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Basic Percentages

Basic Fractions

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% 

⅖ =  = 40%

¾ =  = 75%

 

How do I work out basic percentages of amounts?

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

150 over 1200 cross times 100 equals 12.5

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  
    • 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


  • 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” =
 


 

This is a positive value so is a “percentage increase”
 

A percentage increase of 24%

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