Rates - Practice

What you will learn

distinguish a ratio (same unit) from a rate (different units),
write a rate as a unit rate (e.g. km per hour),
use the speed-distance-time triangle,
compare prices, fuel consumption, exchange rates, and pay rates,
solve multi-step problems that combine rates and percentages.

Worked example 0 Real-world example: road trip fuel cost

Your family is driving Melbourne to Adelaide ( $730$ km). The car uses $7.2$ L per $100$ km and petrol costs $1.85/L. What will the fuel cost?

Fuel needed: $\dfrac{730}{100} \times 7.2 = 7.3 \times 7.2 = 52.56$ L.
Cost: $52.56 \times 1.85 = 97.24$ dollars.
Budget about $100 for fuel each way.

Key idea: two rates interact here — fuel consumption (L/100 km) and fuel price ($/L). Multiplying them in the right order gives dollars per distance.

1. Rate vs ratio

A ratio compares two quantities of the same kind (both times, both amounts of money, both lengths).

A rate compares two quantities of different kinds and keeps the units: km/h, $/kg, L/min.

Unit rate

\text{unit rate} = \dfrac{\text{amount of A}}{\text{amount of B, per 1 unit}}.

Worked example 1 Speed as a unit rate

A car covers $240$ km in $3$ hours. Its average speed is

\dfrac{240 \text{ km}}{3 \text{ h}} = 80 \text{ km/h}.

2. Speed, distance, time

Speed-distance-time

Three forms

\text{speed} = \dfrac{\text{distance}}{\text{time}}, \quad \text{distance} = \text{speed} \times \text{time}, \quad \text{time} = \dfrac{\text{distance}}{\text{speed}}.

Units must match

If speed is in km/h, time must be in hours and distance in km. Convert first if the units do not match.

Worked example 2 Distance from speed and time

A cyclist rides at $18$ km/h for $2$ h $30$ min. How far?

Time in hours: $2.5$ .
Distance: $18 \times 2.5 = 45$ km.

Worked example 3 Time from speed and distance

A train travels $350$ km at $80$ km/h. How long does it take?

t = \dfrac{350}{80} = 4.375 \text{ h} = 4 \text{ h } 22 \text{ min } 30 \text{ s}.

3. Other common rates

Density: $\dfrac{\text{mass}}{\text{volume}}$ (g/cm $^3$ or kg/m $^3$ ).
Pay rate: $/hour.
Fuel consumption: L/100 km.
Exchange rate: units of currency A per 1 unit of currency B.
Interest rate (simple): $ per $100 per year, i.e. a percentage.

Worked example 4 Fuel consumption

A car uses $7$ L per $100$ km. On a $450$ km trip, how much fuel is used?

\dfrac{7}{100} \times 450 = 31.5 \text{ L}.

Worked example 5 Currency conversion

The exchange rate is $1$ AUD = $0.65$ USD. Convert $240 AUD to USD, then $600 USD back to AUD.

AUD → USD: $240 \times 0.65 = 156$ USD.
USD → AUD: $\dfrac{600}{0.65} \approx 923.08$ AUD.

Practice: Year 8 core

Fluency

Unit rates

A car goes $240$ km in $4$ h. Find the average speed.
A tap fills a tank at $600$ L in $20$ min. Find the rate in L/min.
A worker earns $540 for $20$ hours. Find the hourly rate.
A mass of $180$ g has volume $20$ cm $^3$ . Find the density.
A printer prints $60$ pages in $4$ minutes. Find pages per minute.

Fluency

Speed, distance, time

Distance from $65$ km/h × $3$ h.
Time to cover $300$ km at $50$ km/h.
Speed of $200$ m in $25$ seconds (in m/s).
Convert $72$ km/h to m/s.
How long to cover $120$ km at $80$ km/h?
A train covers $400$ km in $2$ h $30$ min. Find its speed.

Fluency

Fuel, pay, exchange

A car uses $8$ L/ $100$ km. Fuel for $350$ km?
A worker earns $22/h. Find pay for $36.5$ hours.
Exchange rate $1$ AUD $= 0.85$ NZD. Convert $150 AUD to NZD.
$1$ USD $= 1.50$ AUD. Convert $200 USD to AUD.
Simple interest: $1000 at $4\%$ for $3$ years. How much interest?

Reasoning

Explain and spot the mistake

Sam writes “the speed is $50$ km in $1$ hour, so $50$ per hour, so $50$ km”. What units are missing? What is the correct way to report speed?
Two cars: A does $60$ km in $1$ hour; B does $30$ km in $30$ minutes. Are they the same speed? Show working.
Explain why a “rate” and a “unit rate” are slightly different ideas. Give an example of each.
Without calculating, compare: a pool fills at $10$ L/min for $20$ min, or at $4$ L/min for $60$ min - which delivers more water?

Problem solving

Real contexts

A road trip is $1600$ km. If the driver averages $100$ km/h (including breaks in planned driving time), how long does the trip take?
A swimming pool holds $120\,000$ L. A hose delivers $120$ L/min. How long to fill (hours)?
A box of $24$ pens costs $7.20. A single pen costs $0.40. Which is better value, and by how much per pen?
Two phone plans: A is $25/month + $0.10/min; B is $35/month with unlimited calls. For what usage does B beat A?
An alloy uses $5$ kg of copper for every $3$ kg of tin. For a $40$ kg alloy, how much of each?

Challenge

Reasoning

Harder problems

A car averages $80$ km/h for $2$ hours and then $60$ km/h for $3$ hours. What is its average speed for the whole trip?
A shopkeeper buys tea at $12/kg and sells it at $15/kg. What percentage profit is this?
A tap fills a tank at $8$ L/min while a drain removes water at $3$ L/min. The tank holds $600$ L; it starts empty. How long to fill?
Currency arbitrage: $1 AUD $=$ $0.65 USD; $1 USD $=$ $0.80 EUR; $1 EUR $=$ $1.60 AUD. Is there a profit in converting $100 AUD → USD → EUR → AUD? If so, how much?