Ratio Problem SolvingEdexcel IGCSE Maths: Revision notes

Section 1

How do you use direct proportion in ratio problems?

Two quantities are in direct proportion if their ratio stays constant as they scale — double one, double the other. If $y$ is directly proportional to $x$ , write $y \propto x$ , which means $y = kx$ for a constant $k$ (the constant of proportionality).

Steps to solve:

Find $k$ using one known pair of values: $k = \dfrac{y}{x}$
Write the formula $y = kx$
Substitute the new value to find the unknown

Quantity type	Example
Direct proportion	Cost of petrol vs litres bought
Direct proportion	Ingredients in a recipe vs number of servings

Key termsdirect proportionconstant of proportionality

Example

If 5 kg of flour costs £4, find the cost of 8 kg. $k = 4 \div 5 = 0.8$ , so cost $= 0.8 \times 8 = £6.40$ .

Exam tip

Always find the 'unit value' (amount per 1 unit) first — it makes scaling to any new amount straightforward.

Section 2

How do you solve best-value problems?

Best-value questions compare prices of different pack sizes to find which gives more for your money. Convert every price to a unit price (cost per single item or per gram/ml) so all options are on the same scale.

Method:

Divide the total price by the total quantity for each option
Compare the unit prices directly — the smallest unit price is the best value
Alternatively, compare quantity per £1 — the largest quantity per £1 is the best value

Be careful with mixed units (e.g. grams vs kilograms) — convert to the same unit before dividing.

Key termsunit pricebest value

Example

A 400 g jar costs £2.40 (60p per 100 g); a 600 g jar costs £3.30 (55p per 100 g). The 600 g jar is better value.

Common mistake

Do not compare total prices alone — a bigger pack costing more is not automatically worse value; always compare per-unit cost.

Section 3

How do you use ratios in scale drawings and maps?

A scale relates a length on a drawing or map to the real-life length it represents, usually written as a ratio, e.g. $1 : 25000$ means 1 cm on the map represents 25000 cm (250 m) in real life.

To find real-life length: map length $\times$ scale factor

To find map length: real-life length $\div$ scale factor

Always convert to the same units first (usually cm), then convert the final answer to a sensible unit (m or km).

Key termsscalescale factor

Example

On a map with scale $1 : 50000$ , a distance of 3 cm represents $3 \times 50000 = 150000$ cm $= 1.5$ km in real life.

Think of it like this

Think of a scale like a photocopier zoom setting — everything on the map has been shrunk by the same factor, so you just reverse the zoom to get real sizes.

Section 4

How do you divide a quantity in a given ratio?

To share an amount in a ratio $a : b$ :

Add the parts: total parts $= a + b$
Find the value of one part: total amount $\div$ total parts
Multiply each ratio part by this value

This extends to three or more parts, and to problems where you are given one share and must find the total or the other shares.

Key termsratioparts

Example

Share £60 in the ratio $2:3:5$ . Total parts $= 10$ , so 1 part $= £6$ . Shares are £12, £18 and £30.

Exam tip

If you're told one share's actual value, divide it by its number of parts to find the value of 1 part, then scale up the rest.

Section 5

How do inverse proportion problems differ from direct ones?

In inverse proportion, as one quantity increases, the other decreases at a matching rate, so their product stays constant: $y = \dfrac{k}{x}$ , or $xy = k$ .

Common exam contexts: workers and time to complete a job, speed and time for a fixed distance.

Method: find $k$ by multiplying a known pair, then use $y = \dfrac{k}{x}$ for the new value.

Key termsinverse proportion

Common mistake

Do not apply the direct proportion method (multiply/divide by the same scale factor) to an inverse situation — check whether the context implies 'more of one means less of the other' first.

Must Know

Direct proportion: $y = kx$ ; find $k$ from one pair, then scale
Inverse proportion: $y = \dfrac{k}{x}$ , so $xy$ stays constant
Best value: always compare unit prices (price per single item or per 100 g/ml), not total prices
Scale drawings: map length $\times$ scale factor $=$ real length; convert units carefully (cm, m, km)
To divide in a ratio: total parts $\to$ value of 1 part $\to$ multiply by each ratio number
Watch units throughout — mixing cm and km, or g and kg, is the most common error

That's the notes covered.

Carry on to the next subtopic.