Compound Measures

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Notes

Compound Measures

A **compound measure** uses more than one measurement (e.g., speed = distance $\div time)$ .
Compound measures measure rates: how one quantity changes per unit of another.
Examples: **speed**, **density**, **pressure**, flow rate, population density, fuel consumption.
Units derive from the formula: e.g., speed in km/h means distance (km) per time (h).
A **formula triangle** helps rearrange: top variable = product of bottom two; bottom variable $= top \div$ other bottom.

Speed, Distance & Time

**Speed** = distance ÷ time (average speed over whole journey).
Common units: m/s, km/h, miles per hour (mph).
Distance = speed × time; Time = distance ÷ speed.
For multi-part journeys, calculate each part separately then combine.
Ensure consistent units: convert minutes to hours, metres to km, etc.

Density, Mass & Volume

**Density** = mass ÷ volume (mass per unit volume).
Common units: $g/cm^{3}, kg/m^{3}$ .
Mass = density × volume; Volume = mass ÷ density.
Check unit consistency: e.g., convert kg to g if density is in $g/cm^{3}$ .
May need to calculate volume from geometry first (e.g., cuboid, cylinder).

Pressure, Force & Area

**Pressure** = force ÷ area (force per unit area).
Common units: $\frac{N}{m}^{2} (Pascals, Pa)$ .
Force = pressure × area; Area = force ÷ pressure.
Weight is a force (different from mass).
Area must be in $m^{2}$ when pressure is in $\frac{N}{m}^{2}$ .

Unit Conversions

Convert speed: m/s to km/h multiply by 3.6; km/h to m/s divide by 3.6.
Convert time: hours to minutes multiply by 60; minutes to seconds multiply by 60.
Convert mass: kg to g multiply by 1000; g to kg divide by 1000.
Convert volume: $m^{3}$ to $cm^{3}$ multiply by 1,000,000; $cm^{3}$ to $m^{3}$ divide by 1,000,000.
Always check final answer units required by the question.

Problem-Solving Tips

Use formula triangles to rearrange equations quickly.
For multi-stage problems, break into steps: find missing variable for each stage.
Watch for mixed units (e.g., minutes and hours) – convert to same unit.
When using density, ensure mass and volume units match.
In pressure problems, remember that force may be given as weight.

Speed, Distance, Time Triangle

Density, Mass, Volume Triangle

Pressure, Force, Area Triangle

Unit Conversion Example: m/s to km/h

Practice questions

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1.The density of a substance is $2.5 g/cm^{3}$ . What is the mass of a block of this substance with volume $400 cm^{3}$ ?
Easy
- A1000 g
- B160 g
- C100 g
- D1000 kg
2.A car travels 120 km in 2 hours. What is its average speed?
Easy
- A60 km/h
- B240 km/h
- C0.0167 km/h
- D120 km/h
3.Pressure is defined as:
Easy
- Aforce × area
- Bforce ÷ area
- Carea ÷ force
- Dmass ÷ volume
4.A force of 200 N acts on an area of $5 m^{2}$ . What is the pressure?
Easy
- A40 Pa
- B1000 Pa
- C0.025 Pa
- D40 N
5.A solid metal cuboid has mass 4 kg and volume $600 cm^{3}$ . Calculate its density in $g/cm^{3}$ .
Medium
- A $6.67 g/cm^{3}$
- B $0.00667 g/cm^{3}$
- C $0.15 g/cm^{3}$
- D $2400 g/cm^{3}$
6.A box exerts a pressure of $42 \frac{N}{m}^{2}$ on a table. The face in contact is 2 m by 1.25 m. Find the force exerted by the box on the table.
Medium
- A105 N
- B84 N
- C52.5 N
- D16.8 N
7.A car travels at 108 km/h for 20 seconds. How far does it travel in metres?
Medium
- A600 m
- B2160 m
- C540 m
- D3000 m
8.A gold bar has mass 12.5 kg and density $19.3 g/cm^{3}$ . Find its volume correct to 3 significant figures.
Medium
- A $648 cm^{3}$
- B $0.648 cm^{3}$
- C $6480 cm^{3}$
- D $241 cm^{3}$

Summary

## Compound Measures (Speed, Density, Pressure) ### Key Formulas - **Speed** = Distance / Time (units: m/s, km/h, etc.) - **Density** = Mass / Volume (units: g/cm³, kg/m³) - **Pressure** = Force / Area (units: N/m² = Pa) ### Speed, Distance & Time - Average speed = total distance / total time. - Use formula triangle: top = Distance; bottom = Speed & Time. - Convert units carefully (e.g., m/s to km/h: multiply by 3.6). - For multi-part journeys, find total distance and total time. ### Density, Mass & Volume - Density = mass / volume. Rearranged: mass = density × volume; volume = mass / density. - Ensure consistent units (e.g., convert kg to g if density in g/cm³). - May need to calculate volume of a shape first. ### Pressure, Force & Area - Pressure = force / area. Rearranged: force = pressure × area; area = force / pressure. - Force is in newtons (N), area in m². - Weight is a force (different from mass). ### Tips - Check units: time in hours/minutes, mass in kg/g, etc. Convert as needed. - Use formula triangles to rearrange. - For average speed, use total distance / total time. ### Example (from past paper) - Density: mass 4 kg, volume 600 cm³ → density = 4000 g / 600 cm³ = 6.67 g/cm³. - Speed: 15 miles in 6 hours → speed = 15/6 = 2.5 mph. - Pressure: force 140 N, pressure 35 N/m² → area = 140/35 = 4 m².

Key terms

Compound measure: A measure calculated using more than one measurement, e.g., speed, density, pressure.
Speed: Distance travelled per unit time. Formula: speed = distance / time.
Density: Mass per unit volume. Formula: density = mass / volume.
Pressure: Force per unit area. Formula: pressure = force / area.
Average speed: Total distance divided by total time.
Formula triangle: A visual aid showing the relationship between three quantities; top variable equals product of bottom two, bottom variable equals top divided by other bottom.

Slides

Compound Measures

A compound measure is calculated using more than one measurement.
Examples include speed, density, pressure, flow rate, and population density.
Units often appear as 'a per $b', e.g., km/h, g/cm^{3}, \frac{N}{m}^{2}$ .

Which of the following is a compound measure?

Mass
Speed
Time
Length

Speed, Distance, Time

Speed = Distance / Time (average speed).
Common units: m/s, km/h.
Use formula triangles: Distance = Speed × Time, Time = Distance / Speed.

A car travels 150 km in 3 hours. Its average speed $is ____ km/h$ .

Density, Mass, Volume

Density = Mass / Volume.
Common units: $g/cm^{3}, kg/m^{3}$ .
Mass = Density × Volume, Volume = Mass / Density.

A block has mass 200 g and volume $50 cm^{3}$ . What is its density?

$0.25 g/cm^{3}$
$4 g/cm^{3}$
$250 g/cm^{3}$
$10000 g/cm^{3}$

Pressure, Force, Area

Pressure = Force / Area.
Common units: $\frac{N}{m}^{2} (Pascals)$ .
Force = Pressure × Area, Area = Force / Pressure.

A force of 100 N acts on an area of $5 m^{2}$ . The pressure $is ____ \frac{N}{m}^{2}$ .

Unit Conversions

Always check units: convert to match before calculating.
Common conversions: 1 $km = 1000 m, 1$ hour $= 3600 s, 1$ $kg = 1000 g$ .
Example: $50 \frac{m}{s} = 50 \times 3.6 = 180 km/h$ .

Which of these are true?

Speed = Distance × Time
Density = Mass / Volume
Pressure = Force × Area
Pressure = Force / Area

Using Formula Triangles

Cover the quantity you want to find.
If the remaining two are side by side, multiply.
If one is above the other, divide top by bottom.

Using the formula triangle for density, how do you find volume?

Mass × Density
Mass / Density
Density / Mass
Mass + Density

Working with Mixed Units

Convert all quantities to consistent units before using formulas.
For example, if speed is in km/h and time in minutes, convert minutes to hours.
Common pitfalls: mixing g and kg, or cm and m.

A train travels 180 km in 1.5 hours. Its speed $is ____ km/h$ .

Problem-Solving Tips

For two-part journeys, calculate each part separately.
Use average speed = total distance / total time.
Check if answer needs specific units (e.g., m/s vs km/h).

Which statements are correct?

Average speed = total distance / total time
Density is measured in $kg/m^{3}$
Pressure = Force × Area
$1 \frac{m}{s} = 3.6 km/h$

Summary

Compound measures combine two different measurements.
Key formulas: Speed $= \frac{D}{T}$ , Density $= \frac{M}{V}$ , Pressure $= \frac{F}{A}$ .
Always check units and convert if necessary.

A box exerts 200 N on an area of $4 m^{2}$ . What is the pressure?

$50 \frac{N}{m}^{2}$
$800 \frac{N}{m}^{2}$
$0.02 \frac{N}{m}^{2}$
$196 \frac{N}{m}^{2}$

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