# Compound Measure Problems

## Topic Overview

During your GCSE maths paper, you may be required to solve a series of compound measure problems.

A '**compound measure**' is a mathematical or scientific measurement which is made up of two or more other measures.

For example, '**speed**' is a compound measure because it is calculated using other measures of 'distance' and ‘**time**’:

Using this compound measure equation, you can solve a variety of questions which concern the measure of speed.

For instance; you can calculate the speed of a motorbike if you know the distance it has travelled and the time it took to do so. If the motorbike travelled 40km in 2 hours, you can calculate it speed in km/h:

Speed = Distance/Time

Speed = 40/2 = 20 km/h

(**Note**: When calculating speed, make sure you use the correct units. In this example, you are dividing kilometres by hours so you would use km/h. However, you may be required to solve compound measure problems which use different measurements; such as metres per second (m/s) or miles per hour (mph). As a result, you should always pay close attention to the specific units of measurements which are outlined in your examination questions).

(**Note**: When calculating speed, make sure you use the correct units. In this example, you are dividing kilometres by hours so you would use km/h. However, you may be required to solve compound measure problems which use different measurements; such as metres per second (m/s) or miles per hour (mph). As a result, you should always pay close attention to the specific units of measurements which are outlined in your examination questions).

## Key Concepts

In the new linear GCSE Maths paper, you will be required to solve various mathematical problems involving compound measures. The specific questions you will be expected to answer will vary depending upon which examination board with which you are registered, but as a rule you will be required to:

- Understand the principles of compound measures
- Use the principles of compound measures to solve various mathemtical problems

Listed below are a series of summaries and worked examples to help you solidify your knowledge about compound measures.

## Worked Examples

**1 - Solving compound measure problems**

As mentioned in the 'Topic Overview', a compound measure consists of two or more different measurements.

One of the most frequently used compound measures is 'speed' . During your GCSE maths exam, it is highly likely that you will be required to use the compound measure formula for speed in order to solve various mathematical problems:

Speed is a compound measure because it is calculated from distance and time, as demonstrated by the diagram below:

This triangle displays the varying relationships between speed, distance and time. By observing the position of the other two measurements in the triangle, you can calculate the value of the other measurements. For example:

*Example*

(a) - A car travels 160 miles in 2 hours and 30 minutes. Calculate the average speed of the car in miles per hour (mph)

*Solution*

(a) - You can substitute the values for distance and time into the compound measure formula in order to calculate the car's speed:

Speed = Distance ÷ Time

Speed = 160 ÷ 2.5 = 64 mph

Therefore the car's average speed is 64mph

*Example*

(b) - A lorry travels at 50 mph for 10 hours. How many miles has it travelled?

*Solution*

(b) - By substituting the necessary values into the compound measure formula for distance, you can calculate the lorry's total distance travelled:

Distance = Speed x Time

Distance = 50 x 10 = 500

Therefore the lorry has travelled 500 miles

*Example*

(c) - A car travels at an average speed of 60km/h. How will it take for the car to travel 250 kilometres?

*Solution*

(c) - By substituting the necessary values into the compound measure formula for time, you can calculate how long it will take the car to travel 270 kilometres:

Time = Distance ÷ Speed

Time = 270 ÷ 60 = 4.5 hours

Therefore it takes 4.5 hours for the car to travel 250 kilometres

**2 - Higher Tier compound measure problems**

If you are sitting the Higher Tier GCSE maths paper, you may be required to answer compound measure problems regarding density. Density is calculated using the measurements of 'mass' and ‘volume’, as displayed in the diagram below:

This triangle displays the varying relationships between density, mass and volume. By observing the position of the other two measurements in the triangle, you can calculate the value of the other measurements. For example:

*Example*

(a) - Calculate the density of a piece of metal which has a mass of 5kg and a volume of 2.75 m^3

Solution

(a) - By substituting the necessary values into the compound measure formula for density, you can calculate its value:

Density = Mass ÷ Volume

Density = 5 ÷ 2.75 = 1.81818181818...

Therefore the piece of metal has a density of 1.82kg per m^3 (to 2 decimal places)

*Example*

(b) - Calculate the mass of a object which has a volume of 4cm^3 and a density of 2g per cm^3.

*Solution*

(b) - By substituting the necessary values into the compound measure formula for mass, you can calculate its value:

Mass = Density x Volume

Mass = 2 x 4 = 8

Therefore the object has a mass of 8 grams

*Example*

(c) - Calculate the volume of a cylinder which has a mass of 100kg and a density of 50 kg per m^3.

*Solution*

(c) - By substituting the necessary values into the compound measure formula for volume, you can calculate its value:

Volume = Mass ÷ Density

Volume = 100 ÷ 50 = 2

Therefore the cylinder has a volume of 2m^3

## Exam Tips

- Remember that a 'compound measure' is a mathematical or scientific measurement which is made up of two or more other measures.
- When calculating compound measure problems, make sure you use the correct units. You should always pay close attention to the specific units of measurements which are outlined in your examination questions.
- When calculating compound measure problems, always approximate your answer to a suitable degree of accuracy.
- Memorise the various relationships in the Speed-Distance-Time triangle: Distance = Speed x Time, Speed = Distance ÷ Time, Time = Distance ÷ Speed
- Memorise the various relationships in the Density-Mass-Volume triangle: Density = Mass ÷ Volume, Mass = Density x Volume, Volume = Mass ÷ Density

When solving compound measure problems, the two most important factors you need to consider are:

- The triangle relationship of the compound measures which you are using
- The specific units of measurement for each value which you are calculating

Many students lose marks because they do not know the correct relationships between compound measures. Subsequently, they will divide values instead of multiplying and miss out on vital marks. Similarly, students will often calculate the correct value, but still lose marks because they have represented their value using an incorrect unit of measurement.

To prevent yourself from making these mistakes, it is helpful to memorise the triangles for speed and density, and to draw these triangles on your exam paper when answering compound measure problems. By doing so you equip yourself with a visual aid from which to work, as well as demonstrating to the examiner that you understand the various relationships of speed, distance and time, as well as density, mass and volume.

Furthermore, you should double check all of your answers and pay close attention to the units of measurement which have been outlined in the question. By doing so, you can earn maximum marks for calculating the correct answer, displaying the correct marking and using the correct unit of measurement.

## Related Topics

- Repeated Proportional Change
- Proportion: Numerical Problems
- Proportion: Algebraic Problems
- Problem Solving with Decimals and Percentages
- Ratio Problems
- Pre-calculus Skills