Estimating

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Estimating Revision

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.

Rounding Numbers

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”.

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:

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

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 $\sqrt{40}$ .

Estimating Example Questions

Question 1: Estimate the value of $\dfrac{9.02 + 6.65}{0.042 \times 11}$

[2 marks]

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

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

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Question 2: Estimate the answer to $\dfrac{57.33-29.88}{8.66-5.55}$

[2 marks]

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:

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

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Question 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 $\pounds$ .

[3 marks]

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.

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

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Question 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.

[3 marks]