Use ratio to share and compare quantities, work with rates, and solve problems involving direct and inverse proportion

What this dot point is asking

SEAB wants you to use ratio to compare and share quantities, to work with rates such as speed or price per unit, and to recognise and solve direct and inverse proportion problems. The skill is deciding which relationship applies and then scaling correctly.

The answer

Ratio

A ratio compares quantities of the same kind, written $a : b$. Simplify by dividing both parts by their highest common factor, so $18 : 24 = 3 : 4$. To share a quantity in a given ratio, add the parts to find the total number of parts, find the value of one part, then multiply.

Equivalent ratios and the unitary method

Ratios stay equal when both parts are multiplied or divided by the same number. The unitary method finds the value of one unit first, which is the most reliable route through most ratio and proportion problems. If $5$ pens cost $3.50$ dollars, then one pen costs $0.70$ dollars, so $8$ pens cost $5.60$ dollars.

Rate

A rate compares two quantities of different kinds, such as distance and time. The most common is speed:

Rates have units attached, for example kilometres per hour or dollars per kilogram, and you must keep units consistent when combining or converting them.

Direct proportion

Two quantities are in direct proportion when one is a constant multiple of the other, so doubling one doubles the other. We write $y \propto x$, meaning $y = kx$ for a constant $k$. The ratio $\dfrac{y}{x}$ stays fixed. Cost per item bought at a fixed price is a direct proportion.

Inverse proportion

Two quantities are in inverse proportion when their product is constant, so increasing one decreases the other in the same ratio. We write $y \propto \dfrac{1}{x}$, meaning $xy = k$. The number of workers and the time to finish a fixed job is an inverse proportion.

Examples in context

Example 1. A recipe scaled up. A recipe for $4$ people needs $300\ \text{g}$ of flour. For $10$ people the flour scales in direct proportion: $\dfrac{300}{4} \times 10 = 750\ \text{g}$. Recipes are everyday direct proportion.

Example 2. Map scale. A map drawn at $1 : 50\,000$ means $1\ \text{cm}$ on the map represents $50\,000\ \text{cm}$, or $0.5\ \text{km}$, on the ground. Reading distances off a map is a direct application of ratio.

Try this

Q1. Simplify the ratio $45 : 60$. [1 mark]

• Cue. Divide both parts by $15$ to get $3 : 4$.

Q2. A runner covers $4\ \text{km}$ in $25$ minutes. Find the average speed in kilometres per hour. [2 marks]

• Cue. $25$ minutes $= \dfrac{25}{60}$ hour, so speed $= 4 \div \dfrac{25}{60} = 4 \times \dfrac{60}{25} = 9.6\ \text{km/h}$.

Q3. If $3$ workers build a wall in $12$ days, how long would $9$ workers take at the same rate? [2 marks]

• Cue. Inverse proportion: total work $= 3 \times 12 = 36$ worker-days, so $9$ workers take $\dfrac{36}{9} = 4$ days.