Density = mass ÷ volume. It tells you how much mass is packed into each unit of volume, and it is a compound quantity just like speed.
Density questions appear with solids, liquids and everyday objects, and they need the unit discipline from converting compound units.
How do you keep the units consistent?
The unit of the answer comes from the units you use. Mass in g and volume in cm³ gives g/cm³. Mass in kg and volume in m³ gives kg/m³.
The safe method is:
- Write the units next to each number before you divide.
- Choose one pair (g with cm³, or kg with m³) and convert anything that does not match.
- Calculate, then write the unit in the answer.
- Sanity check against a known value. Water is about 1 g/cm³, or 1000 kg/m³.
Useful conversions: 1 m³ = 1 000 000 cm³, 1 litre = 1000 cm³, and 1 g/cm³ = 1000 kg/m³.
Worked example
A metal block measures 5 cm by 4 cm by 3 cm and has a mass of 474 g.
Step 1, volume: 5 × 4 × 3 = 60 cm³.
Step 2, density: 474 ÷ 60 = 7.9 g/cm³.
Step 3, convert to kg/m³: 7.9 × 1000 = 7900 kg/m³.
Step 4, reverse check: 7900 kg/m³ is 7.9 times the density of water, which matches 7.9 g/cm³.
Now find a mass. A liquid has density 0.8 g/cm³ and the volume is 250 cm³. Mass = density × volume = 0.8 × 250 = 200 g.
The mistake to watch for
A common slip is to use mass in kg with a volume in cm³ and still label the answer g/cm³.
Mistaken answer: mass 0.474 kg, volume 60 cm³, density = 0.474 ÷ 60 = 0.0079 g/cm³
The mass was in kg but the label says g. The answer is 1000 times too small.
The correction is to convert 0.474 kg to 474 g first, then divide: 474 ÷ 60 = 7.9 g/cm³. Writing the unit beside each number before dividing catches this.
Check yourself
Try these, then open each answer.
1. A stone has mass 360 g and volume 150 cm³. Find its density.
Show answer
360 ÷ 150 = 2.4 g/cm³.
2. Find the mass of 250 cm³ of oil with density 0.8 g/cm³.
Show answer
Mass = 0.8 × 250 = 200 g.
3. A material has density 2700 kg/m³. A block has mass 13.5 kg. Find its volume in m³ and in cm³.
Show answer
Volume = 13.5 ÷ 2700 = 0.005 m³. In cm³: 0.005 × 1 000 000 = 5000 cm³.
Where this leads next
Continue with distinguishing an average rate from an instantaneous reading, or test yourself on the rates mixed practice set. The non-calculator working trainer and the percentage-base explorer let you check your own method.
Some students get the formula right and lose the marks in the unit. A teacher in online one-to-one Mathematics tuition can watch for that and help you fix it.