## What you need to know

### Estimating

The way we estimate answers to calculations is simple – we round every number involved to 1 significant figure, and then calculate with those numbers instead. Just to recall, we start counting significant figures from the __first non-zero__ digit. For some more practise on rounding numbers take a look over here .

In the higher course only, you may also be asked about a slightly different types of estimation, and that is estimating powers and roots. When estimating powers, we can take the same approach e.g. to estimate (9.306)^2, we can round 9.306 to 9, and then work out the estimate to be 9^2 = 81.

### Example 1: Simple Estimating

Estimate the answer to \dfrac{8.21}{3.97} \times 31.59.

Round each number to 1 significant figure:

8.21 \text{ rounds to } 8,

3.97 \text{ rounds to } 4,

31.59 \text{ rounds to } 30.

Therefore, we get

\dfrac{8.21}{3.97} \times 31.59 \approx \dfrac{8}{4} \times 30 = 2 \times 30 = 60.

So, the estimate answer to the calculation is 60 (compared to real answer of 65.328… it’s not too shabby).The \approx symbol means “approximately equal to”.

### Example 2: Estimating with Equations

The formula for the force, F on a moving object is F = ma, where m is the mass and a is the acceleration. Estimate the force on an object which has mass 5.87\text{ kg} and acceleration 24.02\text{m/s}^2.

Round the numbers in the question to 1 significant figure:

5.87 \text{ rounds to } 6,

24.02 \text{ rounds to } 20.

Therefore, we get

\text{Force } = 5.87 \times 24.02 \approx 6 \times 20 = 120

### Example 3: Estimating Square Roots

Find an estimate for \sqrt{40}.

The square root of 40 will be some number that we can square to make 40. We know that

6^2 = 36 \text{ and } 7^2 = 49

So, the answer must fall somewhere between 6 and 7. Since 40 is 4 away from 36 but 9 away from 49, we can conclude the answer will be somewhat closer to 6.

Therefore, 6.3 is the estimate for \sqrt{40}.

### Example Questions

1) Estimate the value of \dfrac{9.02 + 6.65}{0.042 \times 11}

Round each number to 1 significant figure:

9.02 \text{ rounds to } 9,

6.65 \text{ rounds to } 7,

0.042 \text{ rounds to } 0.04,

11 \text{ rounds to } 10.

Therefore we get,

\dfrac{9.02 + 6.65}{0.042 \times 11} \approx \dfrac{9 + 7}{0.04 \times 10} = \dfrac{16}{0.4}

To make this division easier, multiply the top and bottom of the fraction by ten, to find

\dfrac{16}{0.4} = \dfrac{160}{4} = 40

2) Estimate the answer to \dfrac{57.33-29.88}{8.66-5.55}

Rounding each number to 1 significant figure:

57.33\textrm{ rounds to }60

29.88\textrm{ rounds to }30

8.66\textrm{ rounds to }9

5.55\textrm{ rounds to }6

Therefore, we get:

\dfrac{57.33-29.88}{8.66-5.55}\approx\dfrac{60-30}{9-6}=\dfrac{30}{3}=10

3) James wants to buy 5 pens and 3 pencils. The pens cost \pounds1.89 each and the pencils cost 45 p.

Find an estimate for how much this will cost James in \pounds.

Because the answer needs to be in pounds, we should turn the cost of the pencils into pounds first.

45p=\pounds0.45

Now we can start estimating.

1.89\textrm{ rounds to }2

0.45\textrm{ rounds to }0.5

And now we need to multiply these amounts by how many of each he wanted.

\textrm{(Pens) }\pounds2\times5=\pounds10

\textrm{(Pencils) }\pounds0.50\times3=\pounds1.50

And now all we need to do is add them together.

\pounds10+\pounds1.50=\pounds11.50

4) In order to take his family to a show, Sergio will have to purchase 2 adult tickets and 3 child tickets. Given that an adult ticket costs £32.60 and a child ticket costs £17.50, work out an estimate for how much it will cost Sergio to take his whole family to this show.

Round each number to 1 significant figure:

32.60 \text{ rounds to } 30,

17.50 \text{ rounds to } 20,

Therefore, the approximate cost of the 3 child tickets is 3 \times 20 = \pounds 60.

The approximate cost of the 2 adult tickets is 2 \times 30 = \pounds 60.

Thus, the approximate total cost is 60 + 60 = \pounds 120.

5) Find an estimate for \sqrt{98}.

(HIGHER ONLY)

Round the number to 1 significant figure,

98\approx100

Therefore,

\sqrt{98}\approx\sqrt{100}=10

### Worksheets and Exam Questions

#### (NEW) Estimating Exam Style Questions - MME

Level 4-5#### Estimating - Drill Questions

Level 4-5#### Estimating (2) - Drill Questions

Level 4-5#### Estimating (3) - Drill Questions

Level 4-5### Videos

#### Estimating Q1

GCSE MATHS#### Estimating Q2

GCSE MATHS#### Estimating Q3

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