Ratio, proportion & rates of change
Compound Measures and Scale
Year 7 · Year 8 · Year 9
- ✓Understand what a compound measure is and identify examples.
- ✓Calculate speed, distance, or time using the formula Speed = Distance / Time.
- ✓Interpret and use scale in drawings and maps to find actual lengths or distances.
- ✓Create simple scale drawings given actual measurements.
- ✓Convert between different units of time, distance, and speed for calculations.
Key concepts
A compound measure is a measure that combines two or more different units. For example, speed combines units of distance (like kilometres or metres) and units of time (like hours or seconds). Other examples include density (mass and volume) and pressure (force and area).
Speed is a measure of how fast an object is moving. It tells us the distance an object travels in a given amount of time. Common units for speed include kilometres per hour (km/h), metres per second (m/s), and miles per hour (mph).
Distance is the total length covered by an object in motion. It can be calculated if you know the speed and the time taken. Common units for distance include kilometres (km), metres (m), and miles.
Time is the duration for which an object is in motion. It can be calculated if you know the distance covered and the speed. Common units for time include hours (h), minutes (min), and seconds (s).
Scale is the ratio that compares the size of a drawing or model to the size of the real object. It is often expressed as a ratio (e.g., 1:100) or as a statement (e.g., 1 cm represents 5 km). The first number in the ratio refers to the drawing/model measurement, and the second number refers to the actual measurement.
A scale drawing or map is a representation of an object or area where all dimensions are proportionally reduced or enlarged according to a specific scale. This allows large areas or objects (like a country on a map) or small objects (like a cell under a microscope) to be represented on a different size while maintaining accurate relative sizes and proportions.
Key facts to remember
- 1A compound measure combines two or more different units, such as speed (distance and time) or density (mass and volume).
- 2The relationship between speed, distance, and time can be remembered using the formula triangle: Distance at the top, Speed and Time at the bottom. Cover the value you want to find.
- 3Always ensure your units are consistent before performing calculations (e.g., if speed is in km/h, distance should be in km and time in hours).
- 4Common units for speed include km/h (kilometres per hour), m/s (metres per second), and mph (miles per hour).
- 5Scale is a ratio that shows how much smaller or larger a drawing is compared to the actual object. A scale of 1:n means 1 unit on the drawing represents n units in real life.
- 6To find the actual length from a scale drawing, multiply the drawing length by the scale factor (the 'n' in 1:n).
- 7To find the drawing length for a scale drawing, divide the actual length by the scale factor.
- 8Remember common unit conversions: 1 km = 1000 m, 1 m = 100 cm, 1 hour = 60 minutes, 1 minute = 60 seconds.
Worked examples
Example 1
A car travels 240 km in 4 hours. What is its average speed?
Answer
60 km/h
Example 2
On a map with a scale of 1:25,000, the distance between two landmarks is measured as 6 cm. What is the actual distance between the landmarks in kilometres?
Answer
1.5 km
Remember to convert units carefully. 1 km = 1000 m, 1 m = 100 cm.
Example 3
A train travels at an average speed of 120 km/h. How long will it take the train to cover a distance of 300 km? Give your answer in hours and minutes.
Answer
2 hours and 30 minutes
Ensure units are consistent before calculation. Here, distance is in km and speed is in km/h, so time will be in hours.
Common mistakes
- ✗Not converting units before calculating (e.g., using distance in km and time in minutes when speed is required in km/h).
- ✗Confusing the formulas for speed, distance, and time, or incorrectly using the formula triangle.
- ✗Incorrectly applying the scale factor, for example, dividing by the scale factor when you should multiply, or vice versa.
- ✗Forgetting to include the correct units in the final answer for compound measures.
- ✗Misinterpreting scale ratios, especially when converting between different units (e.g., 1 cm to 5 km requires converting km to cm).
Exam tips
- ★Draw the speed-distance-time triangle (Distance on top, Speed and Time on bottom) at the start of your exam to help recall the formulas accurately.
- ★Always write down the units for each value in your working to help you identify if conversions are needed and to ensure your final answer has the correct units.
- ★For scale questions, clearly write out what the scale means (e.g., '1 cm represents 25,000 cm' or '1 cm represents 0.25 km') before starting calculations.
- ★Show all your working steps, especially for unit conversions, as this can earn you method marks even if the final answer is incorrect.
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