Estimating Worksheets | Questions and Revision | MME

Estimating Worksheets, Questions and Revision

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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

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

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

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:

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

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.

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

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 $60 + 60 = \pounds 120$.

Round the number to 1 significant figure,

$98\approx100$

Therefore,

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

Level 4-5

Level 4-5

Level 4-5

Level 4-5

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