Drawing a pie chart

Look at this record of traffic travelling down a particular road.

Traffic Survey 31 January 2008

Type of vehicle	Number of vehicles
Cars	140
Motorbikes	70
Vans	55
Buses	5
Total vehicles	270

To draw a pie chart, we need to represent each part of the data as a proportion of 360, because there are 360 degrees in a circle.

For example, if 55 out of 270 vehicles are vans, we will represent this on the circle as a segment with an angle of: (⁵⁵/₂₇₀) x 360 = 73 degrees.

This will give the following results:

Traffic Survey 31 January 2008

Type of vehicle	Number of vehicles	Calculation	Degrees of a circle
Cars	140	(140/270) x 360	= 187
Motorbikes	70	(70/270) x 360	= 93
Vans	55	(55/270) x 360	= 73
Buses	5	(5/270) x 360	= 7

This data is represented on the pie chart below.

Cumulative increase and decrease

Compound Interest

Here the interest is added to the principal at the end of each year. So the next year the interest is worked out on a larger amount of money than what was originally borrowed.

This means paying interest on the interest of previous years (unlike simple interest, where you only pay interest on the original amount).

This is how it is calculated:

£400 is borrowed for 3 years at 5% compound interest.

Principal at the start = £400

Interest in the 1st year = ⁵/₁₀₀ × 400 = £20

Principal after 1 year = £420

Interest in the 2nd year = ⁵/₁₀₀ × 420 = £21

Principal after 2 years = £441

Interest in the 3rd year = ⁵/₁₀₀ × 441 = £22.05

Principal after 3 years = £463.05

The total interest charged under compound interest will be £63.05.

This is different to the simple interest worked out above.

Four types of question

In percentage questions, read the question carefully and decide what you are being asked to do. You may need to:

• Find a given percentage of an amount.
• Work out a percentage when given 2 amounts.
• Work backwards from a percentage increase or decrease (reverse percentages).
• Find a cumulative change.

Mean, mode and median

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Measures of average in grouped and continuous data

The mean

We already know how to find the mean from a frequency table. Finding the mean for grouped or continuous data is very similar.

The grouped frequency table shows the number of CDs bought by a class of children in the past year.

 

Number of CDs	Frequency (f)
0-4	10
5-9	12
10-14	6
15-19	2
>19	0

• We know that 10 children have bought either 0, 1, 2, 3 or 4 CDs, but we do not know exactly how many each child bought.
• If we assumed that each child bought 4 CDs, it is likely that our estimate of the mean would be too big.
• If we assumed that each child bought 0 CDs, it is likely that our estimate would be too small.
• It therefore seems sensible to use the mid-point of the group and assume that each child bought 2.

Finding the mid-points of the other groups, we get:

 

Number of CDs	f	Mid-point, x	fx
0-4	10	2	20
5-9	12	7	84
10-14	6	12	72
15-19	2	17	34
>19	0	-	0

The mean is 

Remember: This is only an estimate of the mean.

The median

As explained previously, the median is the middle value when the values are arranged in order of size.

As the data has been grouped, we cannot find an exact value for the median, but we can find the class which contains the median.

 

Number of CDs	Frequency (f)
0-4	10
5-9	12
10-14	6
15-19	2
>19	0

There are 30 children, so we are looking for the class which contains the (30 + 1) ÷ 2 = 15¹₂th value. The median is therefore within the 5-9 class.

The mode

The mode is the most common value.

We cannot find an exact value for the mode, and therefore give themodal class. The modal class is 5-9.

Finding the nth term

Sometimes, rather than finding the next number in a linear sequence, you want to find the 41^st number, or 110^th number, say.

Writing out 41 or 110 numbers takes a long time, so you can use a general rule.

To find the value of any term in a sequence, use the n^th term rule.

Question

What is the n^th term of this sequence?

Reveal

For example, to find the 10^th term, work out 5 × 10 = 50. To find the 7^th term, work out 5 × 7 = 35

So the 41^st term is 5 × 41 = 205 and the 110^th term is 5 × 110 = 550