Sheet № 231 · Foundation + Higher · AQA · Edexcel · OCR
Proportional Reasoning –
Proportional reasoning is a fundamental skill in GCSE Maths that underpins many topics including ratio, percentage, speed, density and unit conversions. It involves recognising when two quantities increase or decrease at the same rate and using this relationship to find unknown values. Whether a question asks you to scale up a recipe, com
§Key definitions
Proportional reasoning
is the ability to see that two quantities are connected by a constant multiplier. If one quantity doubles, the other also doubles. If one is halved, the other is halved too.
Question:
5 notebooks cost £8.50. How much do 12 notebooks cost?
Answer:
12 notebooks cost £20.40.
Q1 (Foundation):
8 pens cost £6.00. How much do 5 pens cost?
Q2 (Foundation):
A car uses 6 litres of petrol to travel 78 km. How far can it travel on 10 litres?
§Formulas to memorise
Value for 1 unit = Total value / Number of units
Value for n units = Value for 1 unit x n
Scale factor = New quantity / Original quantity
Proportional reasoning: is the ability to see that two quantities are connected by a constant multiplier. If one quantity doubles, the other also doubles. If one is halved, the other is halved too.
Identify — what you know: a total value and how many units it corresponds to.
Divide — to find the value for one unit.
Multiply — the one-unit value by the number of units you need.
Find the scale factor — by dividing the target quantity by the known quantity.
Apply the scale factor — to the other quantity.
Set up a ratio — from the "for every" statement (e.g., "for every 3 red there are 5 blue" means ratio 3 : 5).
Worked example
5 notebooks cost £8.50. How much do 12 notebooks cost?
Working:
⚠ Common mistakes
- ✗Assuming a relationship is proportional when it is not. Fixed charges (e.g., a delivery fee added to every order) break proportionality. Check whether the relationship genuinely passes through zero.
- ✗Rounding the unitary value too early. Keep full precision when dividing to find the value for one unit, and only round the final answer.
- ✗Dividing when you should multiply (or vice versa). When scaling up, the answer should be larger. When scaling down, it should be smaller. Use this as a common-sense check.
- ✗Forgetting units. If you find a rate like £1.70 per notebook, keep the units attached to avoid confusion.
✦ Exam tips
- →Show the "for 1" step clearly. Writing "cost of 1 = ..." earns a method mark.
- →The unitary method works for almost every proportional reasoning question and is easy for examiners to follow.
- →If the numbers divide neatly using a scale factor, the scaling method can be faster. Use whichever approach suits the numbers.
- →Best-buy questions are proportional reasoning in disguise. Find the price per unit for each option and compare.