Review and use rates and ratios, including identifying rates from a context (such as best buys, fuel consumption, heart rate and pay rates), working with unit rates, simplifying ratios, dividing a quantity in a given ratio, and using scale factors on maps and plans

What this dot point is asking

NESA wants you to recognise a rate or a ratio in a real situation and use it correctly. A rate compares two quantities measured in different units - dollars per litre, beats per minute, litres per $100$ km - while a ratio compares quantities of the same kind with the units cancelled out, such as $2 : 3$. The skills tested are: reading a rate from a context and reducing it to a unit rate so you can compare (the "best buy"), working with the named applications (fuel consumption, heart rate, pay rates), simplifying ratios, dividing a quantity in a given ratio, and using a scale on a map or plan to move between a drawing distance and a real distance. None of the arithmetic is hard. The marks are won by setting the comparison up on a common basis and by deciding whether to multiply or divide.

The answer

A rate is a comparison of two quantities with different units, written "per": $1.50 per litre, $72$ beats per minute. A unit rate has a denominator of $1$ (per one litre, per one minute, per one kilogram), and reducing any rate to a unit rate is what makes two offers comparable. A ratio compares quantities of the same unit, so the units cancel and a ratio is just a pair (or triple) of numbers like $3 : 5$. The three things you do with these are: find a unit rate to compare or to cost a job, simplify a ratio to its lowest terms, and divide a quantity into a given ratio by counting the parts.

Rates and the unit rate

To find a unit rate, divide the two quantities so the second one becomes $1$. A $2$ L carton of milk for $3.20 has unit rate $3.20 \div 2 = 1.60$, that is $1.60 per litre. Once both options are expressed per the same single unit, the smaller unit rate is the better value. The NESA-named rate contexts all use this one idea:

• Best buy: price per unit (per litre, per $100$ g, per kilogram); the lowest wins.
• Fuel consumption: litres per $100$ km, found as $\dfrac{\text{litres}}{\text{km}} \times 100$.
• Heart rate: beats per minute, found by scaling the count up to $60$ seconds.
• Pay rate: dollars per hour, found as total pay divided by hours worked.

Choose a sensible single unit before comparing. For groceries, "per $100$ g" usually gives tidier numbers than "per gram".

Simplifying ratios

A ratio is in simplest form when its numbers are whole numbers with no common factor. Divide every term by the highest common factor: $12 : 18 = 2 : 3$ (dividing by $6$). If a ratio has units or decimals, first convert to the same unit and clear the decimals or fractions. For example $50$ cm $: 2$ m becomes $50 : 200 = 1 : 4$ once both are in centimetres, and $0.5 : 1.5$ becomes $5 : 15 = 1 : 3$ after multiplying both by $10$.

Dividing a quantity in a given ratio

This is the highest-value skill on the page, and the method is always the same three steps. The ratio bar below shows it for sharing $2400 in the ratio $3 : 5$.

The three steps are:

1. Add the parts. For $3 : 5$ the total is $3 + 5 = 8$ parts.
2. Find one part. Divide the quantity by the total parts: $2400 \div 8 = 300$.
3. Scale each share. Multiply one part by each number: $3 \times 300 = 900$ and $5 \times 300 = 1500$, that is $900 and $1500.

Always finish by checking the shares add back to the original: $900 + 1500 = 2400$.

Scale on maps and plans

A scale is a ratio comparing a length on a drawing to the matching real length, written $1 : k$ with no units (both sides in the same unit). A scale of $1 : 50\,000$ means $1$ cm on the map is $50\,000$ cm in reality. So:

• Map to real: multiply the map length by the scale factor $k$.
• Real to map: divide the real length by the scale factor $k$.

Because the answer usually wants kilometres, remember the conversion $1$ km $= 100\,000$ cm. The schematic below works a $1 : 50\,000$ map measurement through to a real distance.

How exam questions ask about rates and ratios

The wording tells you which of the four methods to use:

• "Which is the better buy?" / "best value" means find the unit rate of each on a common basis (per $100$ g, per litre) and pick the smallest. Always show both unit rates - the comparison is where the marks sit.
• "Fuel consumption" / "litres per $100$ km" means $\dfrac{\text{litres}}{\text{km}} \times 100$; "how much fuel for ... km" then scales that rate.
• "... per minute / per hour / per kilogram" is a unit-rate question: divide to make the stated unit equal to $1$.
• "Simplify the ratio" means divide every term by the highest common factor, after converting to the same unit and clearing decimals.
• "Divide / share / in the ratio" is the parts method: add the parts, find one part, scale each share, and check the shares add to the total.
• "Scale of $1 : k$" / "on the map" / "actual distance" is a scale conversion: multiply map$\to$real, divide real$\to$map, then convert units to what is asked.