- Simplify a ratio by dividing every part by the same number, their highest common factor.
- Put both parts into the same units before you simplify.
- For the form 1 to n, divide both parts by the first one.
- A ratio has no units and is not a fraction, though the two are closely linked.
Watch out: Dividing only one part of the ratio by the common factor, instead of dividing every part by the same number.
- Add the ratio parts to find the total number of shares.
- Divide the amount by the total shares to find what one share is worth.
- Multiply one share by each part of the ratio.
- Check that your amounts add back up to the original total.
Watch out: Dividing the amount by the number of shares (for example by 2) instead of by the total number of parts in the ratio.
- In the ratio 2 to 3, the first part is 2/5 of the total and the second is 3/5.
- The denominator of the fraction is the sum of the ratio parts.
- "2/5 are red" means the ratio of red to the rest is 2 to 3.
- Convert between ratio and fraction by comparing a part to the whole.
Watch out: Using the total number of parts as the denominator when the question asks for a fraction of just one other part, rather than of the whole.
- A scale of 1 to 50000 means 1 cm on the map is 50000 cm in real life.
- Convert the real distance into sensible units, usually metres or kilometres.
- Map to real life means multiply by the scale; real life to map means divide.
- A scale factor has no units because both sides measure the same thing.
Watch out: Forgetting to convert units, such as cm to km, before or after applying the scale factor.
- In direct proportion, if one quantity doubles the other doubles: y = k x.
- Find the constant k by dividing a known y by its x, then use it for other values.
- A direct proportion graph is a straight line through the origin.
- The unitary method finds the value of one item first, then scales up.
Watch out: Finding the value of one unit correctly, but then multiplying by the wrong number when scaling up or down.
- In inverse proportion, as one quantity goes up the other goes down: y = k over x.
- The product x y stays constant, so find k by multiplying a known pair.
- More workers means less time; go faster and the journey time falls.
- The graph is a reciprocal curve, not a straight line.
Watch out: Treating inverse proportion like direct proportion, multiplying by a scale factor instead of dividing, or dividing instead of multiplying.
- Work out the price per unit (per gram or per ml) for each option and compare.
- Or work out how much you get for each pound spent.
- The lowest price per unit is the best value, unless the question adds another condition.
- Keep the units the same across all options before comparing.
Watch out: Comparing prices per pack directly, instead of converting each option to a common unit, such as price per 100 g or price per item.
- Speed is distance over time, density is mass over volume, pressure is force over area.
- Rearrange the formula for whichever quantity you need.
- Units must match: convert to consistent units before you substitute.
- A formula triangle helps you rearrange these three-part relationships.
Watch out: Using the wrong rearrangement of the formula, such as dividing distance by time when time is actually the missing value.
- A conversion graph is a straight line used to change between two units.
- Read across then up, or up then across, and always state the units.
- The gradient is the conversion rate between the two quantities.
- Extend the line carefully if you need a value beyond the plotted points.
Watch out: Reading the axes the wrong way round, converting in the opposite direction to the one the question asks for.
- Repeated percentage change uses a multiplier raised to a power.
- Growth uses a multiplier above 1; decay uses a multiplier below 1.
- amount equals start x multiplier to the power of the number of time periods.
- It is the same idea as compound interest, applied to any context.
Watch out: Using the wrong multiplier, such as using 1.08 for an 8% decrease instead of 0.92, or the reverse for a growth problem.
