Practice Maths

Understanding and Graphing Rates

Key Ideas

Key Terms

rate
a comparison of two quantities with different units, written with "per" (e.g. km/h, $/kg, L/min).
unit rate
a rate expressed per 1 unit of the second quantity; e.g. $5 per kg. Found by dividing so the denominator becomes 1.
gradient
the steepness of a line on a graph, equal to rise ÷ run. On a rate graph, the gradient equals the rate of change.
speed
a rate comparing distance and time; speed = distance ÷ time (e.g. km/h). A steeper distance–time graph means faster speed.
fuel consumption
a rate comparing fuel used to distance travelled; often given as litres per 100 km (L/100 km).

Calculating a Unit Rate

Unit rate = total amount ÷ number of units
e.g. 240 km in 3 h → unit rate = 240 ÷ 3 = 80 km/h

Hot Tip A rate always has two different units separated by “per”. Speed = distance ÷ time. If you see two quantities with the same units, it is a ratio, not a rate.

Worked Examples

Example 1: Water flows from a tap at 150 L in 5 minutes. Find the unit rate.

Unit rate = 150 ÷ 5 = 30 L/min

Example 2: A car travels at 80 km/h. How far does it travel in 2.5 hours?

Distance = 80 × 2.5 = 200 km

Example 3: A graph shows a straight line from (0, 0) to (4, 200). What rate does the gradient represent?

Gradient = rise ÷ run = 200 ÷ 4 = 50 km/h

Time (hours) Distance (km) 1 2 3 50 100 150 50 km/h (0,0) (3,150)

What Is a Rate?

A rate compares two quantities with different units. This is what makes it different from a ratio (which compares same-type quantities). Common rates include:

• Speed: kilometres per hour (km/h)
• Price: dollars per kilogram ($/kg)
• Fuel efficiency: litres per 100 km (L/100 km)
• Heartbeat: beats per minute (bpm)
• Data: megabytes per second (MB/s)

The word "per" means "for every one of". So 60 km/h means 60 kilometres for every 1 hour of travel.

Unit Rates

A unit rate is a rate with a denominator of 1. It tells you the amount per single unit of the second quantity, making it easy to compare.

Example: Apples cost $4.80 for 3 kg. Unit rate = $4.80 ÷ 3 = $1.60 per kg.

Example: A cyclist travels 42 km in 1.5 hours. Unit rate = 42 ÷ 1.5 = 28 km/h.

Unit rates make comparisons easy. If Store A sells 2 L of juice for $3.20 and Store B sells 3 L for $4.50, find the unit rate: Store A = $1.60/L, Store B = $1.50/L. Store B is better value.

Converting Rates

You often need to convert between units. The most common conversion is between km/h and m/s.

km/h to m/s: divide by 3.6 (because 1 km = 1 000 m, and 1 hour = 3 600 s, so 1 000/3 600 = 1/3.6)

m/s to km/h: multiply by 3.6

Example: Convert 90 km/h to m/s.   90 ÷ 3.6 = 25 m/s
Example: Convert 15 m/s to km/h.   15 × 3.6 = 54 km/h

For other conversions, always ask: how many of the new unit equals one of the old unit? Then multiply or divide accordingly.

Graphing Rates: Distance-Time Graphs

When you graph a rate, you get a straight line through the origin (if the rate is constant). The gradient of the line equals the rate.

On a distance-time graph: the gradient = distance ÷ time = speed.
A steeper line = faster speed. A flat line = stopped (zero speed).

Example: A car travels at a constant 80 km/h. After 1 hour it has gone 80 km, after 2 hours 160 km, after 3 hours 240 km. Plot these points and connect them — you get a straight line with gradient 80.

If the line is curved, the rate is changing (the car is accelerating or decelerating).

Interpreting Rate Graphs in Context

Distance-time graphs tell a story. A horizontal segment means the object is not moving. A steeper segment means it is moving faster. A line going back down means it returned to the starting point.

Example: A student walks to a bus stop (rising line), waits (flat line), then rides the bus home faster (rising, steeper line to a higher point). Each segment of the graph tells part of the journey story.

Key tip: When reading a distance-time graph, the gradient (steepness) tells you the speed — it does NOT tell you how far. A steeper line means faster, not further. A flat line means the speed is zero (the object is stationary), even if it is high up on the graph (far from home).

Mastery Practice

  1. Calculate the unit rate for each situation. Fluency

    1. $45 for 9 kg of apples
    2. 240 km on 8 L of fuel
    3. $72 for 6 hours of work
    4. 350 words typed in 5 minutes
    5. $3.60 for 4 oranges
    6. 180 L of water in 9 minutes
    7. $84 for 12 kg of meat
    8. 630 km driven in 7 hours

  2. Use the given rate to calculate the unknown quantity. Fluency

    1. At 60 km/h, how far is travelled in 2.5 hours?
    2. At $8.50/kg, what is the cost of 4 kg?
    3. A tap fills at 25 L/min. How much water flows in 6 minutes?
    4. A worker earns $22/h. How much for 8 hours?
    5. A car uses 9 L/100 km. How much fuel for 250 km?
    6. At 80 km/h, how long to travel 200 km? (give answer in hours)
    7. At $12/kg, how many kg can you buy for $90?
    8. A printer prints 15 pages/min. How long for 90 pages? (in minutes)

  3. Convert each rate to the unit shown in brackets. Fluency

    1. 1.5 L/min [L/hour]
    2. 120 km/h [km/min]
    3. $3/min [$/hour]
    4. 500 mL/min [L/hour]
    5. 72 km/h [m/s]
    6. $600/day [$/hour, assume 8-hour day]
    7. 2 m/s [m/min]
    8. 45 L/hour [L/min]

  4. Read and interpret each graph description. State the rate and what it represents. Understanding

    1. A distance–time graph shows a straight line from (0, 0) to (3, 180). What is the speed?
    2. A water tank graph shows volume (litres) on the vertical axis and time (minutes) on the horizontal axis. The line goes from (0, 0) to (5, 200). What is the fill rate?
    3. A cost graph has cost ($) on the vertical axis and mass (kg) on the horizontal axis. The line passes through (0, 0) and (4, 22). What is the cost per kilogram?
    4. A distance–time graph shows a horizontal line at height 60. What does this tell us about the traveller’s speed?
    5. A graph shows two straight lines from the origin. Line A reaches (2, 100) and Line B reaches (2, 60). Which line represents a faster rate? By how much?
    6. A fuel–distance graph goes from (0, 50) to (10, 0). What does the starting point (0, 50) represent?
    7. A graph of earnings vs hours shows a straight line through (0, 0) and (5, 87.50). What is the hourly wage?
    8. A water level graph drops from (0, 80) to (4, 0). At what rate (in L/min) is water leaving the tank?

  5. Compare the rates and decide which is better value. Show your working. Understanding

    1. Store A: $2.40 for 4 bananas. Store B: $3.20 for 5 bananas. Which is cheaper per banana?
    2. Plan A: $45 for 5 GB of data. Plan B: $62 for 8 GB. Which gives more GB per dollar?
    3. Car A: 8 L/100 km. Car B: 12 L/100 km. Which car is more fuel-efficient?
    4. Job A pays $18/h for 6 hours. Job B pays $120 for the whole day (8 hours). Which pays more per hour?
    5. Recipe A uses 200 g flour for 12 biscuits. Recipe B uses 300 g flour for 20 biscuits. Which uses less flour per biscuit?
    6. Swimmer A completes 500 m in 10 min. Swimmer B completes 750 m in 15 min. Who is faster?

  6. Multi-step rate problems. Show all working. Problem Solving

    1. A car travels at 90 km/h for 2 hours, then at 60 km/h for 1.5 hours. What is the total distance? What is the average speed for the whole journey?
    2. Sam earns $24.50/h and works 6 hours on Monday and 8 hours on Friday. How much does Sam earn in total across both days?
    3. A swimming pool holds 48 000 litres. One pump fills it at 400 L/min and another at 200 L/min. How long (in hours) will both pumps take to fill the pool together?
    4. A car uses petrol at 7.5 L/100 km. Petrol costs $1.80/L. What is the fuel cost for a 240 km trip?
    5. A garden hose fills a 1200 L tank at 15 L/min. After 30 minutes, a leak develops that drains water at 5 L/min. How long (from the start) does it take to fill the tank?

  7. Identify the rate being described and write it using correct notation (e.g. km/h). Fluency

    1. A car uses 8 litres of fuel for every 100 kilometres driven.
    2. A worker packs 120 boxes in 4 hours.
    3. A dripping tap wastes 5 litres of water every 2 hours.
    4. A factory produces 600 items per 8-hour shift.
    5. An athlete runs 400 metres in 50 seconds.
    6. A baker uses 250 g of flour to make each loaf of bread.

  8. A distance–time graph shows three lines starting from the origin: Line P reaches (2, 120), Line Q reaches (2, 80), and Line R reaches (4, 120). Answer these questions. Understanding

    1. What is the speed (km/h) represented by Line P?
    2. What is the speed represented by Line Q?
    3. What is the speed represented by Line R?
    4. Which line represents the fastest speed?
    5. Which two lines represent the same distance after 4 hours?
    6. Describe in words what the steepness of a line on a distance–time graph tells you.

  9. A car’s fuel consumption is 9.5 L/100 km. Petrol costs $2.00 per litre. Understanding

    1. How many litres of fuel are needed for a 300 km trip?
    2. What is the total fuel cost for the 300 km trip?
    3. How far can the car travel on a full 50 L tank?
    4. If the driver wants to keep fuel costs under $50, what is the maximum distance they can drive?

  10. Challenging rate problems. Show full working and state units clearly. Problem Solving

    1. A family drives 420 km at 70 km/h, stops for 30 minutes for lunch, then drives the remaining 180 km at 90 km/h. What is the total journey time (in hours and minutes)?
    2. Two taps fill a 900 L pool. Tap A fills at 30 L/min and Tap B fills at 20 L/min. Tap A runs for 10 minutes then is turned off. How long does Tap B need to run on its own to finish filling the pool?
    3. A cyclist averages 24 km/h over a 3-hour ride. In the first hour she travels 30 km, and in the second hour she travels 24 km. How far does she travel in the third hour?
    4. A photocopier prints 80 pages per minute. A second photocopier prints 60 pages per minute. A 2000-page document needs to be printed. If both run simultaneously, how long will it take (in minutes)?
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