Percentages for Finance (AQA Level 3 Mathematical Studies (Core Maths)): Revision Note

Percentage Change & Multipliers

How do I increase by a percentage?

• A percentage increase makes an amount bigger by adding that percentage on to itself

• When you have a calculator, it is efficient to use multipliers

  • A multiplier is the decimal equivalent of a percentage

    • A percentage can be converted to a decimal by dividing by 100

• When increasing by a percentage, we are finding a percentage greater than 100%

  • To increase 80 by 15%

    • We are finding 115% of 80 (as 100% + 15% = 115%)

    • The multiplier is therefore 1.15

    • 1.15 × 80 = 92

How do I decrease by a percentage?

• A percentage decrease makes an amount smaller by subtracting that percentage from itself

• When decreasing by a percentage, we are finding a percentage smaller than 100%

  • To decrease 80 by 15%

    • We are finding 85% of 80 (as 100% - 15% = 85%)

    • The multiplier is therefore 0.85

    • 0.85 × 80 = 68

One Quantity as a Percentage of Another

How do I express one number as a percentage of another?

• Start by writing one number as a fraction of the other

• Find the decimal equivalent of this fraction using your calculator

• Rewrite this as a percentage

• E.g. To find 7 as a percentage of 20

  • Write as a fraction, 

  • This is equivalent to  or 

  • So 7 is 35% of 20

How do I find a percentage change?

• The multiplier that was used for a percentage change can be found using the formula:

  • 

• The value of corresponds to the multiplier for the percentage change

  • 1.05 corresponds to an increase by 5%

  • 0.75 corresponds to a decrease by 25%

Comparing Using Percentages

• Percentages allow quantities to be compared more easily

• For example, if the rate of defects in products produced by two factories were to be compared:

  • Factory A has 389 defective products out of a total of 12 098 produced

  • Factory B has 3111 defective products out of a total of 79 781 produced

• It is not immediately obvious which factory has the lower rate of defects

• Express each quantity as a percentage instead

  • The percentage of defective products from factory A

    • 389 ÷ 12 098 = 0.03215... = 3.2 %

  • The percentage of defective products from factory B

    • 3111 ÷ 79 781 = 0.03899... = 3.9 %

  • Comparing is now much simpler; factory A has the lowest rate of defects

Percentages Over 100%

• Remember that percentages are simply quantities expressed as a proportion of 100

• This means percentages larger than 100% are possible

  • For example 24 is 150% of 16

    • 

  • 360 is 300% of 120

    • 

• Percentages over 100%, and multipliers larger than 1, are used when increasing an amount, as detailed earlier

• A percentage increase of over 100% works in exactly the same way as any other percentage increase

  • To increase 36 by 125%

    • We are finding 225% (125% would be a 25% increase) of 36, so the multiplier is 2.25

    • 2.25 × 36 = 81

Solving Percentage Change Problems

• A common percentage change problem is finding the original amount before a percentage change

• These are also known as "reverse percentage" problems

• You should think about the "before" quantity (even though it is not given in the question)

• Find the percentage change as a multiplier, m (the decimal equivalent of a percentage change)

  • A percentage increase of 4% means m = 1 + 0.04 = 1.04

  • A percentage decrease of 5% means m = 1 - 0.05 = 0.95

• Use "before" × m = "after" to write an equation

  • Get the order right: the change happens to the "before", not to the "after"

• Rearrange the equation to find the "before" quantity...

  • ...by dividing the "after" quantity by the multiplier, m