Estimation

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In order to access this I need to be confident with:

Decimals Rounding numbers to one or more significant figures

The order of operations, BIDMAS

This topic is relevant for:

GCSE Maths Number Rounding Numbers

Estimation

Here we will learn about estimation in maths including its definition and how to estimate a calculation. You’ll also learn about the differences between decimals and significant figures and how to round numbers to one significant figure.

There are estimation worksheets based on Edexcel, AQA and OCR exam questions, along with further guidance on where to go next if you’re still stuck.

What is estimation in maths?

Estimation is when we use approximate values in a calculation to give an approximate, predicted answer rather than an exact answer.

Estimating calculations makes them more manageable by using round numbers.

E.g.
If we wanted to estimate the answer to 46.75 x 3.218 we could round both these numbers and use an estimated calculation of 50 x 3 = 150

Estimation from sample

A population includes everyone within a large group, for example, the population of a country includes everyone that lives within that country.

A sample is a smaller group, taken from the population, for example a sample of 100 people could be taken from the total population.

Samples are used when researching populations because it is usually too costly or too time consuming to carry out research on the entire population. This means that the sample must be selected carefully so that it represents the population as closely as possible.

One of the best methods of selecting a sample is using a stratified sampling technique. This allows the researcher to determine the sample based on the proportional size of different subcategories within the population and then randomly select individuals for participation in the research.

Step-by-step guide: Stratified sampling

Once the individuals within the sample have been selected, we can carry out statistical analysis on the sample such as the mean. The sample mean is the average of the sample, whilst the population mean is the average of the population. If the sample is selected carefully, we can estimate the population mean by calculating the sample mean. Including more individuals within the sample will provide a more accurate value for an estimate of the population mean. This is because the value of the sample mean value tends towards the population mean as the sample size increases.

Estimating calculations

When we estimate the numbers in a calculation, we usually round the numbers to 1 significant figure.

E.g.
Let’s estimate 5.7 ÷ 1.8

First we estimate the value of 5.7 by rounding it to 1 significant figure (1.s.f).
We can see that the first significant figure is 5.
This means we need to use the digit to the right of the 5 which is 7, so we need to round 5.7 up to 6 which is the nearest whole number.

So 5.7 estimated is 6 as shown on the number line.

Then we follow the same process to estimate the value of 1.8 which rounded to 1 significant figure is 2.

Once we have estimated all the values in a calculation we can work out the estimated answer so

5.7 ÷ 1.8 ≈ 6 ÷ 2 = 3

Estimation is used in lots of different areas of maths in order to give an approximate answer to a question, such as estimating the perimeter of a shape or estimating the size of a number written in standard form. We also see examples of estimation all around us, such as estimating the number of bricks will be needed to build a garden wall, or working out if you have enough leftover from £5 in a supermarket to buy a bar of chocolate at the till.

What is estimation?

Key points about estimating calculations

An estimate is an approximate answer; not the exact answer
Estimation relies on using easier numbers that are near to the actual numbers involved
To estimate we usually round to 1 significant figure
We use estimation in many aspects of life and work – for a building job or planning a budget
When we are estimating calculations we still have to consider using the order of operations or BIDMAS principles
≈ means approximately equal to; = means exactly equal to

How to estimate

In order to estimate a number:

Round each number to 1 significant figure.
Construct the new estimated calculation using the ‘approximately equal to’
symbol ≈.
Calculate using the rounded numbers to give an approximate answer and consider units in the answer.

Explain how to estimate a calculation in 3 steps

Estimation worksheet

Get your free estimation worksheet of 20+ questions and answers. Includes reasoning and applied questions.

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Estimation worksheet

Get your free estimation worksheet of 20+ questions and answers. Includes reasoning and applied questions.

DOWNLOAD FREE

Estimation examples

Example 1: (typical foundation GCSE question using the GCSE command word language and no units involved)

Estimate the value of:

\[6.9\times 96\]

Round numbers to one significant figure

6.9 rounded to 1 significant figure is 7

96 rounded to 1 significant figure is 100

2 Write out the new estimated calculation

\[6.9\times 96\approx 7\times 100\]

3 Complete the calculation using estimation

\[7\times 100=700\]

If we compare this to the exact answer which is 662.4 we can see that the estimated answer is an overestimate.

Example 2: worded question

Brian buys $197$ packets of crisps for a party. Each bag costs $45$ p. Use estimation to find the approximate cost in pounds £.

Round numbers to one significant figure

197 is 200, rounded to 1 s.f.

45p is 50p, rounded to 1 s.f.

Write out the new estimated calculation

\[197 \times 45\approx 200 \times 50\]

Complete the calculation using estimation

\[200 \times 50 = 10000\]

In this case we have to consider the units used

10000 is pence so we convert this to pounds (£)

10000 pence is £100

If we compare this to the exact answer which is £88.65 we can see that the estimated answer is an overestimate.

Example 3: multi-step calculations

Round numbers to one significant figure

\[\frac{42 \times 389}{81}\]

42 rounded to 1 s.f. is 40

389 rounded to 1 s.f. is 400

81 rounded to 1 s.f. is 80

Write out the new estimated calculation

\[\frac{42 \times 389}{81} \approx \frac{40 \times 400}{80}\]

Complete the calculation using estimation

\[\frac{16000}{80}=200\]

If we compare this to the exact answer which is 201.7 (1 d.p.) we can see that the estimated answer is an underestimate.

Example 4: estimate using BIDMAS principles

Estimate the value of 9.04 + 19.85 × 2.99 − 5.03

You must show your working.

Round numbers to one significant figure

\[9.04 + 19.85 \times 2.99 \hspace{1mm} – 5.03 \approx 9 + 20 \times 3 \hspace{1mm} – 5\]

Consider the order of operations BIDMAS

Multiply 20 and 3 together first

\[9 + 60 \hspace{1mm} – 5\]

Then follow the calculation from left to right, as it doesn’t matter which operation you do first with addition and subtraction.

\[9 + 60 \hspace{1mm} – 5\]

Complete the calculation using estimation

\[9 + 60 \hspace{1mm} – 5 = 64\]

If we compare this to the exact answer which is 63.3615 we can see that the estimated answer is an overestimate.

Example 5: worded problem requiring reasoning

Estimate how many toys costing £8.48 each can be bought for £425? Is this an over or under-estimate?

Round numbers to one significant figure

£8.48 is £8, rounded to 1 s.f.

£425 is £400, rounded to 1 s.f.

Construct the estimated calculation

\[425 \div 8.48 \approx 400 \div 8\]

Complete the calculation

\[400 \div 8 = 50\]

Consider units within a worded problem in context

The question considered both numbers in pounds but the answer should be a whole number of toys.

The estimate number of toys that could be bought is 50.

If we compare this to the exact answer which is 50.1179… we can see that the estimated answer is an underestimate. .

Example 6: more complex calculation

Write down an estimate for:

\[\sqrt{70}\]

Decide which square numbers are closest to $70$

\[\begin{aligned} 7^{2}&=49 \\\\ 8^{2}&=64 \\\\ 9^{2}&=81 \\\\ 10^{2}&=100 \end{aligned}\]

Decide which two squares $70$ lies between and which one it is closest to

70 lies between 64 and 81

\[\sqrt{70}\]

lies between

\[\sqrt{64} \text { and } \sqrt{81}\]

And

\[\sqrt{64}=8, \sqrt{81}=9\]

70 is closer to 64, being 6 units away, than 81, being 11 units away

So,

\[\sqrt{70}\]

is just under 8.5.

Complete the calculation using estimation

\[\sqrt{70} \approx 8.4\]

Example 7: dividing by decimals

Estimate the value of:

\[\frac{804 \times 2.86}{0.513}\]

Round numbers to one significant figure

804 is 800, to 1 s.f.

2.86 is 3, to 1 s.f.

0.513 is 0.5, rounded to 1 s.f.

Construct the estimated calculation

\[\frac{804 \times 2.86}{0.513} \approx \frac{800 \times 3}{0.5}\]

Complete the calculation using estimation

\[\begin{aligned} 800 \times 3&=2400 \\\\ \frac{2400}{0.5}&=4800 \end{aligned}\]

If we compare this to the exact answer which is 4482.3 (1 d.p.) we can see that the estimated answer is an overestimate.

Common misconceptions

It is common to find the actual calculation rather than use estimation

Estimation relies on rounding to significant figures to simplify the calculation.

An estimate of 5.8 × 2.3 is 12. The actual answer is 13.34.

The first significant figure is the first non-zero digit

E.g.
The first significant figure of 0.00123 is the fourth digit which is 1.

E.g.
The first significant figure of 0.023 is the third digit which is 2.

E.g.
The first significant figure of 0.0000547 is the sixth digit which is 5

Dividing by a decimal gives a larger number

E.g.

\[120 \div 0.2 = 600\]

Significant figures and decimal places

Rounding to one decimal place relies on what happens after the first digit after the decimal point, whereas the first significant figure is the 1st non-zero number. It is important to have a strong understanding of place value to do this.

E.g.
Although 0.02 has 2 decimal places, the 2 is the 1st significant figure.

Estimate with fractions

We can also estimate with fractions by first converting the fraction to a decimal and then rounding the numbers to 1 significant figure.

Estimation with graphs

We can use graphs to estimate values by using a line of best fit and extrapolating.

Did you know?

The word estimation comes from the Middle English word aestimatio which means a valuation.

Estimation is part of our series of lessons to support revision on rounding numbers. You may find it helpful to start with the main rounding numbers lesson for a summary of what to expect, or use the step by step guides below for further detail on individual topics. Other lessons in this series include:

Practice estimation questions

$720$

$700$

$705.6$

$706$

Remember to round each number to one significant figure, then calculate

\begin{aligned} 7.2 \times 98 &\approx 7 \times 100\\\\ 7 \times 100 &= 700 \end{aligned}

$300.33$

$305$

$300$

$302$

Remember to round each number to one significant figure, then calculate

\begin{aligned} \frac{907}{3.02} & \approx \frac{900}{3} \\\\ \frac{900}{3} &=300 \end{aligned}

$\pounds 8.00$

$\pounds 7.00$

$\pounds 9.10$

$\pounds 7.40$

We round the number of litres to one significant figure, and the price per litre to one significant figure.

\begin{aligned} 7.4 \times 123 &\approx 7 \times 100 \\ 7 \times 100&=700 p \text { or } \pounds 7.00 \end{aligned}

$210.407$

$200$

$225.4$

$210$

Remember to round each number to one significant figure, then calculate

\begin{aligned} \frac{23 \times 494}{54} & \approx \frac{20 \times 500}{50} \\\\ \frac{20 \times 500}{50} &=\frac{10000}{50} \\\\ \frac{10000}{50} &=200 \end{aligned}

$11$

$10$

$9$

$10.87$

Remember to round each number to one significant figure, then calculate

\begin{array}{l} 1.84 \approx 2\\20 \div 2 = 10 \text{ packets} \end{array}

$6000$

$3000$

$5529$

$5528$

Remember to round each number to one significant figure, then calculate

\begin{aligned} \frac{600 \times 5}{0.5}&=\frac{3000}{0.5} \\\\ \frac{3000}{0.5}&=6000 \end{aligned}

5.1

5.25

5.87

5.4

The nearest square numbers are $25$ and $36$

\sqrt{25}=5 \text { and } \sqrt{36}=6

so the estimate lies between both.

$30$ is slightly closer to $25$ than $36$ so an estimate will be $5.4$ .

Estimation GCSE questions

1. Anna says that:

472 \div 5 = 9.44

Use estimation to show that Anna cannot be correct.

(3 Marks)

Show answer

472 \approx 500 \\

(1)

500 \div 5 = 100

(1)

The answer should be near $100$

(1)

2. Oliver wants to buy the following items for his garden:

$5$ packs of turf at $\pounds 91$ per pack

$30$ paving slabs at $\pounds 5.81$ each

$4$ fruit trees at $\pounds 16.50$ each

Estimate the total cost.

(3 Marks)

Show answer

\pounds 91 \approx \pounds 90, \pounds 5.81 \approx \pounds 6, \pounds 16.50 \approx \pounds 20

(1)

\begin{aligned} 5 \times \pounds 90 &= \pounds 450 \\30 \times \pounds 6 &= \pounds 180 \\4 \times \pounds 20 &= \pounds 80 \end{aligned}

(1)

\pounds 450 + \pounds 180 + \pounds 80 = £710

(1)

3. (a) Estimate:

\frac{31 \times 3.96}{0.47}

(b) Estimate:

\sqrt{5.3 \times 8.9}

(6 Marks)

Show answer

(a)

\frac{31 \times 3.96}{0.47} \approx \frac{30 \times 4}{0.5}

(1)

\frac{30 \times 4}{0.5} = \frac{120}{0.5}

(1)

\frac{120}{0.5} = 240

(1)

(b)

\sqrt{5.3 \times 8.9} \approx \sqrt{5 \times 9}

(1)

\sqrt{5 \times 9} = \sqrt{45}

(1)

\begin{array}{l} \sqrt{36} = 6, \sqrt{49} = 7\\\\ \sqrt{45} \approx 6.7 \end{array}

(accept $6.6-6.8$ )

(1)

Learning checklist

You have now learned how to:

Estimate by rounding using significant figures
Estimate within the context of a worded problem, considering units
Estimate using multiple steps
Estimate considering BIDMAS principles
Estimate using square roots

The next lessons are

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