# Percentages

This sec­tion builds on Unit 1 Sta­tis­tics and Num­ber, Sec­tion 2 Frac­tions, Dec­i­mals and Per­cent­ages. It’s my kind of maths, mainly because it’s not really math­e­mat­ics it’s just arith­metic. I think if you learn one or two def­i­n­i­tions and do a few prac­tice ques­tions, you  should find this one of the eas­ier parts of the Maths GCSE syllabus.

## Per­cent­age Increase and Decrease

Here you are given a start num­ber and you have to increase or decrease that num­ber by a cer­tain percentage.

There are two ways to solve these type of problems.

The first way is nor­mally how you would do it in your head or on paper. For exam­ple: Fred earns £110 per week and gets a 5% increase, how much will his new weekly wage be? In your head you would work out 5% of £110 and add it to the orig­i­nal sum. 5% of £110 is £5.50, add this to £110 to give the answer of £115.50.

Alter­na­tively, if you were using a cal­cu­la­tor you would add the per­cent­age increase to 100% to give 105%, con­vert this to a dec­i­mal to give 1.05 and then mul­ti­ply it by the orig­i­nal wage, 1.05 x £110 = £115.50.

Per­cent­age Profit or Loss

Is the per­cent­age profit or loss of the cost price.

So if the cost price is £10 and the sales price is £20, then the profit is £10 (£20 — £10) and the per­cent­age profit = £10/£10 x 100% = 100%

In words:-

Per­cent­age Profit (or loss) = actual profit (or loss)/cost price X 100%

Repeated Per­cent­age Change

This is most often used in finance. Usu­ally when you invest money, inter­est is cal­cu­lated using com­pound inter­est. This is where the inter­est is paid on the ini­tial invest­ment plus on any inter­est already earned. Sounds more com­pli­cated than it is! Best to look at an actual example:

John invests £1000 in a savers account which earns 5% inter­est annu­ally com­pound  inter­est. Assum­ing he makes no with­drawals, how much will he have in his account;

a) After 1 year

b) After 2 years

a) £1000 x 5% = £50, so after one year he’ll have £1,000 + £50 = £1,050

b) After 2 years he’ll have £1,050 (from the first year) + 5% inter­est on £1,050=

£1,050 + (£1,050 x 0.05) = £1,050 + £52.50* = £1,102.50

*1% of £1,050 = £10.50, so 5% of £1,050 = 5 x £10.50 = £52.50

Reverse Per­cent­ages

This is another case where the two dif­fer­ent meth­ods, one with­out a cal­cu­la­tor and one with. Best explained with an example.

A shirt is on sale at £60 and the ticket says that it has been reduced by 20%. What was the orig­i­nal price?

If you don’t have a cal­cu­la­tor, fol­low these two steps:

a) Work out what per­cent­age the new price rep­re­sents of the orig­i­nal price:-

In this case 100% — 20% = 80%

In this case £60 x 100/80

At this point could divide by 80 and mul­ti­ple by 100 BUT it’s often a lot sim­pler to use your knowl­edge of fractions:-

so £60 x 100/80 = £60 x 10/8 = £60 x 5/4 = £60 x 1.25 (we know that 0.25 = a quarter)

so £60 x 1.25 = £60 + a quar­ter of £60 = £60 + £15 = £75.

If you do have a cal­cu­la­tor fol­low these steps (note the first step is the same as the “with­out a cal­cu­la­tor method” shown above):

a) Work out what per­cent­age the new price rep­re­sents of the orig­i­nal price:-

100% — 20% = 80%

b) Divide the answer by 100 to get a “multiplier”:-

80/100 = 0.8

c) Divide the new price by the mul­ti­plier to get the orig­i­nal price:-

£60/0.8 = £75

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