Percentage Increase or Decrease Revision Notes for HSC SSCE Mathematics Standard

Percentage Increase or Decrease

What is percentage change?

Percentage change means adjusting a value up or down by a certain percentage of its original amount. This is a really useful skill that you'll encounter in everyday life - think about shop sales, price rises, pay increases, or calculating discounts.

The key idea is that we always start with 100% representing our original value, then adjust from there.

Note

Why 100%?

We use 100% as our starting point because it represents the whole original amount. Any increase or decrease is then calculated relative to this baseline. This makes the method consistent and reliable for all percentage change calculations.

The two methods

When working with percentage changes, there are two scenarios you'll encounter: increasing a value or decreasing it. Both follow a similar pattern, but with one key difference.

Percentage increase

When you need to increase a value by a certain percentage, follow these steps:

Step 1: Start with 100% and add the percentage increase to it.

For example, if you're increasing by $5\%$ , you calculate: $100\% + 5\% = 105\%$

Step 2: Convert this new percentage to a decimal and multiply it by your original amount.

To convert the percentage to a decimal, divide by 100. So $105\% = 1.05$

Then multiply: $1.05 \times \text{original amount}$

Note

Why this method works

This method works because $105\%$ represents the original amount (100%) plus the extra 5% we're adding. By multiplying by $1.05$ , we're essentially getting the original value plus $5\%$ of that value in one calculation - it's a shortcut that combines both steps into one multiplication!

Percentage decrease

Decreasing a value works in a similar way, but you subtract instead of add:

Step 1: Start with 100% and subtract the percentage decrease from it.

For example, if you're decreasing by $20\%$ , you calculate: $100\% - 20\% = 80\%$

Step 2: Convert this new percentage to a decimal and multiply it by your original amount.

Convert to decimal: $80\% = 0.80$

Then multiply: $0.80 \times \text{original amount}$

Here, $80\%$ represents what remains after removing $20\%$ from the original. When you multiply by $0.80$ , you get the reduced value directly.

Worked example: Single percentage increase

Let's apply this to a real-world scenario.

Example

Worked Example: Price Increase Calculation

Question: A toaster is priced at $36 and the price is going to increase by $5\%$ . What will the new price be?

Solution:

First, add the $5\%$ increase to $100\%$ :

$100\% + 5\% = 105\%$

Now we need to find $105\%$ of $36:

$\text{New price} = 105\% \text{ of } \$36$

Convert the percentage to a decimal:

$= 1.05 \times 36$

Calculate the result:

$= \$37.80$

Answer: The new price is $37.80.

Important

Always check your answer makes sense!

Since we increased the price, the new price ($37.80) should be higher than the original ($36). The increase of $1.80 is indeed $5\%$ of $36, confirming our calculation is correct.

Worked example: Consecutive percentage changes

Sometimes you need to apply multiple percentage changes one after another. It's important to remember that these changes don't simply add or subtract - you must work through them step by step.

Example

Worked Example: Consecutive Percentage Changes

Question: Increase $75 by $20\%$ and then decrease the result by $20\%$ .

Solution:

First change - the $20\%$ increase:

Add the $20\%$ increase to $100\%$ :

$100\% + 20\% = 120\%$

Calculate $120\%$ of $75:

\text{New price} = 120\% \text{ of } \$75

\text{New price} = 1.20 \times 75

\text{New price} = \$90

Second change - the $20\%$ decrease:

Now we decrease this new value ($90) by $20\%$ .

Subtract the $20\%$ decrease from $100\%$ :

$100\% - 20\% = 80\%$

Calculate $80\%$ of $90:

\text{New price} = 80\% \text{ of } \$90

\text{New price} = 0.80 \times 90

\text{New price} = \$72

Answer: The final price is $72.

Important

Critical insight about consecutive changes

Notice that we didn't end up back at $75, even though we increased by $20\%$ and then decreased by $20\%$ . This is because the second percentage change was calculated on the new amount ($90), not the original amount ($75).

The $20\%$ decrease was larger in dollar terms ($18) than the $20\%$ increase ($15) because it was based on a larger value. This is a common mistake - percentage changes do not cancel out when applied consecutively!

Remember!

Summary

Key Points to Remember:

Always start with 100% representing your original amount
For increases: add the percentage to 100%, then multiply by the original value
For decreases: subtract the percentage from 100%, then multiply by the original value
Convert percentages to decimals before multiplying (divide by 100 or move the decimal point two places left)
With consecutive percentage changes, work through each step separately - the percentages don't simply cancel out
Check your answer makes sense - increases should give larger values, decreases should give smaller values

Percentage Increase or Decrease (HSC SSCE Mathematics Standard): Revision Notes