How does it work?
The game has two parts:
- Step-by-step lesson – you sweeten your coffee, try to work out what percentage of it is sugar, and discover why that doesn't work straight away. Then you weigh things on a scale, measure the sugar's volume in a measuring cylinder, and calculate both percentages. At the end you can experiment with sliders.
- Practice – 8 questions: mass percent, volume percent, converting with density, and a few "why?" questions. Every answer shows the working.
The key idea: you need the same unit
A percentage shows what part of the whole something is. For that, the part and the whole must be in the same unit. You can't add 10 g of sugar to 200 ml of coffee, or divide one by the other. There are two ways out:
- Mass percent: measure everything in grams. Coffee is almost all water, so 1 ml weighs about 1 g and 200 ml of coffee is 200 g. The solution is 210 g, and the mass percent is 10 ÷ 210 × 100 ≈ 4.8%.
- Volume percent: measure everything in millilitres. Sugar's density is 1.59 g/cm³, so 10 g of sugar has a volume of 10 ÷ 1.59 ≈ 6.3 cm³. The solution is about 206.3 ml, and the volume percent is 6.3 ÷ 206.3 × 100 ≈ 3.1%.
Same coffee, two different numbers. The reason is density: sugar is heavy for its size, so it makes up a bigger share by mass than by volume.
Formulas
- Mass percent = mass of solute ÷ mass of solution × 100
- Volume percent = volume of solute ÷ volume of solution × 100
- Density: ρ = m ÷ V, so V = m ÷ ρ and m = V × ρ
- Solution = solute + solvent (in the coffee: sugar + coffee)
- 1 ml = 1 cm³
A simplification
In the game we assume the volumes of the sugar and the coffee simply add up. In reality the volume of a solution is a little less than the sum of the parts (this is volume contraction, best known from mixing alcohol and water). That is why volume percent is mostly used for mixtures of liquids, such as the alcohol content of drinks (ABV). It doesn't change the main point: why you have to convert with density.
Frequently asked questions
What is the difference between mass percent and volume percent?
Mass percent tells you what percentage of the solution's mass is the solute; volume percent tells you what percentage of its volume is. For the same solution the two numbers are usually different, because the solute and the solvent have different densities.
How do you calculate mass percent?
Mass percent = mass of solute ÷ mass of solution × 100. The mass of the solution is the mass of the solute plus the mass of the solvent. For example, 10 g of sugar in 200 g of coffee: 10 ÷ 210 × 100 ≈ 4.8%.
How do you calculate volume percent?
Volume percent = volume of solute ÷ volume of solution × 100. If you only know the mass of the solute, first work out its volume from its density: V = m ÷ ρ.
Why do you need the density for volume percent?
Because for a percentage the part and the whole must be in the same unit. If you weighed the sugar in grams and measured the coffee in millilitres, you have to turn the mass of the sugar into a volume. That is what density does: it tells you how many grams 1 cm³ of a substance weighs.
Why is the volume percent of sweet coffee lower than its mass percent?
Because sugar's density (1.59 g/cm³) is higher than coffee's (about 1 g/cm³). The same amount of sugar makes up a bigger share by mass than by volume. Alcohol is the other way round: it is less dense than water, so its volume percent is the bigger number.
Who is it for?
Mass and volume percent are usually taught at ages 12–14, but the guided lesson is also a good refresher for older students. It is free, needs no sign-up and works on desktop and mobile.
Want to practise unit conversions? Metric conversion covers grams, kilograms, millilitres and litres.