Revision notes

Every Ratio, Proportion & Rates of Change topic condensed to one page: the facts to know, the mistakes to avoid, and the formulae. Use it as a last-minute check, or print it.

Formulae for this unit

Speed $\dfrac{\text{distance}}{\text{time}}$
Density $\dfrac{\text{mass}}{\text{volume}}$
Pressure $\dfrac{\text{force}}{\text{area}}$

Simplifying Ratios

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.

Sharing in a Given Ratio

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.

Ratio and Fractions

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.

Scale Diagrams & Maps

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.

Direct Proportion

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.

Inverse Proportion

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.

Best Buy Problems

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.

Compound Measures

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.

Conversion Graphs

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.

Growth & Decay

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.

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