Percentage of a Quantity
Find $15\%$ of $\$60$, $8\%$ of $200$ km, or $45\%$ of anything — using the multiplier method or the unitary method.
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$15\%$ of Year 8 students walk to school. If there are 120 Year 8 students, how many walk? Jot down your first reaction — then we'll see who's right.
To find a percentage of a quantity, change the percent to a decimal (or fraction) and multiply. The unitary method finds $1\%$ first, then scales up.
Finding $15\%$ of $\$60$ has two main routes. Multiplier method: $15\% = 0.15$, then $0.15 \times 60 = 9$. Unitary method: $1\%$ of $\$60$ is $\$0.60$, so $15\%$ is $15 \times 0.60 = \$9$. Both give $\$9$. The multiplier method is faster on a calculator; the unitary method is great for mental maths.
Know
- Percentage $\to$ decimal: divide by 100
- $P\%$ of $Q = \tfrac{P}{100} \times Q$
- The unitary method: find $1\%$, then scale
- $10\%$, $5\%$, $1\%$ are mental-maths building blocks
Understand
- Why "of" in maths often means "multiply"
- How $10\%$ and $1\%$ combine to find ANY percentage mentally
- When to use calculator vs mental method
Can Do
- Find any percentage of any quantity with or without a calculator
- Use the unitary method to scale up from $1\%$
- Apply percentages to real money problems
Wrong: "$15\%$ of $\$60$ = $15 \times 60 = 900$" — NO. You forgot to divide by 100. $15\% = 0.15$, not 15.
Right: Convert percentage to decimal FIRST: $15\% = 0.15$. Then $0.15 \times 60 = \$9$.
Wrong: "$25\%$ of $\$80 = 80 \div 25 = 3.2$" — NO. You DIVIDED instead of multiplying.
Right: Use the multiplier: $25\% = 0.25$. $0.25 \times 80 = \$20$.
The fastest way on a calculator. Convert the percentage to a decimal, then multiply.
To find $P\%$ of $Q$, use $\tfrac{P}{100} \times Q$. So $32\%$ of $\$150$ becomes $0.32 \times 150 = \$48$. The decimal form is much easier to feed into a calculator than the fraction form.
Find $1\%$ first by dividing by 100. Then multiply by the percentage you actually want.
Take $23\%$ of $\$400$. Step 1: find $1\%$ — divide $400$ by $100$ to get $\$4$. Step 2: scale up — multiply $\$4$ by $23$ to get $\$92$. The same logic works for ANY percentage. It is brilliant for mental maths because $1\%$ is easy to find.
Watch Me Solve It · 3 examples
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1Find $10\%$ first$10\%$ of $\$120 = \$12$Move the decimal one place left.
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2Find $5\%$$5\% = \tfrac{1}{2}$ of $10\% = \tfrac{1}{2} \times 12 = \$6$Halve the $10\%$ value.
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3Add them$15\% = 10\% + 5\% = 12 + 6 = \$18$$15\%$ of $\$120 = \$18$.
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1Convert $\%$ to decimal$24\% = 0.24$Divide by 100.
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2Multiply$0.24 \times 250$Use the calculator.
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3Compute$0.24 \times 250 = 60$ kgEstimate: $\tfrac{1}{4}$ of $250 = 62.5$. Sensible.
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1$10\%$ of 120$120 \div 10 = 12$$12$ students.
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2$5\%$ of 120$\tfrac{12}{2} = 6$Halve the $10\%$.
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3Add for $15\%$$12 + 6 = 18$$18$ students walk.
Common Pitfalls
Multiplier Method
- $P\% \to$ decimal: $\div 100$
- Multiply: $\tfrac{P}{100} \times Q$
- $25\%$ of $80 = 0.25 \times 80 = 20$
Unitary Method
- $1\%$ of $Q = Q \div 100$
- Scale up by $P$
- $23\%$ of $400$: $4 \times 23 = 92$
Mental Building Blocks
- $10\%$: divide by 10
- $5\%$: half of $10\%$
- $1\%$: divide by 100
Quick Estimates
- $25\% \approx \tfrac{1}{4}$
- $33\% \approx \tfrac{1}{3}$
- $50\% = \tfrac{1}{2}$
How are you completing this lesson?
Brain Trainer · 4 problems
Four drill problems to sharpen your skills. Work each, then reveal the answer.
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1 Find $20\%$ of $\$45$.
$0.20 \times 45 = 9$.$\$9$ -
2 Find $35\%$ of $200$ kg.
$0.35 \times 200 = 70$.$70$ kg -
3 Find $7\%$ of $\$300$ using $1\%$.
$1\% = \$3$, so $7\% = 7 \times 3 = \$21$.$\$21$ -
4 Find $45\%$ of $\$80$ using mental building blocks.
$50\% = \$40$; $5\% = \$4$; $45\% = 40 - 4 = \$36$.$\$36$
Quick Check · 5 questions
Show Your Working · 3 questions
Q6. Calculate each, showing your method: (a) $40\%$ of $\$250$ (b) $7.5\%$ of $\$80$ (c) $12\%$ of $\$45$
Q7. A school has 850 students. $32\%$ catch the bus. How many catch the bus?
Q8. Lucia uses mental maths to find $17\%$ of $\$240$. (a) Show how she could use building blocks ($10\%$, $5\%$, $1\%$) to do it in her head. (b) Verify with the multiplier method. (c) Which method was faster for this problem, and why?
Quick Check
1. B — $0.30 \times 150 = \$45$.
2. C — $\tfrac{1}{4} \times 80 = 20$ km.
3. C — $0.08 \times 200 = 16$ mL.
4. C — $600 \div 100 = \$6$.
5. B — $0.15 \times 80 = 12$ kg.
Show Your Working Model Answers
Q6 (3 marks): (a) $0.40 \times 250 = \$100$ [1]. (b) $0.075 \times 80 = \$6$ [1]. (c) $0.12 \times 45 = \$5.40$ [1].
Q7 (2 marks): $0.32 \times 850 = 272$ students [1, working]. Answer: $272$ catch the bus [1].
Q8 (4 marks): (a) $10\%$ of $240 = \$24$; $5\% = \$12$; $1\% = \$2.40$; so $17\% = 10 + 5 + 1 + 1 = \$24 + \$12 + \$2.40 + \$2.40 = \$40.80$ [2]. (b) Multiplier: $0.17 \times 240 = \$40.80$ ✓ [1]. (c) Multiplier was faster — one calculation instead of four [1].
Sales Tax Surprise
In one country, two taxes apply to a purchase: a $10\%$ goods tax, then a further $5\%$ luxury tax on the new total. A handbag's sticker price is $\$400$ before any tax. (a) What is the final price after both taxes? (b) What single percentage of $\$400$ would give the same final price?
Reveal solution
(a) After $10\%$ tax: $400 \times 1.10 = \$440$. After $5\%$ luxury tax: $440 \times 1.05 = \$462$. (b) Total increase: $\$62$ on $\$400$ = $15.5\%$. (Note: NOT a simple $15\%$ — the taxes compound!)
Multiplier
$P\%$ of $Q = \tfrac{P}{100} \times Q$
Unitary
Find $1\%$, then scale
$10\%$ trick
Divide by 10
$5\%$ trick
Half of $10\%$
Estimate
$\tfrac{1}{4}, \tfrac{1}{3}, \tfrac{1}{2}$ for sanity
"Of" = $\times$
Always multiply, never divide
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