**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.

**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}.

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### 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

**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

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

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.

45p = \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

**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]**

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.

(HIGHER ONLY)

**Question 5:** Find an estimate for \sqrt{98}.

State if your answer is an overestimate or underestimate.

**[2 marks]**

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

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