Percentage Increase and Decrease Revision Notes for VCE SSCE General Mathematics

Percentage Increase and Decrease

Understanding percentage changes

When you increase or decrease a quantity by a given percentage, the percentage change is always calculated as a percentage of the original quantity. This is a crucial rule to remember! Whether you're working out a pay rise, a sale discount, or any other percentage change, you must always start with the original value.

infoNote

The original quantity is always the starting point for any percentage calculation. Think of it as your baseline or reference value.

Calculating a percentage increase

There are two methods you can use to calculate a new amount after a percentage increase. Both methods give the same answer, so choose the one that makes most sense to you.

Method 1: Calculate the increase, then add it

This method involves finding what the percentage increase is worth in actual numbers, then adding this to the original amount.

Steps:

Convert the percentage to a fraction (or decimal)
Multiply by the original amount to find the increase
Add the increase to the original amount

Method 2: Use a multiplier

This method recognises that an increase means you end up with more than 100% of the original. It's often quicker!

Steps:

Add the percentage increase to 100%
Convert this total percentage to a fraction (or decimal)
Multiply by the original amount

chatImportant

When using the multiplier method, remember that a percentage increase means the new amount is more than 100% of the original. For example, a 15% increase gives you 100% + 15% = 115% of the original.

lightbulbExample

Worked Example: Wage Increase

Let's say Sally earns $175 per day, and her wage increases by 15%. What is her new daily wage?

Method 1:

To find 15% of $175:

$15\% \text{ of } 175 = \frac{15}{100} \times 175 = 26.25$

Now add this increase to the original wage:

$175 + 26.25 = 201.25$

Sally's new daily wage is $201.25.

Method 2:

A 15% increase means the new wage is 100% + 15% = 115% of the original wage.

$115\% \text{ of } 175 = \frac{115}{100} \times 175 = 201.25$

Sally's new daily wage is $201.25.

Calculating a percentage decrease

Percentage decreases work similarly to increases, but you subtract instead of add. Again, there are two methods.

Method 1: Calculate the decrease, then subtract it

This method finds the actual value of the decrease, then takes it away from the original amount.

Steps:

Convert the percentage to a fraction (or decimal)
Multiply by the original amount to find the decrease
Subtract the decrease from the original amount

Method 2: Use a multiplier

This method recognises that a decrease means you end up with less than 100% of the original.

Steps:

Subtract the percentage decrease from 100%
Convert this remaining percentage to a fraction (or decimal)
Multiply by the original amount

chatImportant

When using the multiplier method for decreases, the new amount is less than 100% of the original. For example, a 20% decrease gives you 100% - 20% = 80% of the original.

lightbulbExample

Worked Example: Distance Reduction

A school fun run is normally 2.75 km long, but it's decreased by 20% for younger students. What is the new distance?

Method 1:

To find 20% of 2.75 km:

$20\% \text{ of } 2.75 = \frac{20}{100} \times 2.75 = 0.55$

Now subtract this decrease from the original distance:

$2.75 - 0.55 = 2.2$

The new distance is 2.2 km.

Method 2:

A 20% decrease means the new distance is 100% - 20% = 80% of the original distance.

$80\% \text{ of } 2.75 = \frac{80}{100} \times 2.75 = 2.2$

The new distance is 2.2 km.

Calculating percentage discounts

A discount is simply a percentage decrease applied to prices. When shops have sales, they're reducing prices by a certain percentage. The calculation works exactly the same way as any percentage decrease.

lightbulbExample

Worked Example: Clothing Sale

A shop is having a sale with 15% off all items. A pair of jeans originally costs $95. What is the sale price?

Method 1:

Find 15% of $95:

$15\% \text{ of } 95 = \frac{15}{100} \times 95 = 14.25$

Subtract the discount from the original price:

$95 - 14.25 = 80.75$

The sale price would be $80.75.

Method 2:

A 15% discount means the sale price is 100% - 15% = 85% of the original price.

$85\% \text{ of } 95 = \frac{85}{100} \times 95 = 80.75$

The sale price would be $80.75.

Finding percentage change

Sometimes you know both the original and new values, and you need to work backwards to find the percentage change. This is where we use a formula.

The percentage change formula

The core formula for finding percentage change is essential to master:

chatImportant

The Key Formula:

$\text{Percentage change} = \frac{\text{change}}{\text{original}} \times 100$

This general formula can be applied to specific situations:

$\text{Percentage discount} = \frac{\text{discount}}{\text{original}} \times 100$

$\text{Percentage increase} = \frac{\text{increase}}{\text{original}} \times 100$

The key is to:

Find the actual change (increase or decrease) in numbers
Divide by the original amount
Multiply by 100 to convert to a percentage

lightbulbExample

Worked Example: Percentage Increase

A university had 35,000 students last year. This year it has 38,000 students. By what percentage did the student population increase? (Give your answer to two decimal places.)

Step 1: Find the increase

$\text{Increase} = 38000 - 35000 = 3000$

Step 2: Apply the formula

$\text{Percentage increase} = \frac{\text{increase}}{\text{original}} \times 100$

$= \frac{3000}{35000} \times 100$

$= 8.5714...$

Step 3: Round to two decimal places

The student population increased by 8.57%.

lightbulbExample

Worked Example: Percentage Discount

A calculator normally costs $38 but is on sale for $32. What is the percentage discount? (Give your answer to the nearest whole per cent.)

Step 1: Find the discount amount

$\text{Discount} = 38 - 32 = 6$

Step 2: Apply the formula

$\text{Percentage discount} = \frac{\text{discount}}{\text{original}} \times 100$

$= \frac{6}{38} \times 100$

$= 15.7895...$

Step 3: Round to the nearest whole per cent

The percentage discount is 16%.

bookmarkSummary

Key Points to Remember:

Percentage changes are always calculated from the original value, not the new value.
For increases: You can either calculate the increase and add it, or find the total percentage (100% + increase%) directly.
For decreases and discounts: You can either calculate the decrease and subtract it, or find the remaining percentage (100% - decrease%) directly.
To find percentage change when given both values: Use the formula $\frac{\text{change}}{\text{original}} \times 100$
Always check your answer makes sense - if something increases by 20%, the new value should be bigger than the original!

Percentage Increase and Decrease (VCE SSCE General Mathematics): Revision Notes