## What you need to know

Sometimes the calculations we come up against are complicated, and it’s helpful to be able to come up with an easy estimate for what the answer is. The answer probably won’t be perfect, but that doesn’t mean the answer is meaningless to us.

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 this topic, take a look over here (https://mathsmadeeasy.co.uk/gcse-maths-revision/rounding-numbers-gcse-math-revision/).

Example: 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”.

Sometimes it can feel strange not getting exact answers when a lot of maths you’ve seen has been all about your answers being precise and exact. As long as you don’t think about it too hard and remember that estimating just means simplifying everything to 1 significant figure, then it becomes something quite straightforward.

Example: 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 sf:

$5.87 \text{ rounds to } 6$,

$24.02 \text{ rounds to } 20$.

Therefore, we get

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

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 as with all the stuff we’ve seen already, e.g. to estimate $(9.306)^2$, we can round 9.306 to 9, and then work out the estimate to be $9^2 = 81$.

However, roots are a little bit different. Let’s see an example.

Example: 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 $\sqrt{40}$.

So, the process is to find the nice square numbers either side of the number you’re trying to square root, see how far the number you’re square rooting is from each of them, and then select a 1 decimal place answer that reflects how much closer it is to the one of the numbers. See the third question below to have another go at this.

## Example Questions

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

Answer

Round the numbers in the question to 1 sf:

$9.02 \text{ rounds to } 9$,

$6.65 \text{ rounds to } 7$,

$0.042 \text{ rounds to } 0.04$,

$11 \text{ rounds to } 10$.

Therefore, the top of the fraction is approximately equal to

$9 + 7 = 16$

The bottom of the fraction is approximately equal to

$0.04 \times 10 = 0.4$

So, we get

$\dfrac{9.02 + 6.65}{0.042 \times 11} \approx \dfrac{16}{0.4}$

To make this division easier, multiply the top and bottom of the fraction by ten. Then we get

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

So, the estimate answer to the calculation above is 40.

#### 2) 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.

Answer

Round the numbers in the question to 1sf (the numbers 2 and 3 are already there):

$32.60 \text{ rounds to } 30$,

$17.50 \text{ rounds to } 20$,

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 $60 + 60 = \pounds 120$.

#### 3) (HIGHER ONLY) Find an estimate for $\sqrt{98}$.

Answer

We know that $9^2 = 81$ and $10^2 = 100$.

98 is only 2 away from 100 compared to being 17 away from 81, so the square root of 98 will be very close to 10.

Therefore, the estimate for $\sqrt{98}$ is 9.9.

GCSE Maths tutors in Leeds, Manchester, London and everywhere else, you can use the Maths Made Easy estimating worksheets as part of tuition. You can also access many more GCSE Maths revision resources through our main page.