Practice Maths

Finding Percentages of Quantities

Key Ideas

Key Terms

percentage of a quantity
the result of finding what a given percentage of a number or measurement is; e.g. 30% of 80 = 24.
multiplier
the decimal form of a percentage used to calculate a percentage of a quantity; e.g. 35% → multiplier 0.35.
mental strategy
a quick calculation approach; e.g. build any percentage from 10% (divide by 10), 5% (half of 10%), and 1% (divide by 100).
unitary method
finding 1% first (divide by 100), then multiplying to reach any required percentage.

Common percentage shortcuts

50% = ½ of the amount.   25% = ¼.   10% = 1⁄10.   20% = 1⁄5.   75% = ¾.

Hot Tip Build up percentages from 10%: “15% of 60” = 10% + 5% = 6 + 3 = 9. This is fast and avoids calculator errors!

Worked Example

Question: Find 35% of 240.

Method 1 — Decimal multiplier.
35% of 240 = 0.35 × 240 = 84

Method 2 — Mental strategy.
10% of 240 = 24
30% of 240 = 3 × 24 = 72
5% of 240 = 12
35% = 72 + 12 = 84 ✓

The Core Method: Decimal × Quantity

To find a percentage of a quantity, convert the percentage to a decimal and then multiply. This works every single time, for any percentage and any quantity.

Example: Find 15% of $80.
Step 1: Convert 15% to a decimal → 15 ÷ 100 = 0.15
Step 2: Multiply → 0.15 × $80 = $12

Example: Find 8.5% of 200 kg.
Step 1: 8.5 ÷ 100 = 0.085
Step 2: 0.085 × 200 = 17 kg

The decimal method works on a calculator or by hand. Practise it until it feels automatic.

Mental Strategies Using 10%

For quick mental maths, use 10% as your building block because dividing by 10 is easy.

• 10% of any amount = divide by 10. e.g. 10% of $250 = $25
• 5% = half of 10%. e.g. 5% of $250 = $12.50
• 20% = double 10%. e.g. 20% of $250 = $50
• 1% = divide by 100. e.g. 1% of $250 = $2.50
• 15% = 10% + 5%. e.g. 15% of $250 = $25 + $12.50 = $37.50

These strategies are especially useful when you want to tip at a restaurant or estimate a discount in your head while shopping.

Common Fraction Shortcuts

Some percentages are exact fractions that are easier to work with directly:

50% = 12,   25% = 14,   75% = 34,   33.3...% ≈ 13,   66.6...% ≈ 23

Example: Find 75% of 48.
75% = 34, so find 14 first: 48 ÷ 4 = 12, then multiply by 3: 12 × 3 = 36.
Check: 0.75 × 48 = 36. ✓

Finding the Original Amount (Reverse Percentage)

Sometimes you know the percentage amount and need to find what 100% was. For example: "A shirt costs $34 after a 15% discount. What was the original price?"

Wait — this is a reverse percentage problem, which is different from just finding 15% of something. Here, the $34 is 85% of the original (since 100% − 15% = 85%).

Method: Original = amount ÷ percentage as decimal
Original = $34 ÷ 0.85 = $40

Check: 15% of $40 = $6, and $40 − $6 = $34. ✓

Real-Life Applications

Finding percentages of quantities comes up constantly: calculating GST on a purchase (10% of the price), working out how much protein is in a food (8.5% of 200 g = 17 g), finding commission a salesperson earns (3% of $15 000 = $450), or calculating how much of a school fundraising goal has been reached.

Example: A gym membership costs $600 per year. Members get a 12.5% discount for paying upfront. Discount = 12.5% of $600 = 0.125 × $600 = $75. Upfront price = $600 − $75 = $525.

Key tip: Always convert the percentage to a decimal before multiplying — never multiply by the percentage number itself. 15% of 80 is NOT 15 × 80 = 1200. It is 0.15 × 80 = 12. That factor-of-100 error is the most common mistake in this topic.

Mastery Practice

  1. Find each percentage of the given quantity. Fluency

    1. 50% of 80
    2. 25% of 120
    3. 10% of 350
    4. 75% of 200
    5. 20% of 60
    6. 5% of 180
    7. 30% of 90
    8. 40% of 150

  2. Calculate using the decimal multiplier method. Fluency

    1. 35% of 140
    2. 65% of 80
    3. 12% of 250
    4. 8% of 75
    5. 45% of 360
    6. 7.5% of 200
    7. 3% of 400
    8. 110% of 50

  3. Find the original amount given the percentage and result (unitary method). Understanding

    1. 10% of a number is 7. What is the number?
    2. 25% of a number is 18. What is the number?
    3. 5% of a number is 12. What is the number?
    4. 40% of a number is 56. What is the number?
    5. 20% of a number is 35. What is the number?
    6. 15% of a number is 24. What is the number?

  4. Solve these percentage problems. Understanding

    1. A class of 32 students has 75% present. How many students are present?
    2. A jacket costs $180. It is on sale for 30% off. How much is the discount?
    3. Tom earns $620 per week. He saves 15% each week. How much does he save?
    4. A school has 840 students. 5% are absent today. How many students are absent?

  5. Apply percentage calculations to real-world situations. Problem Solving

    1. A phone is priced at $960. GST of 10% is added. What is the total price including GST?
    2. Maria got 78% on a test worth 50 marks. How many marks did she get?
    3. A recipe uses 40% of a 250 mL bottle of cream. Later, 25% of the remaining cream is used. How much cream remains?
    4. Which is more: 35% of 160 or 40% of 140? Show your calculations.

  6. Calculate these special percentages. Fluency

    1. 125% of 80
    2. 200% of 45
    3. 0.5% of 600
    4. 150% of 120
    5. 0.25% of 800
    6. 250% of 36
    7. 175% of 200
    8. 0.1% of 5000

  7. Use mental strategies (build from 10%) to find each percentage. Show your working. Fluency

    1. 15% of 80
    2. 35% of 200
    3. 45% of 40
    4. 5% of 360
    5. 22% of 50
    6. 12% of 150

  8. Use the unitary method to find the unknown. Understanding

    1. 30% of a number is 45. What is the number?
    2. 8% of an amount is $36. What is the full amount?
    3. 60% of a quantity is 150 kg. What is the full quantity?
    4. 75% of the class passed a test. If 24 students passed, how many students are in the class?
    5. A tax of 7% on an item came to $14. What was the pre-tax price?
    6. 12.5% of a journey has been completed; the distance covered is 25 km. How long is the full journey?

  9. Compare and reason. Understanding

    1. Is 20% of 250 greater or less than 25% of 200? Show calculations.
    2. School A has 400 students and 60% own a bicycle. School B has 550 students and 45% own a bicycle. Which school has more bicycle owners?
    3. A shopkeeper marks up item A (cost $40) by 30% and item B (cost $65) by 20%. Which item has the larger dollar mark-up?
    4. Two investments earn interest: Investment X earns 5% on $1600, and Investment Y earns 8% on $1000. Which earns more interest in one year?

  10. Multi-step percentage problems. Problem Solving

    1. A factory produces 1200 units per day. Quality control rejects 4% of units. Of the remaining units, 15% are sent for export. How many units are exported per day?
    2. Leo earns $850 per week. He pays 22% in tax, saves 10% of his after-tax income, and spends the rest. How much does he spend per week?
    3. A school survey found that 45% of 200 students walk to school, 30% catch the bus, 20% are driven by a parent, and the rest cycle. How many students cycle? What percentage is this?
    4. A drink is 12% fruit juice and 88% water. If a bottle contains 375 mL, how many mL of fruit juice does it contain? If you want at least 60 mL of juice, what is the minimum bottle size you should buy?
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