Percentage increase and decrease practice | Year 8

Start with a single change, then examine two successive changes. Always state which amount represents the original 100%.

Use multipliers above or below one for increases and decreases.
Calculate a percentage change relative to the original amount.
Reverse a percentage change by dividing by its multiplier.

The idea

A percentage change is measured against the original amount. Increasing by p% multiplies by 1 + p/100; decreasing by p% multiplies by 1 − p/100. To find a percentage change from two amounts, divide the change by the original amount, then multiply by 100%.

Before you begin: Convert percentages to decimals and multiply or divide decimal numbers.

Worked examples

01 Start with the method

An $80 ticket increases in price by 15%. Find its new price.

80 \times (1 + 0.15)

Step 1

The original price represents 100%. After a 15% increase, the new price is 115% of the original.

100 % + 15 % = 115 % = 1.15

Step 2

Multiply the original amount by 1.15.

80 \times 1.15 = 92

Step 3

Check the increase separately. Fifteen percent of $80 is $12, so the new price is $92.

0.15 \times 80 = 12,

80 + 12 = 92

A mistake to avoid

For percentage change, divide by the starting amount. A 15% increase followed by a 15% decrease does not return to the start because the second percentage uses a new base.

02Use a decrease multiplier

A $250 bag is discounted by 12%. Find the sale price.

250 \times (1 - 0.12)

Step 1

After removing 12%, 88% of the original price remains.

100 % - 12 % = 88 % = 0.88

Step 2

Multiply the original price by the remaining proportion.

250 \times 0.88 = 220

Step 3

The discount is $30. Subtracting it independently confirms the sale price.

0.12 \times 250 = 30,

250 - 30 = 220

Result

$ 220

03Two equal percentages do not cancel

A $50 item rises by 10%, then its new price falls by 10%. Find the final price.

50 \times 1.10 \times 0.90

Step 1

Apply the increase to the original $50.

50 \times 1.10 = 55

Step 2

The decrease uses $55 as its base, so it removes $5.50 rather than $5.

55 \times 0.90 = 49.50

Step 3

Combining the multipliers gives 0.99, so the final price is 99% of the original.

1.10 \times 0.90 = 0.99

Result

$ 49.50

Practice questions with answers

Work these out first, then open each solution to compare your method. These questions are untimed.

Question 1

A $120 bicycle accessory increases in price by 20%. Find its new price.

120 \times 1.20

Show solution 1

120 \times 1.20 = 144

The new amount is 120% of $120. The increase is $24, so the new price is $144.

Question 2

Decrease $90 by 30%.

90 \times 0.70

Show solution 2

$ 63

Seventy percent remains. Multiplying 90 by 0.70 gives 63.

Question 3

An amount grows from 60 to 75. Find the percentage increase.

\frac{75 - 60}{60} \times 100 %

Show solution 3

25 %

The increase is 15. Divide by the original 60, not the new 75: 15/60 = 0.25.

Question 4

An amount falls from 200 to 170. Find the percentage decrease.

\frac{200 - 170}{200} \times 100 %

Show solution 4

15 %

The decrease is 30 out of the starting 200. This is 0.15, or 15%.

Question 5

After a 20% discount, a book costs $64. Find its original price.

0.80 \times original price = 64

Show solution 5

original price = 64 \div 0.80 = $ 80

The sale price is 80% of the original. Divide by 0.80; adding 20% to $64 would use the wrong base.

Common questions

What is the base for percentage change?

It is the amount before the change. Divide the increase or decrease by that starting amount, then multiply by 100%.

Why do successive increases and decreases not cancel?

Each change acts on the amount at that stage. For example, 10% off $55 removes more money than 10% of $50 added.

How do I recover an original amount?

Identify the multiplier that produced the final amount, then divide the final amount by it. For a 20% discount the multiplier is 0.80.