What is a compound unit?
A unit is what we measure in, like metres (m) or seconds (s). A compound unit is made by joining two units.
- Quotient (divide) units use "per": m/s, km/h, g/cm³, kg/m³, L/min, coins/kg. They tell you how much in ONE unit.
- Product (multiply) units join by multiplying: kilowatt-hours (kWh), person-days, newton-metres. 3 workers × 4 days = 12 person-days of work.
The word per means "for each one" and is written with a slash (/).
Speed, density and pressure
Speed = distance ÷ time. Units: m/s or km/h. Turn it round: distance = speed × time, time = distance ÷ speed.
Density = mass ÷ volume. Units: g/cm³ or kg/m³. Water is about 1 g/cm³; iron is about 7.9 g/cm³. Things less dense than water float.
Pressure = force ÷ area. Unit: N/m² (also called pascal, Pa). A sharp knife has a tiny area, so the same push makes a big pressure.
Tip: draw a triangle with the top quantity on top (distance, mass, force) and the two others below. Cover what you want.
Rates in daily life: unit price, flow and concentration
Unit price = cost ÷ amount. It lets you compare packs of different sizes: work out the price per 1 kg (or per 100 g) for each, and the smaller number is the better buy.
Flow rate = volume ÷ time (L/min). Concentration = mass ÷ volume of liquid (g/L). Pay rate = money ÷ hours. Population density = people ÷ area (people/km²).
Converting compound units
Convert the top and the bottom separately.
- Write the rate as a fraction: 72 km/h = 72 km ÷ 1 h.
- Change the top: 72 km = 72 000 m.
- Change the bottom: 1 h = 3600 s.
- Divide: 72 000 ÷ 3600 = 20 m/s.
Shortcut: km/h → m/s divide by 3.6; m/s → km/h multiply by 3.6. For density, 1 g/cm³ = 1000 kg/m³.
Check by units: if your answer to "find a speed" comes out in m·s instead of m/s, you multiplied when you should have divided.
Try it at home
Walk 20 big steps and time yourself with a phone. Measure one step with a ruler. Work out your walking speed in m/s, then turn it into km/h by multiplying by 3.6. Most people walk about 1.2–1.5 m/s.
Key formulas and definitions
- speed = distance ÷ time
- density = mass ÷ volume
- pressure = force ÷ area
- unit price = cost ÷ amount
- flow rate = volume ÷ time
- km/h ÷ 3.6 = m/s
- 1 g/cm³ = 1000 kg/m³
Worked examples
1. A cyclist rides 45 km in 3 hours. Find the average speed.
Speed = 45 ÷ 3 = 15 km/h.
2. A block has mass 540 g and volume 200 cm³. Find its density.
Density = 540 ÷ 200 = 2.7 g/cm³ (this is aluminium).
3. Rice: 2 kg for ₹130 or 5 kg for ₹310. Which is better value?
130 ÷ 2 = ₹65 per kg; 310 ÷ 5 = ₹62 per kg. The 5 kg bag is better value.
4. A car travels at 25 m/s for 2 minutes. How far does it go?
Time = 2 × 60 = 120 s. Distance = 25 × 120 = 3000 m = 3 km.
5. Convert 54 km/h into m/s.
54 km = 54 000 m; 1 h = 3600 s. 54 000 ÷ 3600 = 15 m/s (or 54 ÷ 3.6 = 15).
6. A tap fills a 300 L tank at 12 L/min. How long does it take?
Time = volume ÷ rate = 300 ÷ 12 = 25 minutes.
Common mistakes
- Multiplying instead of dividing. "Per" always means divide: m/s = m ÷ s.
- Mixing units: using minutes with km/h. Change minutes to hours first (30 min = 0.5 h).
- Converting only the top: 72 km/h is not 72 000 m/s. The hours must become seconds too.
- Comparing pack prices without working out the price for the same amount (per 1 kg or per 100 g).