Ratio, proportion and rates

• Write ratios in their simplest form (20 : 30 : 40 = 2 : 3 : 4), including with different units and with fractions.
• Divide a quantity in a given ratio, and use a given difference or a given part.
• Use proportional reasoning: adapt recipes, use map scales and find the best value.
• Use rates: pay, exchange rates, flow rates, fuel consumption, pressure, density and population density.
• Solve average speed problems; the speed–distance–time formula must be known.

Divide every part by the HCF

Divide all parts of a ratio by their highest common factor. Make the units the same first, and clear decimals or fractions by multiplying every part by the same number.

Total parts, value of one part, then each share

Add the parts, divide the quantity by the total to get one part, then multiply. If the question gives a difference or one share instead of the total, use that to find one part.

Find one unit, then scale

For a recipe, find the amount for one person (or one item), then multiply. A map scale 1 : 50 000 means 1 cm on the map is 50 000 cm in real life. For best value, compare the price per unit (or units per dollar).

An amount per unit

A rate is one quantity per unit of another. Multiply by the rate to go one way; divide by it to go back. Read the units: “per” means divide.

Formulas are given in the question

For these rates the exam gives the formula in the question. Units are written like g/cm³ (grams per cubic centimetre).