Basic Percentages (OCR GCSE Maths)

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Basic Percentages

What is a percentage?

  • “Per-cent” simply means “ ÷ 100” (or “out of 100”)
  • You can think of a percentage as a standardised way of expressing a fraction – by always expressing it “out of 100”
  • That means it is a useful way of comparing fractions e.g.

½ = 50 over 100 = 50% 

⅖ = 40 over 100 = 40%

¾ = 75 over 100 = 75%

 

How do I work out basic percentages of amounts?

  • You can use simple equivalences to calculate percentages of amounts without a calculator
    • 50% = 1 half so you can divide by 2
    • 25% = 1 fourthso you can divide by 4
    • 20% = 1 fifth so you can divide by 5
    • 10% = 1 over 10 so you can divide by 10
    • 5% = 1 over 20 so you can find 10% then divide by 2
    • 1% = 1 over 100 so you can divide by 100 etc.

  • You can then build up more complicated percentages such as 17% = 10% + 5% + 2 x 1%

How do I find any percentage of an amount?

  • Method 1: You can find any percentage of an amount by dividing by 100 and multiplying by the given %
    • 23% of 40 is 40 ÷ 100 = 0.4, multiply by 23:  0.4 × 23 = 9.2
  • Method 2: To find “a percentage of X”: multiply X by the "decimal equivalent" of that percentage (percentage ÷ 100)
    • for example,  23% of 40 is 40 x 0.23 = 9.2
  • To find “A as a percentage of B”: do A ÷ B to get a decimal, then x 100, e.g.
    • for example, to find 26 as a percentage of 40 first do 26 ÷ 40 = 0.65, then x 100 to get 65%
      • 26 is 65% of 40

Worked example

Jamal earns £1200 for a job he does and pays his agent £150 in commission.

Express his agent's commission as a percentage of Jamal's earnings.

Commission over Earnings cross times 100

150 over 1200 cross times 100 equals 12.5

bold 12 bold. bold 5 bold percent sign

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Percentage Increases & Decreases

How do I increase or decrease by a percentage?

  • Identify “before” & “after” quantities
  • If working in percentages, add to (or subtract from) 100 
    • a percentage increase of 25% is 100 + 25 = 125% of the "before" price
      • Add 25% to the original amount
    • a percentage decrease of 25% is 100 - 25 = 75% of the "before" price
      • Subtract 25% from the original amount
  • A multiplier, p, is the decimal equivalent of a percentage increase or decrease
    • The multiplier for a percentage increase of 25% is p = 1 + 0.25 = 1.25
      • Multiply the original amount by the multiplier, 1.25, to find the new amount
    • The multiplier for a percentage decrease of 25% is p = 1 - 0.25 = 0.75
      • Multiply the original amount by the multiplier, 0.75, to find the new amount
  • The amount "before" and the amount "after" a percentage change are related by the formula "before" × p = "after"
    • where p is the multiplier

How do I find a percentage change?

  • Method 1: rearrange the formula "before" × p = "after" to make p (the multiplier) the subject
    • p after over before 
    • Calculate p and interpret its value
      • p = 1.02 shows a percentage increase of 2%
      • p = 0.97 shows a percentage decrease of 3%
  • Method 2: Use the formula that the "percentage change" is  fraction numerator after space minus space before over denominator before end fraction cross times 100
    • A positive value is a percentage increase 
    • A negative value is a percentage decrease
  • The same formula can be used for percentage profit (or loss)
    • the "cost" price is the price a shop has to pay to buy something and the "selling" price is how much the shop sells it for


Percentage space profit space left parenthesis or space loss right parenthesis space equals space fraction numerator selling space price space minus space cost space price over denominator cost space price end fraction cross times 100

  • You can identify whether there is a profit or loss 
    • cost price < selling price = profit (formula gives a positive value)
    • cost price > selling price = loss (formula gives a negative value)

Worked example

(a) Increase £200 by 18%

Write 18% as a percentage (by dividing by 100)
 

18 ÷ 100 = 0.18
 

Find the decimal equivalent of an 18% increase (by adding 1 to 0.18)
 

1 + 0.18 = 1.18
 

Multiply £200 by 1.18
 

200 × 1.18

£236

(b) Decrease 500 kg by 23%

Write 23% as a percentage (by dividing by 100)
 

23 ÷ 100 = 0.23
 

Find the decimal equivalent of a 23% decrease (by subtracting 0.23 from 1)
 

1 - 0.23 = 0.77
 

Multiply 500 by 0.77
 

500 × 0.77

385 kg

(c) The number of students in a school goes from 250 to 310. Describe this as a percentage change.

Use the formula "percentage change" = fraction numerator after space minus space before over denominator before end fraction cross times 100
 

fraction numerator 310 space minus space 250 over denominator 250 end fraction cross times 100 space equals space 24 percent sign
 

This is a positive value so is a "percentage increase"
 

A percentage increase of 24%

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Mark

Author: Mark

Expertise: Maths

Mark graduated twice from the University of Oxford: once in 2009 with a First in Mathematics, then again in 2013 with a PhD (DPhil) in Mathematics. He has had nine successful years as a secondary school teacher, specialising in A-Level Further Maths and running extension classes for Oxbridge Maths applicants. Alongside his teaching, he has written five internal textbooks, introduced new spiralling school curriculums and trained other Maths teachers through outreach programmes.