Ratio & Proportion

Video coming soon3 worked examples

Theory

The Golden Rule: The most critical step in any proportion question is stopping to ask yourself: "Is this Direct or Inverse proportion?" If you apply the wrong rule, you will lose almost all the marks for that question.

1. Simplifying Ratios

Ratios must often be written in their simplest form. Just like fractions, you must find the highest common factor and divide both (or all) sides of the ratio by the same number.

2. Sharing in a Ratio (When given the TOTAL amount)

To split a total quantity into a given ratio:

Step 1: Add the numbers in the ratio together to find the total number of parts.
Step 2: Divide the total amount by this number to find the value of one part.
Step 3: Multiply the value of one part by each specific number in the ratio.

3. Sharing in a Ratio (When given ONE PERSON'S share)

Sometimes you are given the amount one person received, rather than the total.

Step 1: Divide their monetary amount by their specific number of ratio parts to find the value of 1 part.
Step 2: Multiply this value by the total number of parts to find the overall total.

4. Direct Proportion

In direct proportion, as one quantity increases, the other increases at the exact same rate. (For example, buying 5 apples costs more than buying 1 apple).

The Method: Divide to find the value of ONE unit, then multiply by the number of units you need. Ensure your units are consistent before calculating.

5. Inverse (Indirect) Proportion

In inverse proportion, as one quantity increases, the other decreases. (For example, if you hire more painters, the time it takes to paint a room decreases).

The Method: Multiply the two given numbers together to find the "total effort" (e.g., worker-hours). Then, divide this total by the new number of workers/machines to find the new time.

⚠️ Common Examiner Traps

The "Direct vs. Inverse" Trap: Candidates frequently read an inverse proportion question (like workers building a wall) and treat it as direct proportion. If your calculation results in 6 workers taking longer to build a wall than 3 workers, you have fallen into the trap.
The "Extra Workers" Trap: a common question states that a company sent "2 extra workers" to help with a job. Candidates frequently divide the total worker-hours by 2, completely forgetting that they must add the 2 extra workers to the original team first.
The "Given Share" Trap: When given a ratio (e.g., 2:3:4) and told that the last person received £120, candidates often mistakenly divide £120 by the total parts (9). You must only divide it by their specific parts (4).

Worked examples

Example 1

Alice, Ben, and Charlie are business partners. They share the annual profits of their business in the ratio 4:3:5. This year, Charlie received a share of £1,850. Calculate the total profit the business made this year. (2 marks)

Step 1: Calculate the value of exactly 1 part. Since Charlie gets 5 parts and his share is £1,850, we divide his amount by his parts.

\pounds1,850 \div 5 = \pounds370 \text{ (This is 1 part)}

Step 2: Calculate the total number of parts in the ratio.

4 + 3 + 5 = 12 \text{ parts total}

Step 3: Multiply the value of 1 part by the total number of parts.

\pounds370 \times 12 = \pounds4,440

Final Answer: The total profit was £4,440.

Example 2

A gardener uses liquid plant food to feed a large greenhouse. The instructions state that 8 ml of plant food must be used for every 5,000 ml of water. Calculate the volume of plant food required for 12 litres of water. (3 marks)

Step 1: Ensure units match. Convert 12 litres into millilitres.

12 \text{ litres} = 12,000 \text{ ml}

Step 2: Find out how much water 1 ml of plant food treats (Divide).

5,000 \div 8 = 625 \text{ ml of water per 1 ml of food}

Step 3: Divide the total required water by this unit rate.

12,000 \div 625 = 19.2 \text{ ml}

(Alternative method: $12,000 \div 5,000 = 2.4$ , then $2.4 \times 8 = 19.2$ ).

Final Answer: 19.2 ml of plant food is required.

Example 3

A school hires a contractor to refurbish the science labs. It normally takes a team of 4 workers exactly 15 hours to complete this task. To finish the job faster, the contractor sends 2 additional workers. All workers work at the exact same rate. Calculate how long it will take the new team to complete the refurbishment. (3 marks)

Step 1: Recognise this is inverse proportion (more workers = less time) and calculate the "total worker-hours" required for the job.

4 \text{ workers} \times 15 \text{ hours} = 60 \text{ worker-hours}

Step 2: Calculate the new total number of workers. Avoid the trap—you must add the extra workers to the original team!

4 + 2 = 6 \text{ workers}

Step 3: Divide the total worker-hours by the new number of workers.

60 \div 6 - 10 \text{ hours}

Final Answer: It will take the new team 10 hours to complete the task.

Ready to practise?

10 Ratio and Proportion questions with answers and video solutions