Estimation is about finding an answer to a problem that is *close enough *to the exact answer.

In many real life situations calculating an approximate answer will be faster and easier and this may be more important than having a high degree of accuracy.

For example, if you are checking you have been charged the correct amount in a shop or working out your fair share of the bill in a restaurant, you may not be concerned about the pennies but you would want to pay the right amount in pounds.

We often rely on computers or calculators to make precise calculations for us but human error when using machines can lead to a precise but incorrect answer. In these situations, having a *feel for the numbers*, which comes from having good estimation skills, is important for checking that the answer is about the right size.

When estimating, we replace numbers in a calculation with other numbers which are much easier to work with. We can do this by rounding each number either to the nearest whole number, 10, 100, 1000 *etc* or by rounding to a given number of significant figures. The more detail we keep in our numbers when we round, the more accurate our answer will be, but of course the more difficult the calculation will be.

A good approach is to **round off all the numbers in a calculation to 1 significant figure**. We can then work out our answer using our easy numbers.

We use the symbol ‘≈’ which means ‘approximately equal to’.

4658 × 0.314

SolutionIf we round off these values to 1 sig. fig.

4698 ≈ 5000 and 0.314 ≈ 0.3

So 4658 × 0.314 ≈ 5000 × 0.3 = 1500

207.56 ÷ 397.63

SolutionIf we round off these values to 1 sig. fig.

207.56 ≈ 200 and 397.63 ≈ 400

207.56 ÷ 397.63 ≈ 200 ÷ 400 = 0.5

A kitchen is 8.9m wide and 11.4m long, and it will cost £7.85 per square metre for new flooring. Peter estimates that it will cost roughly £8000, and Paul estimates that it is nearer £800. Whose estimate is better?

SolutionIf we round our measurement to the nearest whole metre,

8.9m ≈ 9m and 11.4m ≈ 11m.

So the area of the room ≈ 9m × 11m = 99 square metres

Let’s say this is approximately 100 square metres

Rounding the cost of flooring to the nearest pound gives £8 per square metre.

The total cost will be in the region of £8 × 100 = £800.

Paul’s estimate is better. Peter’s estimate is ten times too big, we could say that it is the wrong *order of magnitude*.

You have carried out a scientific experiment and the result is given by the formula below. The person you are working with has a calculator and uses it to get the answer 159.121, but you aren’t sure if they used their calculator correctly. How can you use estimation techniques to decide whether you accept their answer or not?

Solution

Replacing each value with an approximate value gives

Your estimate is about 10 times bigger than the value from the calculator. It would be a good idea to ask your friend to retry the calculation or ask to use the calculator yourself.

**When adding or subtracting, very small numbers can be approximated by zero.**

1999 + 0.0001

Solution1999 + 0.0001 ≈ 2000 + 0 = 2000

389 – 0.034

Solution389 – 0.034 ≈ 400 – 0 = 400

**When multiplying or dividing, you must never approximate a small number with zero.** Round the number to 1 significant figure or replace with 0.1, 0.01, 0.001 etc.

0.0099 × 813

SolutionIf we round these values to 1 sig. fig.

0.0099 × 813 ≈ 0.01 × 800 = 8

If we had approximated with zero here, we would have got the answer zero.

5 ÷ 0.00879

SolutionIf we round 0.00879 to 0.01 then

5 ÷ 0.00879 ≈ 5 ÷ 0.01 = 500

If we had approximated with zero here, we would have got 5 ÷ 0, which can’t be calculated. If you try this on your calculator, you will find you get an error message.

In a café you order 3 Lattes at £2.15 a Cappuccino at £2.05 and 2 teas at £1.85. You pay with a £20 note and get £5.75 in change. Have you been given the correct amount?

You are calculating 245 × 48 on your calculator and you get the answer 1200. Is this the right answer?

A baguette costs £1.95 at the café and there are 9 of us in the office who would like a baguette. How much money will I need to take to the shop?

Homer brings home a bonus of $102.50. He wants to divide it equally between himself, Marge, Bart, Lisa, and Maggie. He calculates that he can give each family member $15.20. Lisa protests that he is short-changing them. Roughly how much should each person expect to receive?

100 + 0.00031

987.9 – 0.03

0.098 × 24

251 ÷ 0.0095

48965 × 0.00638