Approximations

You should always do a quick estimate in your head when doing arithmetic so you can see if your answer is reasonable.

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!

If you can't perform your estimate in your head then it's too complicated - think again!

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).

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.

Start with whole numbers.

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!

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

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.

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.

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!