Unit 1 Sec 3.3

Unit 1 sec 3.3 Percentages

25 September 2015

23:32

You may have noticed that there are a few types of fractions used in the media. You don’t usually come across fraction like  or, because fractions like these are difficult to visualise and compare instead of percentages are often used.

 Some basic techniques for calculating with percentages, and shows you how they can be used to make comparisons and describe changes. To make a news story seems more dramatic than it really is.

The term “per cent” means “per 100”. If the percentage of a cake tells you about 20% of a cake is fat, then it means that  (or ) of the is fat.

To do this you first write the percentage in the form of a fraction with the denominator 100, then you simplify this to get the fraction, or divide out the decimal.

Example.

45%=  =  and 45% =  = 45÷100=0.45

Similarly

100% =  = 1

To convert the other way from fraction or to percentage you just need to multiply by 100%. Because 100% =1, this does not change the value of the fraction or decimal; it just allows it to be written as a percentage

Example

0.015= 0.015×100%=1.5%

Activity 21.

Percentage	Decimal	Fraction
60%	60÷100=0.6	60%= =
=  ×100%=7÷8×100%= 87.5%	7÷8=0.875
1.35=1.35×100=135%	1.35	1.35=135% =  =  = 1

Convert 3.8% to decimal

3.8% =  = 0.038

It is often helpful to express one number as a percentage of another. This is done by expressing the first number as a fraction of the second number.

 Strategy to express as a percentage of another number

Calculate

  × 100%

Example 11 expressing a number as a percentage of another number.

In a survey of 1500 mature students, 465 agreed with the statement that higher education is vital for getting a new career. What percentage of the group is this?

Solution

Write down the fraction and convert it to a percentage.

The fraction of students who agreed with the statement is

.

So the percentage of student who agreed with the statement is

× 100% = 31%

ACTIVITY 22

The fraction of students is

So the percentage of students is

× 100%=28%

Using percentages to make comparisons

While  of the year group attained the nationally expected standard in mathematics, then might not be able to tell immediately which subject had the better performance.

It you were informed 78% and 75% of the year group attained the expected levels in English and mathematics, respectively, then you would know immediately that the performance in English is slightly better.

A comparison which takes account of underlying numbers in this way is called a relative comparison.

ACTIVITY 23

The percentage of pupils at school A who achieved the standard is

× 100% = 62.4% (to 1d.p)

The percentage of pupils at school B who achieved the standard is

× 100% = 66.2% (1 d.p)

Even when it is clear that a relative comparison is fairer than an absolute one, it is not always clear what the comparison should be relative to.

Activity 23 shows which of the two school had a better performance, but it certainly does not show which school had the better teaching.

 A fair relative comparison would need to take figures for this issue, and probably others into account.

A good way to compare schools is a value added measures of performance, which take into account the attainment of pupils at the time when they start at the school, are better, but it is difficult to devise a truly fair method of comparison.

You might like to watch out for examples of absolute and relative comparisons in the media, as different viewpoints can be put forward depending on the comparison used.

Percentages of quantities.

Sometimes you need to work out a percentage of a quantity. Calculations like this can be worked out using the strategy below.

Strategy to calculate a percentage of a quantity

Change the percentage to a fraction or a decimal, and multiply by the quantity.

Example 12

What is 3% of £300,000?

Solution

3% of £300,000 =  × £300,000=£9000.

Activity 24

(A1) 30% of 150g

Answer  × 150g = 45g

(A2) 110% of 70ml

Answer  × 70ml = 77ml

(A3) 0.5% of £220

Answer  ×£220 = £1.10

Percentages increases and decrease

Another common use of percentages is in indicating how quantities have changed.

A percentage increase or decrease is calculated by expressing the increase or decrease as a fraction of the original value.

Strategy to calculate a percentage increase or decrease.

Calculate  × 100%

Example 13

Last year 1450 students enrolled on maths courses. This year 1870 students have enrolled. What is the percentage increase in the number is students.

Solution

The actual increase is 1870 – 1450 = 420

So the increase as a percentage of the original number is

 ×100% =29% (to 2 s.f)

Hence there is a 29% increase in the number of students

Activity 25

The number of complaints received by a customer services department has fallen from 145 to 125 over the last month. What is the percentage decrease?

145 -125 = 20

 × 100% = 14% (to 2 s.f)

Hence there is a 14% decrease in the number of complaints.

Often you know about a percentage increase or decrease in the value of something, and you want to work out the new value.

Example 14

A computer originally priced at £599 is reduced by 15% in a sale. What is the new price?

Solution

First method

Calculate the decrease in price and subtract it from original price

The decrease in price is

15% of £599= 0.15 × £599 = £89.85

So the reduced price is

£599 - £89.86 = £509.15

Second method

Use the fact that you have 100% -15% of the original price.

The reduced price is 100% - 15% =85% of the original price. So the reduced price is

85% of £599 = 0.85 ×£599 = £509.15.

You have to multiply the original value by a percentage greater than 100%.

Example 15

The rent on a flat is £800 per month and is to be raised by 5%. What is the new rent?

Solution

The new rent is 100% + 5% = 105% of the original rent. So the new rent is

105% of £800= 1.05×£800= £840

Activity 26

(A)       Work out the new price of a car if the original price was £15,400 and the price has been reduced by 20%

Answer 0.20×15,400= £12.320

(B)      If a weekly wage of £360 is increased by 2.5%, what is the new weekly wage?

Answer 1.025×360= £369

(C)      If a barrel of oil cost £90 and the price rise by 100%, what is the new price?

Answer 2×90 = £180

If something increases by 100% then it doubles, you can also work out that id something increases by 200% then it triples, and if something increases by 300% then is quadruples, and so on.

The average price of a house in the UK rose from £165,000 to £200,000. The actual increase was £200,000- £165,000= £35,000, so the percentage increase was

 ×100%= 21% (to 2 s.f)

By the forth quarter of 2008, the average house price had fallen back to about £165,000 again. This is a percentage decrease of

 ×100%=18% (to 2 s.f)

So the average house price rose by 21%. This is because the rise started from a smaller value than the fall did.

In 2008, the UK government reduced the rate of value added tax (vat) from 17.5% to 15%, in response to the economic situation at the time. Many people thought that this meant prices should drop by 2.5%, but in fact the reduction was smaller: only about 2.1%. to see why, consider an item cost £100 exclusive of VAT. When the VAT rate was 17.5%, the item cost £117.50, and when the VAT rate was 15%, it cost £115. So the VAT cut caused the priced to decrease by £2.50 from an original price of £117.50, and hence the percentage decrease was

 ×100%= 2.1% (to 2 s.f)

You also need to know that the population of England was about 50 million in 2007.

Activity 27

(A)       Explain how the figures of 27% and £100 in cutting on the left of figure 17 were derived.

Answer: The spending rose from £18.7 billion to £23.7bn, which is a 5% increase. The percentage increase is

  ×100%=27%.

Since the increase spending of £5 bn. There were 50 million people in England in 2007. The spending for each person was approximately

 = =  =£100

(B)      For each of 2002-3 and 2006-7 calculate the percentage of total expenditure that was spent on public order. Check that these percentages correspond to the amounts to 6.8p and 6.6p in the cutting on the right of figure.17

Answer:

   × 100% =6.8%

The percentage of total expenditure that was spent on public order in 2006-7 was

 × 100%= 6.6% (to 2 s.f)

(C)      How much would the government have spent on public order in 2006-7 if it had spent the same percentage of total expenditure as in 2002-3 give answer to 3s.f.

Answer:

The percentage of total expenditure spent on public order in 2002-3 was approximately 6.8% or more precisely 6.81983…% if this percentage of total expenditure had been spent on public order in2006-7 then the spending on public order would have been

6.81983…% of £359.2 bn

 ×£359.2 =0.0681983…×£359.2bn=£24.5bn

(D)      Use your rounded answer to part C to explain how the figure of £800 million in the second cutting has derived.

Answer:

The difference between the amount in part C and the amount that the government actually spent on public order 2006-7 is

£24.5bn-£23.7bn =£0.8bn= £800bn.  It shows how it was worked out.

(E)       What criticism could you make of each article?

Answer:

In the first article there is absolute increase in the amount spent, but ignores the fact that prices will have risen over the four year period as well. So some of the extra £5bn would be spent just maintaining the level of support that the public received in 2002. The key question here is what new support is provided for the public and neither they nor the amount spent on new support is started in the article.

By using the relative comparison the second ignores the fact that there was significant absolute increase. There were large increases in health and education without any loss to spending to the public order. The percentage spent has dropped but again the key question is what effect has that had on the service provided has there been an overall increase or decrease in those.