Ratio, proportion and rates of change

A ratio compares quantities. Simplify it by dividing every part by their highest common factor, and keep the parts in the order the question gives them.

Ratios must be in the same units before simplifying: 50 cm : 2 m is not 50 : 2 but 50 : 200, which simplifies to 1 : 4.

Writing a ratio in the form 1 : n means dividing both parts by the first, which is how ratios are compared — the one with the larger n has proportionally more of the second quantity.

Sharing in a given ratio has one reliable method: add the parts to get the total number of parts, divide the quantity by that to find the value of one part, then multiply for each share. Check by adding the shares back to the original total.

A ratio can also be read as a fraction of the whole: a : b means the first share is a⁄(a+b), which is what turns most ratio questions into a single multiplication. Expressing one quantity as a fraction of another works the same way, and the fraction can be greater than 1 if the first is larger.

Scale factors and maps use ratio directly. A scale of 1 : 25 000 means 1 cm on the map is 25 000 cm on the ground, which is 250 m. Convert the units at the end, not during, and remember from Measures and accuracy that areas scale by the square of the length scale factor.

Proportion is equality of ratios: if two ratios are equal, the quantities are in proportion. That is the basis of every scaling and recipe question — set up the two ratios, and solve for the missing part.

Percentage means parts per hundred. The multiplier method handles every percentage question and is worth using exclusively, because it makes the harder types no harder than the easy ones.

To find a percentage of an amount, multiply by the decimal: 15% of 60 is 0.15 × 60 = 9.

To increase by a percentage, multiply by 1 plus the decimal — a 20% rise is × 1.2. To decrease, multiply by 1 minus the decimal — a 20% fall is × 0.8.

To express one quantity as a percentage of another, divide and multiply by 100. Percentage change is the change divided by the original, times 100 — and it is always the original, not the new value.

Percentages above 100% are allowed, and mean more than the whole: a multiplier of 2.5 is a 150% increase.

Simple interest is calculated on the original amount every year, so it adds the same sum each time. Compound interest is calculated on the running total, so the amount grows faster each year — and it is the one examined most.

For compound growth or decay over n periods:

final amount = starting amount × (multiplier)ⁿ

A depreciating car losing 12% a year is × 0.88 each year, so after 4 years it is worth 0.88⁴ of its original value. The same formula covers population growth, bacterial growth and radioactive decay — only the multiplier changes.

Every proportion question follows three steps: write the equation with k in it, substitute the given pair of values to find k, then use the completed equation to answer what was asked. Finding k is the step students skip, and without it nothing else works.

Compound units combine two measures, and each has a formula triangle worth reconstructing rather than memorising — cover the quantity you want and the arrangement of the other two tells you whether to multiply or divide.

• Speed = distance ÷ time, so distance = speed × time • Density = mass ÷ volume, so mass = density × volume • Pressure = force ÷ area, so force = pressure × area

Also in this group are rates of pay, unit pricing and exchange rates. Unit pricing answers "which is better value" questions: work out the cost of one unit for each option and compare — the same amount of each, compared like with like.

Units must be consistent. A speed in km/h with a time in minutes needs one of them converting first, and that conversion is where the marks are lost.

The gradient of a straight-line graph is a rate of change, and what it represents depends on the axes — distance against time gives speed, cost against quantity gives price per unit.

On a curve the rate of change is not constant, so the specification distinguishes two things.

The average rate of change between two points is the gradient of the chord joining them — the straight line between the two points on the curve.

The instantaneous rate of change at a single point is the gradient of the tangent at that point — the straight line touching the curve there. Drawing the tangent by eye makes this an estimate, and the mark scheme allows a range.

On a velocity–time graph the tangent gradient is the acceleration at that instant, and on a distance–time graph it is the speed at that instant. None of this uses calculus.