Percentages

Key Concepts

• A percentage means “out of 100”.

  • The symbol for percentage is %.
  • For example:
    • 25% means 25 out of 100
    • 50% means 50 out of 100
    • 100% means the whole amount

• Percentages can be written in different forms:

  • Percentage: 40%
  • Fraction: 40/100 = 2/5
  • Decimal: 0.4

• To change a percentage to a decimal:

  • divide by 100
  • Example: 35% = 35/100 = 0.35

• To find a percentage of a quantity:

  • Change the percentage to a fraction or decimal
  • Then multiply by the quantity
  • Formula:
    • Percentage of a quantity = Percentage × Total quantity

• Discount means the amount taken off the original price.

  • It is a reduction in price.
  • Formula:
    • Discount = Discount rate × Original price
    • Selling price after discount = Original price − Discount

• GST stands for Goods and Services Tax.

  • It is a tax added to the price of goods and services.
  • Formula:
    • GST amount = GST rate × Price before GST
    • Price including GST = Price before GST + GST amount

• A percentage increase happens when an amount becomes larger.

  • Formula:
    • Increase = Percentage increase × Original amount
    • New amount = Original amount + Increase

• A percentage decrease happens when an amount becomes smaller.

  • Formula:
    • Decrease = Percentage decrease × Original amount
    • New amount = Original amount − Decrease

• When finding the original amount, work backwards.

  • If a value is increased or decreased by a percentage, the final amount is only a part of the original.
  • Example:
    • After a 20% decrease, 80% remains
    • After a 15% increase, 115% of the original remains as the new amount

• Profit and loss compare the selling price and the cost price.

  • Cost price = amount paid to buy or make an item
  • Selling price = amount the item is sold for

• If selling price is more than cost price:

  • there is a profit
  • Formula:
    • Profit = Selling price − Cost price

• If selling price is less than cost price:

  • there is a loss
  • Formula:
    • Loss = Cost price − Selling price

• To find profit percentage:

  • Profit % = (Profit ÷ Cost price) × 100%

• To find loss percentage:

  • Loss % = (Loss ÷ Cost price) × 100%

• Important idea:

  • Profit and loss percentages are usually based on cost price, not selling price.

Expressing One Quantity as a Percentage of Another

• Formula:

  $$\text{Percentage} = \frac{\text{Part}}{\text{Whole}} \times 100\%$$

• Both quantities must be in the same unit before dividing.

• Worked example 1: “Express 45 cm as a percentage of 3 m.”

  • Convert: 3 m = 300 cm
  • $\frac{45}{300} \times 100\% = 15\%$

• Worked example 2: “In a class of 40 students, 24 are girls. What percentage are girls?”

  • $\frac{24}{40} \times 100\% = 60\%$

• Common mistake: forgetting to convert to the same unit first.

Percentage Change

• Percentage change measures how much something has increased or decreased as a percentage of the original value.

  • Formula:
    • Percentage change = (change ÷ original) × 100%
  • The change is the difference between the new value and the original value.

• Percentage increase: the new value is greater than the original.

  • Change = New value − Original value
  • Percentage increase = (increase ÷ original) × 100%

• Percentage decrease: the new value is less than the original.

  • Change = Original value − New value
  • Percentage decrease = (decrease ÷ original) × 100%

• Important note: Always use the ORIGINAL value as the denominator, not the new value.

Important Definitions

• Percentage: a number or ratio expressed as a fraction of 100.

• Quantity: the amount or number of something.

• Original amount: the starting amount before any increase, decrease, discount, or GST is applied.

• Discount: the amount taken away from the original price.

• Discount rate: the percentage of the original price that is reduced.

• GST (Goods and Services Tax): a tax added to the price of goods and services.

• Increase: the amount by which something becomes more.

• Decrease: the amount by which something becomes less.

• Percentage increase: the increase written as a percentage of the original amount.

• Percentage decrease: the decrease written as a percentage of the original amount.

• Cost price: the amount paid to buy or produce an item.

• Selling price: the amount for which an item is sold.

• Profit: the amount gained when the selling price is more than the cost price.

• Loss: the amount lost when the selling price is less than the cost price.

• Profit percentage: the profit expressed as a percentage of the cost price.

• Loss percentage: the loss expressed as a percentage of the cost price.

Worked Examples

Example 1: Percentage of a Quantity

A class has 40 pupils. 35% of them are boys. How many boys are there?

Step 1: Change the percentage to a decimal or fraction

• 35% = 35/100 = 0.35

Step 2: Multiply by the total number of pupils

• Number of boys = 35% of 40
• = 0.35 × 40
• = 14

Answer: There are 14 boys.

Example 2: Discount and GST

A bag costs $80. It is sold at a 10% discount. Then 9% GST is added to the discounted price. Find the final price paid.

Step 1: Find the discount

• Discount = 10% of $80
• = 0.10 × 80
• = $8

Step 2: Find the price after discount

• Price after discount = $80 − $8
• = $72

Step 3: Find the GST

• GST = 9% of $72
• = 0.09 × 72
• = $6.48

Step 4: Find the final price paid

• Final price = $72 + $6.48
• = $78.48

Answer: The final price paid is $78.48.

Example 3: Finding the Original Amount

After a 20% discount, a shirt costs $48. What was the original price?

Step 1: Understand what remains

• If there is a 20% discount, then:
  • 100% − 20% = 80%
• So $48 is 80% of the original price.

Step 2: Find 1%

• 80% = $48
• 1% = $48 ÷ 80
• = $0.60

Step 3: Find 100%

• 100% = $0.60 × 100
• = $60

Answer: The original price was $60.

Example 4: Profit

A shopkeeper bought a toy for $24 and sold it for $30. Find:

1. the profit
2. the profit percentage

Step 1: Find the profit

• Profit = Selling price − Cost price
• = $30 − $24
• = $6

Step 2: Find the profit percentage

• Profit % = (Profit ÷ Cost price) × 100%
• = (6 ÷ 24) × 100%
• = 1/4 × 100%
• = 25%

Answer:

1. Profit = $6
2. Profit percentage = 25%

Example 5: Percentage Increase

A price rose from $40 to $50. Find the percentage increase.

Step 1: Find the increase

• Increase = $50 − $40 = $10

Step 2: Apply the formula

• Percentage increase = (increase ÷ original) × 100%
• = (10 ÷ 40) × 100%
• = 0.25 × 100%
• = 25%

Answer: The percentage increase is 25%.

Example 6: Percentage Decrease

A shirt was $60. It now costs $45. Find the percentage decrease.

Step 1: Find the decrease

• Decrease = $60 − $45 = $15

Step 2: Apply the formula

• Percentage decrease = (decrease ÷ original) × 100%
• = (15 ÷ 60) × 100%
• = 0.25 × 100%
• = 25%

Answer: The percentage decrease is 25%.

Remember: Always divide by the original value, not the new value.

Common Mistakes to Avoid

• Mixing up percentage and amount.

  • Example: 20% is not $20 unless the question says so.

• Forgetting that % means out of 100.

• Using the wrong base when calculating percentages.

  • For profit and loss, percentage is usually based on cost price.
  • For discount, percentage is based on original price.

• Adding and subtracting percentages wrongly.

  • A 20% decrease means 80% remains, not 20% remains.

• Finding GST from the wrong amount.

  • GST is usually added to the price before GST, not after.

• Applying discount and GST in the wrong order.

  • Read the question carefully.

• Forgetting to convert the percentage to a decimal or fraction before multiplying.

• Writing money answers without correct units.

  • Always write $ when needed.
  • Money is usually written to 2 decimal places.

• Confusing original amount with final amount.

  • If the final amount is given after an increase or decrease, work backwards.

• Using selling price instead of cost price for profit or loss percentage.

Exam Tips

• Circle important keywords in the question:

  • discount
  • GST
  • increase
  • decrease
  • original price
  • cost price
  • selling price
  • profit
  • loss

• Ask yourself:

  • “What is the 100% in this question?”
  • This helps you choose the correct base.

• For percentage of a quantity:

  • write clearly:
    35% of 40 = 0.35 × 40 = 14

• For original amount questions:

  • state what percentage remains first.
  • Example:
    • “After a 15% discount, 85% remains.”

• For profit and loss:

  • always show:
    1. cost price
    2. selling price
    3. difference
    4. percentage based on cost price

• For questions involving money:

  • check if the final answer should be in dollars and cents.

• Use mark-earning phrases such as:

  • “100% represents the original amount.”
  • “After the discount, 90% of the price remains.”
  • “Profit is calculated by subtracting cost price from selling price.”
  • “Loss percentage is based on the cost price.”

• Check whether your answer makes sense:

  • After a discount, the price should be less
  • After GST, the price should be more
  • After a percentage increase, the new amount should be greater

Quick Summary

• Percentage means out of 100.
• To find a percentage of a quantity, multiply the quantity by the percentage.
• Convert percentages to fractions or decimals before calculating.
• Discount is an amount taken off the original price.
• GST is a tax added to the price.
• Percentage increase means the amount becomes more.
• Percentage decrease means the amount becomes less.
• To find the original amount, work out what percentage remains first.
• After a discount or decrease, subtract from 100% to find the remaining percentage.
• Profit = Selling price − Cost price.
• Loss = Cost price − Selling price.
• Profit and loss percentages are usually based on the cost price.