# Approximations

## Estimating

#### Arithmetic Operations

Sometimes an exam question will test your ability to do this!

Generally, you should round each number involved to one significant figure (see Rounding Learn-it) and then it's easy to estimate by using the single digits and moving the point around.

Let's have a look at one:

936 x 27 - this is difficult to do in your head but if we round both numbers to one significant figure it becomes 900 x 30.

Now this is easy to do in your head by doing 9 x 3 = 27 then moving the point 3 times (putting three noughts on!) giving the answer 27 000 which is a good estimate of the real answer 25 272.

Here's some more!

45 x 72 becomes 50 x 70 which is 3500

317 x 23 becomes 300 x 20 which is 6000

Check these are reasonable estimates of the real answers!

Here's a more difficult one; click the boxes to reveal the estimates:

Tip!

#### Error

It is important to remember that most measurement is approximate.

If you say your garden is 8 metres long you are rounding to the nearest metre and it could be anything from 7.5 to 8.5 metres long.

Upper and Lower Bounds

The real value can be as much as half the rounded unit above or below the value given.

So, if you are given 5.4cm the upper bound is 5.45cm and the lower bound is 5.35cm.

For 6.0kg you need to go 0.05kg either way so the upper bound is 6.05kg and the lower bound is 5.95kg.

Maximum and Minimum Values

For calculations you must use the upper or lower bounds of each measurement depending on what calculation you are doing.

Addition - For the maximum use the upper bound of each measurement, for the minimum use the lower bound of each measurement.

For example:

If a piece of wood measuring 15cm is joined to another piece measuring 12cm you can see the maximum and minimum values of the addition by clicking below.

Subtraction - For the maximum you need the biggest difference between the two measurements i.e. the upper bound of the first number and the lower bound of the second and for the minimum it's the other way round.

For example:

David and Steven were given seeds to plant in Biology and decided to see whose would grow the highest. After two weeks they measured them to the nearest centimetre and David's had grown to 11cm whereas Steven's had grown to 15cm. What are the maximum and minimum values of Steven's victory?

Multiplication - Same as for Addition

Division - Same as for Subtraction

Tip!

If it is a complicated calculation e.g. (32.3 x 42.6) - 12.7 then remember the rules for each separate operation. For a maximum this would be (32.35 x 42.65) - 12.65 (Notice the lower bound was used for 12.7 as it was a subtraction).

#### Trial and Improvement

Sometimes we have to find the answer to something by simply guessing! We may then try another guess to see if it is better and so on until we are happy with our answer.

This is called Trial and Improvement and there are two main rules:

1. Use tables to display your guesses and the answer they gave.

2. Be methodical. Don't guess randomly!

For example:

a rectangle has an area of 100cm2 and its base is 1cm more than it's height. Find its height to 2 decimal places.

 Height (cm) Base (cm) Area (cm2) 9 10 90 10 11 110

Now we know the height must be between 9 and 10 so we move on to the first decimal place. We can start with 9.5 if we want!

 Height (cm) Base (cm) Area (cm2) 9.5 10.5 99.75 (too low!) 9.6 10.6 101.76 (too high!)

Now we know the height is between 9.5 and 9.6 so we can move to the second decimal place.

 Height (cm) Base (cm) Area (cm2) 9.55 10.55 100.7525 (too high!) 9.54 10.54 100.5516 (too high!) 9.53 10.53 100.3509 (too high!) 9.52 10.52 100.1504 (too high!) 9.51 10.51 99.9501 (too low!)

Now we know it's between 9.51 and 9.52. We only want two decimal places so the answer has to be one of these!

We try the middle i.e. 9.515. The answer is too high so we now know it is nearer to 9.51

To two decimal places the height is 9.51cm.

Try to guess Tom's height:

Tom's is somwhere between 100cm and 200cm tall. Enter your guess in the box and click on the 'GO' button.

After each guess you can see if it is too low or too high, use this to narrow down your guesses.

See how many tries it takes you.

## Exam-style Questions

1. Graham has a plank of wood of length 610cm, correct to the nearest 10cm. He uses a cutting machine to cut the plank into pieces, without any wastage. Each piece of wood is of length 15 cm, correct to the nearest half centimetre.

Find the maximum number of pieces of wood that Graham can be certain of getting.

(Marks available: 3)

Answer outline and marking scheme for question: 1

605 ÷ 15.25 = 39 or

605 ÷ 15.249 recurring = 40

(Total = 3 marks)

2. a) The diagram shows a goalkeeper standing between goal posts. Estimate the height, in metres, of a goal post.

(1 mark)

b) The length of a football pitch is 95 metres.

(i) Write this length in centimetres.

(1 mark)

(ii) By how many metres is the length of the football pitch less than one tenth of a kilometre?

(2 marks)

(iii) 1 yard = 0.9144 metres

Work out the length of the football pitch in yards.

(2 marks)

c) The transfer fee for a footballer was £2,300,000

(i) Round this figure to the nearest million.

(1 mark)

(ii) Round this figure to the nearest half million.

(1 mark)

(Marks available: 8)

Answer outline and marking scheme for question: 2

a) The height is between or equal to 2 and 3 metres.

b) (i) 9500cm

(ii) 5m

(iii) 104, 103.9 or 103.89 yards.

c) (i) 2,000,000 or 2m

(ii) 2,500,000 or 2.5m

(Total = 8 marks)

## Rounding

#### Decimal Places

Often you are asked to write an answer to a given number of decimal places (be careful to read the question properly!).

What you need to do:

1. Count the number of decimal places you need.

2. Look at the next digit. If it's 4 or below just write down the answer with the right amount of decimal places. If it's 5 or above write down the number but put your last decimal place up by one.

For example: 2.3635 to two decimal places

For example: 53.586 to two decimal places

What if the last digit is a 9?

A 9 goes up to a 10 so you need to put a zero in the last column and add one to the previous number.

For example: 8.6397 to three decimal places

Try this one! Type in what you think the answer is and click the button to see whether you are right:

Tips!

If you are not told how many places to write just be sensible! Generally, you should go to one more place than the numbers used in the question.

With angles, no more than one decimal place should be used unless told otherwise.

#### Significant Figures

These involve all digits, not just decimal places. Zeros are only "significant" if they separate two other non-zero digits!

What you need to do:

1. Start counting at the first non-zero digit until you have the number of digits that you need.

2. Look at the next digit. If it's a 4 or below just write the number down leaving the last digit the same. If it's a 5 or above put the last digit up by one.

3. If you are rounding whole numbers (i.e. to the left of the decimal point) put zeros in all the other columns after your last digit until you reach the decimal point.

e.g. 12 736 to three significant figures is 12 700

e.g. 6530 to one significant figure is 7000

e.g. 0.576 to two significant figures is 0.58

Try this one! Type in the answer and click the button to see if you are right:

Tip!

In real situations, use common sense to decide on your accuracy. E.g. Length of a back garden would not be written as 8.5632 metres. It would be more sensible to write 8.6 metres!

## Rounding

Decimal Places - count from the decimal point

Significant Figures

1. Count from the first digit.
2. Don't count zeros unless they're in-between non-zero digits.

Estimating

1. Use one significant figure unless you can be more accurate and still do it in your head.

Error

1. Measurement is only approximate.
2. The real answer can be half the rounded unit either way.
3. We call them upper and lower bounds.

Maximum values of calculations:

1. Addition and Multiplication - use all upper bounds.
2. Subtraction and Division - first number upper bound, second number lower bound.

Vice versa for minimum values of calculations.

Trial and Improvement

1. Use tables to display guesses and answers. Say whether too high or too low!
2. Be methodical.
3. If you know which two numbers the answer is between try the middle and that will tell you which one it is closer to.