# Estimating

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## Estimating

The way we estimate answers to calculations is simple – we round every number involved to $1$ significant figure, unless stated otherwise, and then perform the calculation with those numbers instead.

Make sure you are happy with the following topics before continuing.

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## Type 1: Simple Estimating

These types of questions are the easiest you will see.

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

Step 1: Round each number to $1$ significant figure:

$8.21$ rounds to $8$,

$3.97$ rounds to $4$,

$31.59$ rounds to $30$.

Step 2: Put the rounded numbers into the equation and calculate:

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

Note: The $\approx$ symbol means “approximately equal to”.

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## Type 2: Estimating with Equations

Estimating with equations is a little bit more difficult, since we also have to interpret the question.

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 $5.87$ kg and acceleration $24.02$ m/s$^2$.

Step 1: Round the numbers in the question to $1$ significant figure:

$5.87$ rounds to $6$,

$24.02$ rounds to $20$.

Step 2: Put the rounded numbers into the equation and calculate:

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

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## Type 3: Estimating Square Roots

Estimating square roots is the hardest type of estimating question you will see, and is only for HIGHER students.

Example: Find an estimate for $\sqrt{40}$.

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

Step 1: Find $2$ square numbers, one on each side of the number we are given:

We know that

$6^2 = 36$ and $7^2 = 49$

So, the answer must fall somewhere between $6$ and $7$.

Step 2: Choose an estimate based on which square number it is closest to:

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 a suitable estimate for $\sqrt{40}$.

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## Estimating Example Questions

Round each number to $1$ significant figure:

$9.02$ rounds to $9$,

$6.65$ rounds to $7$,

$0.042$ rounds to $0.04$,

$11$ 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

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

Gold Standard Education

Rounding each number to $1$ significant figure:

$57.33$ rounds to $60$
$29.88$ rounds to $30$
$8.66$ rounds to $9$
$5.55$ rounds to $6$

Therefore, we get:

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

Gold Standard Education

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

$45$p $= \pounds0.45$

Now we can start estimating.

$1.89$ rounds to $2$
$0.45$ 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.

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

Gold Standard Education

Round each number to $1$ significant figure:

$32.60$ rounds to $30$,

$17.50$ 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 $60 + 60 = \pounds 120$.

Gold Standard Education

First, we need to find $2$ square numbers either side of $98$.

We know that

$9^2 = 81$ and $10^2 = 100$

So the answer must be between $9$ and $10$.

Since $98$ is only $2$ away from $100$, but $17$ away from $81$, we can conclude that the solution is going to be much closer to $10$.

Therefore, the estimate is

$\sqrt{98} \approx 9.9$

Gold Standard Education

## Estimating Worksheet and Example Questions

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

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## Estimating Drill Questions

Level 4-5GCSE

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### Estimating (3) - Drill Questions

Level 4-5GCSE

Gold Standard Education

Level 1-3GCSEKS3