Year 9AC9M9M05

Direct Proportion, Rates and Scale

Use mathematical modelling to solve practical problems involving direct proportion, rates, ratio and scale, including financial contexts.

If doubling one doubles the other, they are directly proportional. That single test decides the method.

Method

Direct proportion

y = kx

Their ratio stays constant. The graph is a straight line through the origin.

Find $k$ from one known pair

k = \frac{y}{x}

Then use it for every other pair.

A rate is a proportion with units

\text{km/h},\ \$/\text{kg}

Scale factors: length $k$, area $k^2$, volume $k^3$

The commonest error is using $k$ for all three.

Not everything is proportional

If the graph does not pass through the origin, it is not.

e.g.

\text{5 kg of apples cost } \$17{,}50. \text{ Find the cost of } 12 \text{ kg.}

Check it is proportional. No apples means no cost, so yes. $\text{passes through the origin}$
Find the constant from the known pair. $k = \frac{17{,}50}{5} = 3{,}50$
Write the rule. $C = 3{,}50m$
Substitute the new value. $C = 3{,}50 \times 12$
Work it out. $C = \$42{,}00$
Check the ratio held. $\frac{42}{12} = 3{,}50 \quad\checkmark$

Test for proportion first, then find $k$ .

\text{3 m of rope costs } \$7{,}50. \text{ Cost of 8 m?}

Answer

\$20{,}00

\text{A car uses } 24 \text{ L for } 300 \text{ km. How far on } 40 \text{ L?}

Answer

500 \text{ km}

\text{A map scale is } 1:25\,000. \text{ How far is } 6 \text{ cm?}

Answer

1{,}5 \text{ km}

\text{A photo is enlarged } \times 4. \text{ What happens to its area?}

Answer

\times 16

\text{Is a taxi fare with a flag fall directly proportional?}

Answer

\text{No. It does not start at zero.}

\text{Two similar solids have length ratio } 2:3. \text{ Volume ratio?}

Answer

8:27