy to work with. For example, in Example 2 above, we could have said that

14.1 × 18.3 ≈ 14 × 20 = 280

which is closer to the correct answer 258.03

The important thing to remember is that you must be able to do the estimation calculation in your head.

Example 3

Estimate the value of:

(a)

≈ = = 2

or

≈ = = = = 2.6

Note that the actual value is 2.462 to 3 d.p.

(b)

≈ = = =

Note that is between 7 and 8, but will be closer to 7 since 7 = .
We can therefore take 7 as our estimate.

The actual value is 7.659 to 3 d.p.

You might also need to approximate when converting between different units. The following list reminds you of some of the important approximations between metric and imperial units.

8 km≈5 miles
1 m≈40 inches
30 cm≈1 foot
2.5 cm≈1 inch
1 kg≈2.2 lbs
1 litre≈1.75 pints
1 gallon≈4.5 litres
1 acre≈0.4 hectare
450 g≈1 lb

Exercises

Question 1

Estimate the answer to each of the following calculations:

(a)3.6 × 14.3
(b)47 × 192
(c)33.6 × 403
(d)11.25 × 76.3
(e)12.84 × 3.94
(f)103.6 ÷ 21.72
(g)44.32 ÷ 1.987
(h)68.39 ÷ 7.48
(i)12.021 ÷ 5.917
Question 2

Estimate the value of each of the following:

(a)
e.g. = = 4
(b)
e.g. = ≈ = 11
(c)
e.g. = = 8 × 2 = 16
(d)
e.g. = ≈ = 7
Question 3

James writes down this statement:

14.62 × 401 = 586.262

(a)

Use estimation to describe why James must be wrong.

14.62 × 401 ≈
e.g. 15 × 400 = 6000 means that James' answer is approximately one tenth of the correct value.
(b)

Use a calculator to determine the correct answer.

James has put the decimal point in the wrong place.
Question 4

Use estimation to decide which of the following calculations are definitely incorrect:

(a)

15.2 × 6120 = 930240

This calculation because 15.2 × 6120 ≈
(b)

65.224 ÷ 12.4 = 5.26

This calculation because 65.224 ÷ 12.4 ≈
(c)

192 × 4587 = 880704

This calculation because 192 × 4587 ≈
(d)

346.92 ÷ 2.36 = 14.7

This calculation because 346.92 ÷ 2.36 ≈
Question 5

For each of the following calculations three possible answers are given, but only one is correct. Use estimation to select the correct answer.

(a)
3.712 × 805 ≈ so the correct answer must be
(b)
2955.82 ÷ 9.82 ≈ so the correct answer must be
(c)
≈ so the correct answer must be
(d)
≈ so the correct answer must be
Question 6

The month of December has 31 days. Estimate the number of seconds in December. Compare your estimate with the correct value.

Number of seconds in December = × × × ≈

The correct number of seconds in December =

Question 7

Estimate the circumference of a circle with a radius of 20.5 cm.

cm
π × 20.5 ≈ 3 × 20 = 60 cm
Question 8

A book has 326 pages with an average of 268.4 words on each page.
Estimate the number of words in the book.

words
Number of words in book = 326 × 268.4 ≈ 300 × 300 = 90 000 words
Question 9

At a school the average daily amount spent by each child in the canteen is £1.42. There are 1264 pupils in the school. Estimate the total amount spent in the canteen each day.

£
Total spent in canteen = 1264 × £1.42 ≈ 1200 × £1.50 = £1800
Question 10

Estimate the area of a circle of radius 6.27 cm.

cm²
Area = π × 6.27² ≈ 3 × 6² = 3 × 36 = 108 cm²
Question 11

Carl has an old recipe for egg custard with raisins.

(a)

The custard must be cooked at 320 degrees Fahrenheit. Carl has a rule to change the temperature to degrees Celsius.

Rule:
To change the temperature to degrees Celsius, subtract 32 from the temperature in degrees Fahrenheit, then multiply the answer by 5, then divide by 9.

Use Carl's rule to change 320 degrees Fahrenheit to degrees Celsius. You must show each step in your calculation.

Temperature in degrees Fahrenheit = () × ÷
= × ÷ = ÷ = °F
(b)

Carl is using this recipe:

Egg Custard with Raisins ¼ pound of raisins
1 pint of milk
3 eggs
Put the raisins in an 8 inch bowl.
Mix the eggs and milk, and pour over the raisins.
Bake in the oven at 320° Fahrenheit for about an hour.

He starts to change the amounts into metric measures. Write down the words or numbers missing from the sentences below.

Egg Custard with Raisins About 100 grams of raisins
About 0.5 of milk
3 eggs
Put the raisins in a centimetre bowl.