A compound measure combines two different measurements into one rate. You will always spot it because its units contain the word "per" or a slash.
| Measure | Formula | Units |
|---|---|---|
| Speed | $S = \dfrac{D}{T}$ | m/s, km/h, mph |
| Density | $\rho = \dfrac{M}{V}$ | g/cmยณ, kg/mยณ |
| Pressure | $P = \dfrac{F}{A}$ | N/mยฒ (pascal), N/cmยฒ |
| Rate of pay | $\dfrac{\text{pay}}{\text{hours}}$ | ยฃ/hour |
| Unit price | $\dfrac{\text{cost}}{\text{amount}}$ | p/g, ยฃ/litre |
| Population density | $\dfrac{\text{people}}{\text{area}}$ | people/kmยฒ |
| Flow rate | $\dfrac{\text{volume}}{\text{time}}$ | litres/min |
A formula triangle helps you rearrange. Cover the quantity you want, and the triangle shows you what to do with the other two.
Cover $S$ and you see $D$ over $T$, so $S = \dfrac{D}{T}$.
Cover $T$ and you see $D$ over $S$, so $T = \dfrac{D}{S}$.
- Write down what you know and what you want.
- Check the units match (convert minutes to hours if the speed is in km/h).
- Choose the right arrangement of $S = \dfrac{D}{T}$.
- Calculate, then convert the answer to the units asked for.
A train travels $273$ km in $3$ hours $30$ minutes. Find its average speed in km/h.
Answer: $78$ km/h
A cyclist rides $63$ km at an average speed of $18$ km/h. How long does the ride take, in hours and minutes?
Answer: $3$ hours $30$ minutes
A driver travels $80$ km at $80$ km/h, then $80$ km at $40$ km/h. Find the average speed for the whole journey.
A steel bar has volume $250\text{ cm}^3$. Steel has density $7.8\text{ g/cm}^3$. Find the mass of the bar in kilograms.
A solid cylinder of radius $4$ cm and height $10$ cm has mass $1.6$ kg. Find its density in g/cmยณ, to 2 decimal places.
Answer: $3.18\text{ g/cm}^3$
A box exerts a force of $840$ N on the floor through a base measuring $1.2$ m by $0.5$ m. Find the pressure in N/mยฒ.
Answer: $1400\text{ N/m}^2$ (1400 Pa)
A machine must not exert more than $250\text{ N/cm}^2$ on a surface. It presses with a force of $6000$ N. What is the smallest area of contact allowed?
A smaller area would push the pressure above the limit.
Jamal is paid ยฃ$11.60$ per hour for the first $35$ hours each week, and time-and-a-half after that. One week he works $42$ hours. Find his pay.
A county has a population of $486\,000$ and an area of $2700\text{ km}^2$. Find its population density.
Answer: $180$ people per kmยฒ
Water flows into a tank at $18$ litres per minute. The tank is a cuboid measuring $1.5$ m by $0.8$ m by $0.75$ m. How long does it take to fill, to the nearest minute?
Speed
$S = \dfrac{D}{T}$, $D = ST$, $T = \dfrac{D}{S}$.
Density
$\rho = \dfrac{M}{V}$, $M = \rho V$, $V = \dfrac{M}{\rho}$.
Pressure
$P = \dfrac{F}{A}$, $F = PA$, $A = \dfrac{F}{P}$.
Average speed
Total distance $\div$ total time โ never average the speeds.
Time as a decimal
Minutes $\div 60$. $1$ h $45$ min $= 1.75$ h.
Units first
Match the units to the rate before calculating.
km/h โ m/s
$\div 3.6$ to go to m/s; $\times 3.6$ to come back.
Read the unit
"per" means divide โ the unit is the formula.
A coach travels $153$ km in $2$ hours $15$ minutes. Find its average speed in km/h.
โถ Show solution
$T = 2 + \dfrac{15}{60} = 2.25$ hours.
$S = \dfrac{153}{2.25} = 68$
Answer: $68$ km/h
A runner keeps a steady speed of $4.5$ m/s for $12$ minutes. How far does she run, in kilometres?
โถ Show solution
$12$ minutes $= 12 \times 60 = 720$ seconds.
$D = S \times T = 4.5 \times 720 = 3240$ m
$= 3.24$ km
A block of wood has mass $432$ g and volume $540\text{ cm}^3$. Find its density, and say whether it will float in water (density $1\text{ g/cm}^3$).
โถ Show solution
$\rho = \dfrac{432}{540} = 0.8\text{ g/cm}^3$
$0.8 < 1$, so the block is less dense than water and it will float.
Gold has a density of $19.3\text{ g/cm}^3$. Find the volume of a gold bar of mass $2$ kg, to 1 decimal place.
โถ Show solution
$2$ kg $= 2000$ g.
$V = \dfrac{M}{\rho} = \dfrac{2000}{19.3} = 103.626\ldots$
Answer: $103.6\text{ cm}^3$
A crate weighing $1560$ N stands on a square base of side $0.4$ m. Find the pressure it exerts in N/mยฒ.
โถ Show solution
Area $= 0.4 \times 0.4 = 0.16\text{ m}^2$
$P = \dfrac{1560}{0.16} = 9750$
Answer: $9750\text{ N/m}^2$
Convert a speed of $25$ m/s into km/h.
โถ Show solution
$25 \times 3.6 = 90$
Answer: $90$ km/h
Full working: $25$ m/s $= 25 \times 3600 = 90\,000$ m per hour $= 90$ km/h โ
A driver travels $120$ km at $60$ km/h and then $120$ km at $120$ km/h. Find the average speed for the whole journey.
โถ Show solution
Stage 1 time $= 120 \div 60 = 2$ hours.
Stage 2 time $= 120 \div 120 = 1$ hour.
Total distance $= 240$ km; total time $= 3$ hours.
Average speed $= 240 \div 3 = \mathbf{80}$ km/h.
(Note this is not $90$ km/h โ the average of $60$ and $120$.)
Sofia is paid ยฃ$13.20$ per hour for a $37$-hour week, plus double time for weekend hours. One week she works $37$ hours plus $6$ hours at the weekend. Find her total pay.
โถ Show solution
Basic $= 37 \times 13.20 = ยฃ488.40$
Weekend rate $= 13.20 \times 2 = ยฃ26.40$ per hour
Weekend pay $= 6 \times 26.40 = ยฃ158.40$
Total $= 488.40 + 158.40 = ยฃ646.80$
A swimming pool is $25$ m long, $10$ m wide and has a uniform depth of $1.4$ m. Water flows in at $450$ litres per minute. How long, to the nearest hour, does it take to fill?
โถ Show solution
Volume $= 25 \times 10 \times 1.4 = 350\text{ m}^3$
$350\text{ m}^3 = 350 \times 1000 = 350\,000$ litres
Time $= 350\,000 \div 450 = 777.78$ minutes
$777.78 \div 60 = 12.96$ hours
Answer: about $13$ hours
An alloy is made by mixing $300\text{ cm}^3$ of metal A (density $8.9\text{ g/cm}^3$) with $200\text{ cm}^3$ of metal B (density $7.1\text{ g/cm}^3$). Assuming the volumes simply add, find the density of the alloy.
โถ Show solution
Mass of A $= 8.9 \times 300 = 2670$ g
Mass of B $= 7.1 \times 200 = 1420$ g
Total mass $= 2670 + 1420 = 4090$ g
Total volume $= 300 + 200 = 500\text{ cm}^3$
Density $= \dfrac{4090}{500} = \mathbf{8.18\text{ g/cm}^3}$
Sense check: the answer lies between $7.1$ and $8.9$, and closer to $8.9$ because there is more of metal A โ