Why This Matters for Grade 10#
Ratio and proportion appear across multiple Grade 10 topics:
- Trigonometry: $\sin\theta = \frac{\text{opposite}}{\text{hypotenuse}}$ — a ratio of side lengths
- Probability: $P(A) = \frac{n(A)}{n(S)}$ — the ratio of favourable to total outcomes
- Geometry: Similar triangles have sides in the same ratio
- Finance: Interest rate $i = \frac{r}{100}$ is a ratio
1. What is a Ratio?#
A ratio compares two or more quantities of the same kind.
$3 : 5$ means “for every 3 of the first, there are 5 of the second.”
Simplifying Ratios#
Divide all parts by their HCF (highest common factor):
$12 : 18 = 2 : 3$ (divide both by 6)
$20 : 30 : 50 = 2 : 3 : 5$ (divide all by 10)
Key: A ratio has no units. $200\text{ml} : 500\text{ml} = 2 : 5$
2. Sharing in a Given Ratio#
To share an amount in the ratio $a : b$:
- Find the total number of parts: $a + b$
- Find the value of one part: $\frac{\text{Total}}{a + b}$
- Multiply each share
Worked Example#
Share $R600$ in the ratio $2 : 3$.
Total parts: $2 + 3 = 5$
One part: $\frac{600}{5} = R120$
First share: $2 \times 120 = R240$
Second share: $3 \times 120 = R360$
Check: $240 + 360 = 600$ ✓
3. Direct Proportion#
Two quantities are directly proportional if when one doubles, the other doubles too.
$$ \frac{y}{x} = k \quad \text{(constant)} \qquad \text{or} \qquad y = kx $$Example#
If 5 books cost $R150$, how much do 8 books cost?
$k = \frac{150}{5} = 30$ (cost per book)
$\text{Cost of 8} = 30 \times 8 = R240$
The graph of a direct proportion is a straight line through the origin — exactly like the linear function $y = mx$ in Grade 10.
4. Inverse Proportion#
Two quantities are inversely proportional if when one doubles, the other halves.
$$ xy = k \quad \text{(constant)} \qquad \text{or} \qquad y = \frac{k}{x} $$Example#
6 workers take 10 days to build a wall. How long would 15 workers take?
$k = 6 \times 10 = 60$ (total worker-days)
$\text{Days} = \frac{60}{15} = 4$ days
The graph of an inverse proportion is a hyperbola — exactly like $y = \frac{a}{x}$ in Grade 10 Functions!
5. Rate#
A rate compares quantities of different kinds (unlike ratio which compares same kinds).
| Rate | Example |
|---|---|
| Speed | $60 \text{ km/h}$ = 60 km per hour |
| Price | $R15/\text{kg}$ = R15 per kilogram |
| Interest | $8\%$ per year = $0.08$ per year |
Using Rate#
If a car travels at $80$ km/h for $3$ hours:
$\text{Distance} = \text{speed} \times \text{time} = 80 \times 3 = 240$ km
Grade 10 Finance connection: The interest rate $i$ is a rate — “how much interest per rand per year”.
🚨 Common Mistakes#
- Ratio order matters: $3 : 5$ is NOT the same as $5 : 3$.
- Mixing units in ratios: $2\text{ m} : 50\text{ cm}$ — convert first! $200\text{ cm} : 50\text{ cm} = 4 : 1$.
- Direct vs inverse confusion: More workers = less time (inverse). More items = more cost (direct). Ask: “Does increasing one increase or decrease the other?”
- Forgetting to check: After sharing in a ratio, add the shares — they must equal the original total.