Unit 1 sec 2.4

Unit 2 sec 2.4 Rounding numbers

21 September 2015

23:28

It is sometimes helpful to round your answer.

Another situation where you often need round numbers is when you are doing calculations since the answers provided by your calculator can consist of long strings of digits.

Decimal places.

Numbers arising from calculations are sometimes rounded to a particular number of decimal places.

In a calculation involving money, the answer might be rounded to two decimal places, so that it can be intercepted in pounds and pence.

Once you have decided where to round a number, you should use the following rule to decide whether to round up or down

Round up if the number is above 5 and down otherwise.

Example 4

(A) 0.0582 to three decimal places

(B)7.05683 to one  decimal place

(C) 2.3971 to two decimal places

Solution

(A) look at the digit after the first three decimal places: 0.0582.

It is 2 which is less than 5, so round down

0.0582= 0.058 (to 3 d.p.)

(B) look at the digit after the first decimal places: 7.05683. It is 5, which is 5 or more so round up.

7.05683= 7.1 (to 1 d.p.)

(C) lookat the digit after the first two decimal places: 2.3971. It is 7, which is is 5 or more, so round up.

2.3971= 2.40 (to 2 d.p.)

Activity 7

(A) 2.24

(B)0.005

(C) 42.5982

(D) 8.0

 Significant figures

Another way of specifying a number should be rounded involves looking at its significant figures.

The second figure shows the next most important digit for telling you how big the number is and so on

The usual 5 or more rule in the strategy is used when rounding to a particular number of significant figure.

 EXAMPLE 5

Round the following numbers indicated

A. 36.9572 to four significant figure

B. 0.000349 to one significant figure

C. 56.0463 to one significant figure

D.               0.0198 to two significant figure

SOLUTION

A. Look at the digit after the first four significant figures: 36.9572.

It is 7 which is greater than 5 so round up.

36.9572= 36.96 (to 4 s.f.)

( B )  Look at the digit after the first significant figure 0.000349.

It is four which is less than 5 so round down.

0.000349= 0.0003 (to 1 s.f.)

( C )  Look at the digit the first significant figure 56.0463.

It is 6 which is greater than 5 so round up

56.0463=60

( D ) Look at the digit the first two significant figure: 0.0198.

It is the 8 which is greater than 5 so round up

0.0198=0.020 (to 2 s.f.)

0 is included after the two to make it clear that the number is rounded to two significant figures. You should likewise when you round numbers yourself.

For example the final zero in the number 0.020 in the solution to example 5d is significant.

In contrast the zero in the number 60 in the solution.

The number 3700 could be the result of rounding 3684 to two significant figure, 3697 to three significant figures or 3700 to four significant figures.

This is one reason why it is important to state how a number has been rounded

Whether or how the number has rounded, you can usually assume that any zeros at the end are not significant.

Activity 8 rounding to a number of significant figures.

Round following numbers as indicated

a)  23650 to 2 significant figures = 24

b)  0.00547 to 1 significant figure = 0.005

c)  42.59817 to 4 significant figures = 42.60

Other types of rounding

Numbers are also sometimes rounded to the nearest 10, or hundred, or thousand, and so.

similary, you can also round to the nearest meter or the nearest 10 kg, and so on.

Choosing which type of rounding to use

Rounding to a number of significant figures is often the most useful type of rounding to use.

You need to know the height of a woman who is 1.65m tall then an approximation to the nearest meter is not useful. In each case however rounding to 3 significant figures gave a useful approximation.

Rounding answers appropriately

The measurements that you have used in a calculation give you an indication of the amount of rounding that you should use for your answer.

For example, the role of distance from Paris to Lyon 465KM. Suppose that you want to convert this distance into miles. You can use the fact that

1Km= 0.621371192 miles (to 9 s.f)

multiplying the distance in km by the conversion factor gives the distance in miles as

465×0.621371192 = 288.937 6043.

It was given as 465KM to the nearest kilometre, so it could be anything from 464.5km up to 465.5KM.

You should round to no more significant figures than the number of significant figures in at least precise number in the calculation.

A full analysis of rounding is outside the scope of the module, so activities and TMA questions more often state what to rounding to use in your answer.

The number of significant figures and answer is stated to is known as the precision to the answer.

 Activity 9 rounding an answer appropriately.

In this activity you are asked to convert 465KM into miles again, but this time using the following less precise conversion factor.

1km is approximately equal to 0.62 miles.

a)  Do the calculation and round your answer to the nearest mile. Compare your answer to the answer found in the calculation on page 22, and comments on why they are not the same.

b)  Round your answer appropriately.

Answer (a) 465 × 0.62= 288.3

rounding to the nearest mile games 288, the correct answer to the nearest mile 289.

Answer (b) rounding to 2 significant figures get the answer 290 miles

Activity 10 rounding at different stages of a calculation.

Suppose that you want to calculate the length, in miles, of a return journey to a time 36KM away. Use the conversion factor given earlier to carry out the following calculations.

a)  Convert 36km to miles, and round your answer to the nearest mile. Use this answer to find the total length of the journey.

b)  Convert 36km to miles. When the unrounded answer still in your calculated display type ‘× 2’ into your calculator, and press the ‘=’ key to obtain the total length of the journey. Round your answer to the nearest mile.

c)  coment on which of the parts (a) and (B) is the better way of carrying out the calculation.

Answer (a) 36×0.621= 22

2×22= 44

answer (B) 2 × 22.3693 6291= 44.738725 82= 45 (2 the nearest mile)

answer (c) the answer in part b is more accurate. In part a, rounding to early led to inaccurate fine answer.

If you do a calculation in 2 or more steps and round your answer after one of the steps, then your final answer may be inaccurate.

This is known as using full calculator precision.

You do not need to include all the digits in the working that you write down. You can write down just a few of them usually at least 3 digits after the decimal point. Some of the digits of a number just before you round it, you should make sure that you have written digits so that someone reading your working can see that the rounding is correct.

Considering the context

it is also important to consider the context. You are calculating how many cupboards will fit along dual kitchen wall and your answer is 7.9, then you should round by almost 7 cupboards, because 8 cupboards wouldn’t fit. On the other hand, if you are painting your kitchen and need 1.2 tins of paint, then you should round up and by 2 tins.

Rounding answers

v useful calculator precision throughout calculations, to avoid rounding errors.

v Round your answer appropriately, taking account of the measurements used and the context.

v Check that you have fold any instructions on rounding given in a question.