Percentage Increases & Decreases (Edexcel IGCSE Maths)

Spec point — 1.6E

Percentage increases & decreases

How do I increase by a percentage?

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

• With a calculator you can 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, so the multiplier is 1.15
    • 1.15 × 80 = 92

• You can also use the basic percentages methods to find the percentage you are increasing by

  • Then add this on to the original amount

  • To increase 30 by 10%

    • 10% of 30 is 3
    • 30 + 3 = 33
    • This is equivalent to finding 110% of 30

How do I decrease by a percentage?

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

• With a calculator you can use multipliers

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

  • To decrease 80 by 15%

    • We are finding 85% of 80, so the multiplier is 0.85

      • Because 100% - 15% = 85%

    • 0.85 × 80 = 68

• You can also use the methods outlined in Basic Percentages to find the percentage you are decreasing by

  • Then subtract this from the original amount

  • To decrease 30 by 10%

    • 10% of 30 is 3

    • 30 - 3 = 27

    • This is equivalent to finding 90% of 30

      • Because 100% - 10% = 90%

Worked Example (a) Increase 200 kg by 21%.

Answer:

An increase by 21% is equivalent to finding 121% of the original amount So the multiplier is 1.21

1.21 × 200

Final answer: 242 kg

(b) An item that costs $500 is discounted by 35%.

Find the new price of the item.

Answer:

A discount of 35% means the price decreases by 35%

A decrease of 35% is equivalent to finding 65% of the original amount (100 - 35 = 65) So the multiplier is 0.65

500 × 0.65

$325

How do I deal with repeated percentage changes?

• In some problems there may be several changes by a percentage

• For example,

  • A shop increases the price of a product costing £80 by 10%,

    • equivalent to a multiplier of × 1.10

  • and then discounts the product by 15%,

    • equivalent to a multiplier of × 0.85

  • and then discounts the product by a further 20%

    • equivalent to a multiplier of × 0.80

• You can either:

  • Multiply the starting amount by each multiplier in turn

    • ( ( ( 80 × 1.10 ) × 0.85 ) × 0.80 ) = £59.84

  • Or combine the multipliers first and then multiply by the "combined multiplier"

    • 1.10 × 0.85 × 0.80 = 0.748

      • This shows it is equivalent to 74.8% of the original amount, or a discount of 25.2%

    • 80 × 0.748 = £59.84

• In general, for $n$ multipliers of values $m_{1},m_{2},...,m_{n}$

  • The combined multiplier is $m_{1}\times m_{2}\times...\times m_{n}$

How do I find a percentage change?

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

  • $m=\frac{\mathrm{Amount} \mathrm{after}}{\mathrm{Amount} \mathrm{before}}$

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

  • A value greater than 1 is a percentage increase

    • 1.05 corresponds to an increase by 5%

  • A value less than 1 is a percentage decrease

    • 0.75 corresponds to a decrease by 25%

• Alternatively you can use the formula:

  • Percentage Change = $\frac{\mathrm{After}-\mathrm{Before}}{\mathrm{Before}}\times100$

  • A positive value is a percentage increase

    • An answer of 12 means an increase of 12%

  • A negative value is a percentage decrease

    • An answer of -28 means a decrease of 28%

How do I find a percentage profit or loss?

• Similar strategies to the above can be used to find the percentage profit or loss

• Shops buy or produce items at a "cost price" and sell them at a "selling price"

• Using a multiplier method:

  • $m=\frac{\mathrm{Selling} \mathrm{Price}}{\mathrm{Cost} \mathrm{Price}}$

  • A value greater than 1 is a profit

    • **1.05 **corresponds to a 5% profit

  • A value** less than 1 is a loss**

    • 0.75 corresponds to a 25% loss

• Alternatively you can use the formula:

  • Percentage Profit = $\frac{\mathrm{Selling} \mathrm{Price}-\mathrm{Cost} \mathrm{Price}}{\mathrm{Cost} \mathrm{Price}}\times100$

  • A positive value is a profit

    • An answer of 12 means a 12% profit

  • A negative value is a loss

    • An answer of -28 means a 28% loss

Exam Hint

• Use "common sense" to check your answer!

  • If an item is sold for more than it was bought for, you are expecting a profit, not a loss

Worked Example The number of students in a school changes from 250 to 310.

Describe the percentage change in number of students.

Answer:

Method 1 Use the formula $m=\frac{\mathrm{Amount} \mathrm{after}}{\mathrm{Amount} \mathrm{before}}$

$\frac{310}{250}=1.24$* *

This multiplier is greater than 1, so corresponds to a percentage increase

Final answer: A percentage increase of 24%

Method 2 Use the formula Percentage Change = $\frac{\mathrm{After}-\mathrm{Before}}{\mathrm{Before}}\times100$

$\frac{310-250}{250}\times100=24$

The value is positive, so this is a percentage increase

Final answer: A percentage increase of 24%

Worked Example Sophie purchases a car for $8000 and sells it several years later for $5600.

Describe the percentage profit or loss on the car.

Answer:

Method 1 Use the formula $m=\frac{\mathrm{Selling} \mathrm{Price}}{\mathrm{Cost} \mathrm{Price}}$

$\frac{5600}{8000}=0.7$

The value is less than 1 so means it is a percentage loss

The selling price was 70% of the cost price, so a loss of 30%

Final answer: A loss of 30%

Method 2 Use the formula Percentage Profit = $\frac{\mathrm{Selling} \mathrm{Price}-\mathrm{Cost} \mathrm{Price}}{\mathrm{Cost} \mathrm{Price}}\times100$

$\frac{5600-8000}{8000}\times100=-30$

The value is negative, so this is a percentage loss

Final answer: A loss of 30%