Compound Measures

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Notes

Compound Measures Overview

A **compound measure** uses more than one measurement (e.g., speed = distance $\div time)$ .
Compound measures often describe **rates** – how one quantity changes per unit of another.
Examples: **speed**, **density**, **pressure**, flow rate, population density, fuel consumption.
Units are derived from the formula, e.g., km/h means distance (km) per time (h).
Use **formula triangles** to rearrange: cover the variable you need, then multiply or divide the other two.

Speed, Distance & Time

**Speed** = distance ÷ time (average speed over a journey).
Common units: **m/s** or **km/h**.
Distance = speed × time; Time = distance ÷ speed.
Check units: if distance is in km and time in minutes, speed is km/min.
For mixed units (e.g., minutes and hours), convert to consistent units before calculating.
In multi‑part journeys, treat each part separately or combine total distance and total time.

Density, Mass & Volume

**Density** = mass ÷ volume.
Common units: **g/cm³** or **kg/m³**.
Mass = density × volume; Volume = mass ÷ density.
An object floats if its density is **less than** the density of water $(1 g/cm^{3})$ .
Ensure consistent units: convert kg to g or m to cm as needed.

Pressure, Force & Area

**Pressure** = force ÷ area.
Common units: **N/m²** (also called Pascals, Pa).
Force = pressure × area; Area = force ÷ pressure.
Remember: **weight is a force** (measured in newtons), not mass.
Area units must match: if force is in N and area in $m^{2}$ , pressure is $\frac{N}{m}^{2}$ .

Unit Conversions

Always check if the answer requires different units (e.g., m/s to km/h).
To convert km/h to m/s: multiply by $\frac{1000}{3600} (or$ divide by 3.6).
To convert m/s to km/h: multiply by $\frac{3600}{1000} (or$ multiply by 3.6).
For time: 1 hour $= 60$ minutes $= 3600$ seconds.
For mass: 1 $kg = 1000$ g; for length: 1 $km = 1000$ m; for volume: $1 m^{3} = 1 000 000 cm^{3}$ .

Worked Examples – Speed

Example: George drives 315 km in 3 h 30 min. Average speed $= 315 \div 3.5 =$ **90 km/h**.
Example: A speedboat travels 84 km/h. Convert to $m/min: 84 \times  1000 \div 60 =$ **1400 m/min**.
Example: Ivan walks 1.5 km in 20 min. Speed $= 1.5 \div (\frac{20}{60}) =$ **4.5 km/h**.
Example: A car travels at 20 m/s for 10 km. Time $= 10 000 \div 20 = 500 s =$ **8 min 20 s**.

Worked Examples – Density & Pressure

Density example: Wood block $8 \times  20 \times  10$ cm, mass 1360 g. Volume $= 1600 cm^{3}$ , density $= \frac{1360}{1600} =$ **0.85 g/cm³** (floats).
Pressure example: Force 300 N on area $5 m^{2}$ . Pressure $= \frac{300}{5} =$ **60 N/m²**.
Pressure example: Force 300 N, pressure $0.6 N/cm^{2}$ , width 20 cm. Area $= \frac{300}{0.6} = 500 cm^{2}$ , length $= \frac{500}{20} =$ **25 cm**.
Always state units in final answer.

Exam Tips

**Memorise** the three formulas: speed, density, pressure.
Watch for mixed units (e.g., minutes and hours) – convert before calculating.
Use formula triangles to avoid rearranging errors.
Show all working clearly, especially when converting units.
For floating questions, compare density to water $(1 g/cm^{3})$ .

Speed, Distance, Time Triangle

Density, Mass, Volume Triangle

Pressure, Force, Area Triangle

Unit Conversion Flowchart

Practice questions

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1.A car travels 180 km in 3 hours. What is its average speed in km/h?
Easy
- A60 km/h
- B540 km/h
- C30 km/h
- D90 km/h
2.What is the formula for pressure?
Easy
- A $P = F / A$
- B $P = F \times A$
- C $P = A / F$
- D $P = F + A$
3.A substance has a mass of 50 g and a volume of $25 cm^{3}$ . What is its density?
Easy
- A $2 g/cm^{3}$
- B $0.5 g/cm^{3}$
- C $1250 g/cm^{3}$
- D $75 g/cm^{3}$
4.A force of 200 N is applied over an area of $4 m^{2}$ . What is the pressure?
Easy
- A $50 \frac{N}{m}^{2}$
- B $800 \frac{N}{m}^{2}$
- C $0.02 \frac{N}{m}^{2}$
- D $200 \frac{N}{m}^{2}$
5.George drives 315 km in 3 hours 30 minutes. What is his average speed in km/h?
Medium
- A90 km/h
- B105 km/h
- C80 km/h
- D110.25 km/h
6.A speedboat travels at 84 km/h. Convert this speed into metres per minute.
Medium
- A1400 m/min
- B5040 m/min
- C1.4 m/min
- D84000 m/min
7.Ivan walks 1.5 km in 20 minutes. What is his average speed in km/h?
Medium
- A4.5 km/h
- B3 km/h
- C0.075 km/h
- D30 km/h
8.A cuboid block of wood measures 8 $cm \times 20$ $cm \times 10$ cm and has mass 1360 g. The density of water is $1 g/cm^{3}$ . Will the block float?
Medium
- AYes, because its density is $0.85 g/cm^{3}$
- BNo, because its density is $1.18 g/cm^{3}$
- CYes, because its density is $1 g/cm^{3}$
- DNo, because its density is $0.85 g/cm^{3}$

Summary

## Compound Measures - **Compound measures** use two or more measurements to calculate a rate (how one quantity changes per unit of another). - Common examples: speed, density, pressure, flow rate, population density, fuel consumption. - Units are derived from the formula: e.g., speed = distance/time → units: km/h or m/s. ### Speed, Distance & Time - **Formula:** $ \text{Speed} = \frac{\text{Distance}}{\text{Time}} $ (average speed). - Use formula triangle: $ D = S \times T $, $ S = D \div T $, $ T = D \div S $. - Watch for mixed units (e.g., minutes to hours, metres to km). ### Density, Mass & Volume - **Formula:** $ \text{Density} = \frac{\text{Mass}}{\text{Volume}} $. - Units: g/cm³ or kg/m³. - An object floats if its density is less than water (1 g/cm³). ### Pressure, Force & Area - **Formula:** $ \text{Pressure} = \frac{\text{Force}}{\text{Area}} $. - Units: N/m² (Pascals). - Force (weight) is different from mass. ### Key Tips - Always check unit consistency before calculation. - Convert time: 1 hour = 60 minutes, 1 minute = 60 seconds. - Convert length: 1 km = 1000 m, 1 m = 100 cm. - For multi‑step problems, break into parts and apply the correct formula. ### Example (from past paper) - George drives 315 km in 3h 30min. Average speed = $ \frac{315}{3.5} = 90 $ km/h. - Pressure = Force/Area: 300 N on 5 m² → pressure = 60 N/m².

Key terms

Compound measure: A measure calculated using more than one measurement, often a rate (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 taken.
Flow rate: Volume of fluid passing a point per unit time; formula: Flow rate = Volume / Time.
Population density: Number of people per unit area; formula: Population density = Number of people / Area.
Fuel consumption: Distance travelled per unit volume of fuel; formula: Fuel consumption = Distance / Volume.
Formula triangle: A diagram showing the relationship between three quantities in a compound measure; used to rearrange formulas.

Slides

Compound Measures

A compound measure uses more than one measurement, e.g., speed = distance ÷ time.
Common compound measures: speed, density, pressure.
Units like km/h or $g/cm^{3}$ tell you the formula (e.g., km/h means distance per time).

Which of the following is a compound measure?

Length
Speed
Mass
Time

Speed, Distance & Time

Speed = Distance ÷ Time (average speed).
Common units: km/h, m/s.
Use formula triangles: D on top, S and T on bottom.

To find distance when speed and time are known, use: distance = speed ____ time.

Density, Mass & Volume

Density = Mass ÷ Volume.
Common units: $g/cm^{3}, kg/m^{3}$ .
If density < water $(1 g/cm^{3})$ , the object floats.

A block has mass 500 g and volume $250 cm^{3}$ . What is its density?

$0.5 g/cm^{3}$
$2 g/cm^{3}$
$125 g/cm^{3}$
$750 g/cm^{3}$

Pressure, Force & Area

Pressure = Force ÷ Area.
Units: $\frac{N}{m}^{2} (Pascals)$ .
Force is often weight; area is contact area.

Which of these are true about pressure?

Pressure = Force × Area
Pressure is measured in $\frac{N}{m}^{2}$
Force is measured in Newtons
Area must be in $m^{2}$ for pressure in $\frac{N}{m}^{2}$

Converting Units

Always check units: e.g., if distance is in km and time in minutes, speed is km/min.
Convert to consistent units before calculating.
Example: 20 m/s to km/h: multiply by 3.6 → 72 km/h.

To convert m/s to km/h, multiply $by ____$ .

Worked Example: Speed

George drives 315 km in 3 hours 30 minutes.
Average speed = distance ÷ time $= 315 \div 3.5 = 90 km/h$ .
Remember: 30 minutes $= 0.5$ hours.

Ivan walks 1.5 km in 20 minutes. What is his average speed in km/h?

4.5 km/h
0.075 km/h
1.5 km/h
30 km/h

Worked Example: Density

A wood block: 8 $cm \times 20$ $cm \times 10$ cm, mass 1360 g.
Volume $= 8\times 20\times 10 = 1600 cm^{3}$ .
Density $= 1360 \div 1600 = 0.85 g/cm^{3} < 1$ , so it floats.

Which statements about the wood block are true?

Its volume is $1600 cm^{3}$
Its density is $0.85 g/cm^{3}$
It will sink because density $< 1$
Mass is 1360 g

Worked Example: Pressure

A force of 300 N on an area of $5 m^{2}$ .
Pressure $= 300 \div 5 = 60 \frac{N}{m}^{2}$ .
Use formula triangle: $P = \frac{F}{A}$ .

Pressure = Force ÷ ____.

Exam Tips

Watch for mixed units: e.g., km and minutes, or g and kg.
Convert all to same units before using formula.
Show working clearly, especially in multi-step problems.

A car travels 10 km at 20 m/s. How long does it take in minutes?

8 min 20 s
500 min
0.5 min
8.33 min

Flashcards

What is a compound measure?

A compound measure is calculated using more than one measurement, e.g., speed, density, pressure.

Write the formula for speed.

Speed = Distance / Time

Write the formula for density.

Density = Mass / Volume

Write the formula for pressure.

Pressure = Force / Area

A car travels 315 km in 3 hours 30 minutes. Calculate its average speed.

Average speed $= 315$ $km / 3.5 h = 90 km/h$

Convert 84 km/h into metres per minute.

$84 km/h = 84000 m / 60$ $min = 1400 m/min$

Ivan walks 1.5 km in 20 minutes. Find his average speed in km/h.

20 $min = \frac{1}{3} h$ , so speed $= 1.5$ $km / (\frac{1}{3} h) = 4.5 km/h$

An object exerts 300 N on an area of $5 m^{2}$ . Find the pressure.

Pressure $= 300 N / 5 m^{2} = 60 \frac{N}{m}^{2} (or 60 Pa)$

Ramond walks 2460 m in 33 minutes. Find average speed in km/h.

$2460 m = 2.46$ km, 33 $min = \frac{33}{60} h = 0.55 h$ , speed $= 2.46 / 0.55 = 4.4727$ ... ≈ 4.47 km/h

Kiran leaves at 9:45 am, drives 135 km, arrives at 11:15 am. Find average speed.

Time $= 1.5 h$ , speed $= 135$ $km / 1.5 h = 90 km/h$

A car travels at 20 m/s. How long to travel 10 km? Give answer in minutes and seconds.

10 $km = 10000 m$ , time $= 10000 / 20 = 500 s = 8$ minutes 20 seconds

A rectangle 0.8 m by 1.2 m has pressure $200 \frac{N}{m}^{2}$ . Find the force F.

Area $= 0.8 \times 1.2 = 0.96 m^{2}, F =$ Pressure × Area $= 200 \times 0.96 = 192 N$

A wood block 8 $cm \times 20$ $cm \times 10$ cm has mass 1360 g. Density of water is $1 g/cm^{3}$ . Will it float?

Volume $= 8\times 20\times 10 = 1600 cm^{3}$ , density $= \frac{1360}{1600} = 0.85 g/cm^{3} < 1$ , so it floats.

A plane flies 8352 km at 783 km/h. Leaves Melbourne 09:52 local. Tokyo is 2 hours behind. Find arrival time in Tokyo.

Time $= \frac{8352}{783} = 10.666$ ... $h = 10 h 40$ min. Depart 09:52 Melbourne $= 07:52$ Tokyo. Arrival $= 07:52 + 10:40 = 18:32$ Tokyo time.

A car travels at 45 km/h for 5 minutes. Wheel radius 25 cm. Find number of complete revolutions.

Distance $= 45 km/h \times (\frac{5}{60}) h = 3.75$ $km = 375000$ cm. Circumference $= 2\pi \times 25$ ≈ 157.08 cm. Revolutions $= 375000 / 157.08$ ≈ 2387.

A plane flies 375 km at 450 km/h. Leaves at 14:45. Find arrival time.

Time $= \frac{375}{450} = 0.8333$ ... $h = 50$ min. Arrival $= 14:45 + 50$ $min = 15:35$ .

A force of 300 N on a rectangle of width 20 cm and length x cm gives pressure $0.6 N/cm^{2}$ . Find x.

Area $= Force/Pressure = \frac{300}{0.6} = 500 cm^{2}$ . Area $= 20 \times x$ , so $x = \frac{500}{20} = 25$ cm.

A wood block $0.2 m \times 0.2 m \times 0.1 m$ has mass 3 kg. Density of water is $1 g/cm^{3}$ . Will it float?

Volume $= 0.2\times 0.2\times 0.1 = 0.004 m^{3} = 4000 cm^{3}$ . Mass $= 3$ $kg = 3000 g$ . Density $= \frac{3000}{4000} = 0.75 g/cm^{3} < 1$ , so it floats.

What is the formula triangle for speed, distance, and time?

Distance on top, Speed and Time on bottom. Distance = Speed × Time; Speed = Distance / Time; Time = Distance / Speed.

What are typical units for speed?

Metres per second (m/s) or kilometres per hour (km/h).

What are typical units for density?

Grams per centimetre cubed $(g/cm^{3})$ or kilograms per metre cubed $(kg/m^{3})$ .

What are typical units for pressure?

Newtons per square metre $(\frac{N}{m}^{2})$ , also called Pascals (Pa).

How do you convert km/h to m/s?

Multiply by 1000 and divide by 3600 (or divide by 3.6).

How do you convert m/s to km/h?

Multiply by 3600 and divide by 1000 (or multiply by 3.6).

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