Lesson 4 ~25 min Unit 1 · Financial Maths +85 XP

Quantity as a Percentage

Turn "$34$ out of $40$" into a percentage. Compare test scores, profit margins, and survey results.

Today's hook: You scored 34 out of 40. Your friend scored 43 out of 50. Who had the better percentage score?

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Think First

warm-up

You scored 34 out of 40. Your friend scored 43 out of 50. Who had the better percentage score? Jot down your first reaction — then we'll see who's right.

Record your answer in your workbook.

The Big Idea

+5 XP

To express one number as a percentage of another, write them as a fraction (part over whole), then multiply by 100. The units MUST match.

To find $A$ as a percentage of $B$: write $\tfrac{A}{B}$, then multiply by $100$. So $34$ out of $40$ = $\tfrac{34}{40} \times 100 = 85\%$. Same units, same direction: if $A$ is in cm, $B$ must be in cm too. If $A$ is the part, $B$ is the whole. Don't mix them.

$A \text{ as }\%\text{ of } B = \dfrac{A}{B} \times 100\%$

Part over whole

The part is always on TOP; the whole goes on the BOTTOM.

Then $\times 100$

Turn the fraction into a percentage by multiplying by 100.

Units must match

Convert to the same unit BEFORE making the fraction. cm with cm, kg with kg.

What You'll Master

objectives

Know

Percentage of a quantity: $\tfrac{\text{part}}{\text{whole}} \times 100\%$
Both quantities must be in the same units
Common contexts: test scores, profit margins, survey results

Understand

Why the same units rule must apply
How to recognise which value is the part and which is the whole
Why $34/40$ and $43/50$ can be directly compared as $\%$

Can Do

Calculate one quantity as a percentage of another
Compare two ratios by converting to percentages
Convert mixed units (e.g., cm to m) before computing

Words You Need

vocabulary

PartThe smaller portion being measured against the whole.

WholeThe total amount — the reference.

ComparisonUsing percentages to compare apples with apples.

Same unitsBoth part and whole must be in the same unit before dividing.

Percentage scoreA test result expressed out of 100 for easy comparison.

MarginThe proportion something contributes — like profit margin.

Spot the Trap

heads-up

Wrong: "I scored $17$ out of $20$. That's $\tfrac{20}{17} \times 100$." — Wrong way up.

Right: Part on top: $\tfrac{17}{20} \times 100 = 85\%$.

Wrong: "$50$ cm out of $2$ m = $\tfrac{50}{2} \times 100 = 2500\%$" — units don't match!

Right: Convert: $2$ m = $200$ cm. Now $\tfrac{50}{200} \times 100 = 25\%$.

Same Units First

+5 XP

You can't compare seconds with hours or grams with kilograms without converting them first. Same units are non-negotiable.

Suppose a runner finishes $300$ m of a $2$ km race. To express this as a percentage of the race, both must be in the SAME unit. Convert $2$ km to $2000$ m. Then $\tfrac{300}{2000} \times 100 = 15\%$. If you forget to convert, you'll get $\tfrac{300}{2} \times 100 = 15000\%$ — clearly nonsense.

$300$ m of $2$ km $= \dfrac{300}{2000} \times 100 = 15\%$

Pick the bigger unit

Convert smaller to bigger (or vice versa). Just be consistent.

Always check size

If your answer is over $1000\%$, units probably don't match.

Money is already same unit

$\$$ with $\$$ — easy.

Comparing Scores Fairly

+5 XP

Two scores on differently sized tests can't be compared directly. Convert both to percentages.

Tahlia scored $34/40$ on her science test. Jay scored $43/50$ on a longer one. Who did better? Compute: $\tfrac{34}{40} \times 100 = 85\%$ and $\tfrac{43}{50} \times 100 = 86\%$. Jay edged out — by just $1$ percentage point.

Convert each $\dfrac{\text{score}}{\text{total}} \times 100$

Higher percentage = higher score

As long as both totals are positive.

Tiny differences matter

$85\%$ vs $86\%$ — Jay wins.

Try as fractions first

$\tfrac{34}{40} = \tfrac{17}{20}, \tfrac{43}{50}$. Cross multiply to compare.

Watch Me Solve It · 3 examples

Watch Me Solve It · The race

+15 XP per step

PROBLEM

Tahlia scored $34$ out of $40$. Jay scored $43$ out of $50$. Who had the better percentage?

1

Convert Tahlia's score

$\tfrac{34}{40} \times 100 = 85\%$

Divide then multiply.
2

Convert Jay's score

$\tfrac{43}{50} \times 100 = 86\%$

Same method.
3

Compare

$86\% > 85\%$

Jay won by 1 percentage point.

AnswerJay (86% vs 85%)

Watch Me Solve It · Unit conversion

+15 XP per step

PROBLEM

A water bottle holds $750$ mL. Tahlia drinks $300$ mL. What percentage has she drunk?

1

Same units?

Both in mL — yes!

No conversion needed.
2

Part over whole

$\tfrac{300}{750}$

Tahlia drank 300 mL of the 750 mL.
3

Multiply by 100

$\tfrac{300}{750} \times 100 = 40\%$

Simplify: $\tfrac{300}{750} = \tfrac{2}{5} = 0.4$.

Answer$40\%$

Watch Me Solve It · Profit margin

+15 XP per step

PROBLEM

A market stall sells a candle for $\$15$. It cost the seller $\$12$ to buy. What is the profit as a percentage of cost?

1

Find the profit

$\$15 - \$12 = \$3$

Selling minus cost.
2

Fraction: profit over cost

$\tfrac{3}{12}$

Profit margin uses COST as the whole.
3

$\times 100$

$\tfrac{3}{12} \times 100 = 25\%$

$\tfrac{3}{12} = \tfrac{1}{4} = 0.25$.

Answer$25\%$ profit on cost

Common Pitfalls

heads-up

Wrong-way-up fraction

Writing $\tfrac{40}{34}$ instead of $\tfrac{34}{40}$.

Fix: Part on top, whole on bottom. Always.

Mismatched units

$30$ minutes out of $2$ hours: $\tfrac{30}{2}$ gives nonsense.

Fix: Convert. $2$ hours = $120$ min. $\tfrac{30}{120} = 25\%$.

Forgetting $\times 100$

Stopping at $\tfrac{34}{40} = 0.85$ when the question wants a percentage.

Fix: Always finish with $\times 100$. $0.85 \times 100 = 85\%$.

Copy Into Your Books

Formula

$\tfrac{\text{part}}{\text{whole}} \times 100\%$
Part on top
Always $\times 100$ at end

Unit Conversions

cm $\to$ m: $\div 100$
min $\to$ h: $\div 60$
g $\to$ kg: $\div 1000$

Test Scores

$\tfrac{\text{score}}{\text{total}} \times 100$
$34/40 = 85\%$
$43/50 = 86\%$

Profit Margin

Profit $=$ SP $-$ CP
% profit $= \tfrac{\text{profit}}{\text{cost}} \times 100$
Compares fairly

How are you completing this lesson?

Brain Trainer · 4 problems

Brain Trainer · Quantity as a Percentage

4 problems

Four drill problems to sharpen your skills. Work each, then reveal the answer.

1 Express $14$ as a percentage of $20$.

$\tfrac{14}{20} \times 100 = 70\%$.$70\%$
2 Express $\$60$ as a percentage of $\$240$.

$\tfrac{60}{240} \times 100 = 25\%$.$25\%$
3 Express $25$ cm as a percentage of $1$ m.

$1$ m = $100$ cm; $\tfrac{25}{100} \times 100 = 25\%$.$25\%$
4 Tahlia ran $4$ km of a $5$ km event. What % did she finish?

$\tfrac{4}{5} \times 100 = 80\%$.$80\%$

Complete in your workbook.

Quick Check · 5 questions

$15$ out of $60$ as a percentage is:

+10 XP

A test has 40 questions. Lucia got 36 right. Her percentage is:

+10 XP

$500$ mL is what percentage of $2.5$ L?

+10 XP

A stall buys mangoes for $\$2$ each and sells them for $\$3$. Profit as % of cost is:

+10 XP

A 90-minute movie has $18$ minutes of ads. What % is ads?

+10 XP

Show Your Working · 3 questions

Show Your Working

9 marks total

Apply Medium 3 MARKS

Q6. Express each as a percentage (round to 1 dp if needed): (a) $27$ out of $45$ (b) $\$80$ out of $\$320$ (c) $450$ g out of $1.5$ kg

Answer in your workbook.

Understand Easy 2 MARKS

Q7. Mia's phone battery had $1200$ mAh. After a day she has $360$ mAh left. What percentage has she used?

Answer in your workbook.

Reason Hard 4 MARKS

Q8. Two students compare maths marks. Asha scored $47/60$. Eli scored $74/95$. (a) Convert each to a percentage to 1 dp. (b) Who scored higher? (c) Asha argues "I got more questions right" — explain why this doesn't make her score better.

Answer in your workbook.

Comprehensive Answers

Quick Check

1. C — $\tfrac{15}{60} \times 100 = 25\%$.

2. D — $\tfrac{36}{40} = 90\%$.

3. C — $\tfrac{500}{2500} = 20\%$.

4. B — $\tfrac{1}{2} = 50\%$.

5. B — $\tfrac{18}{90} = 20\%$.

Show Your Working Model Answers

Q6 (3 marks): (a) $\tfrac{27}{45} \times 100 = 60\%$ [1]. (b) $\tfrac{80}{320} \times 100 = 25\%$ [1]. (c) $1.5$ kg $= 1500$ g; $\tfrac{450}{1500} \times 100 = 30\%$ [1].

Q7 (2 marks): Used = $1200 - 360 = 840$ mAh [1]. $\tfrac{840}{1200} \times 100 = 70\%$ used [1].

Q8 (4 marks): (a) Asha: $\tfrac{47}{60} \times 100 = 78.3\%$ [1]. Eli: $\tfrac{74}{95} \times 100 = 77.9\%$ [1]. (b) Asha scored higher [1]. (c) Eli answered MORE questions correctly in absolute terms (74 vs 47), but Asha's proportion was slightly higher — percentage controls for test length so we can compare fairly [1].

Stretch Challenge · +25 XP, +10 coins

The Sneaky Mark Boost

A teacher scales test marks by adding 5 marks to every student. If Asha originally scored $\tfrac{47}{60}$, her new mark is $\tfrac{52}{65}$ (assuming the test total also increases). (a) What is her old percentage? (b) What is her new percentage? (c) Did the scaling actually change her percentage, and why or why not?

Reveal solution

(a) Old: $\tfrac{47}{60} \times 100 = 78.3\%$. (b) New: $\tfrac{52}{65} \times 100 = 80.0\%$. (c) Yes — her percentage went UP by $1.7$ points. Adding to both numerator and denominator moves any fraction TOWARDS 100%. So this kind of scaling helps students with lower marks more than higher marks.

Quick Review

Part/whole

Part on top, whole on bottom

$\times 100$

Then finish by multiplying by 100

Units must match

Convert first if needed

Compare scores

Both as %, then compare

Profit margin

Profit ÷ cost × 100

Sanity check

Above $100\%$ is unusual — verify

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