Proportional Reasoning 1
Proportional reasoning is about comparing quantities and scaling them up or down fairly. This first Class 8 chapter on the topic covers ratios and equivalent ratios, what it means for quantities to be in proportion, the unitary method, and direct proportion — the basis of recipes, maps, speeds and prices.
Learning objectives
- Write and simplify ratios.
- Test whether quantities are in proportion using cross products.
- Solve problems by the unitary method.
- Recognise and use direct proportion.
Key concepts
Ratio
A ratio compares two quantities of the same kind by division, written a : b. Ratios can be simplified like fractions, so 8 : 12 = 2 : 3, and equivalent ratios are found by multiplying or dividing both terms by the same number. Order matters: 2 : 3 is not the same as 3 : 2.
Proportion
Two equal ratios form a proportion, written a : b :: c : d, meaning a/b = c/d. Four numbers are in proportion when the cross products are equal: a × d = b × c. For example 2 : 3 = 8 : 12 because 2 × 12 = 3 × 8.
Unitary method
In the unitary method we first find the value of one unit, then multiply for the required number. If 5 pens cost ₹60, one pen costs ₹60 ÷ 5 = ₹12, so 8 pens cost ₹12 × 8 = ₹96. It is a reliable way to scale quantities.
Direct proportion
Two quantities are in direct proportion if they increase or decrease together so that their ratio stays constant: x/y = k. More petrol means more distance, and doubling one doubles the other. Many everyday relationships — cost and quantity, distance and time at fixed speed — are directly proportional.
Important formulas
Proportion (cross product)
a : b :: c : d ⇒ a × d = b × c
Direct proportion
x₁ ÷ y₁ = x₂ ÷ y₂ (ratio stays constant)
Unitary method
value of many = (value of one) × number
Key definitions
- Ratio
- A comparison of two like quantities by division, written a : b.
- Proportion
- A statement that two ratios are equal, a : b :: c : d.
- Unitary method
- Finding the value of one unit first, then scaling to the required number.
- Direct proportion
- A relation where two quantities change together keeping a constant ratio.
Solved examples
Q1. Simplify the ratio 18 : 24.
Solution: Divide both by 6: 18 : 24 = 3 : 4.
Q2. Are 4, 6, 10, 15 in proportion?
Solution: Check cross products: 4 × 15 = 60 and 6 × 10 = 60. Equal, so yes, 4 : 6 :: 10 : 15.
Q3. If 3 kg of rice costs ₹150, find the cost of 7 kg.
Solution: One kg costs ₹150 ÷ 3 = ₹50, so 7 kg costs ₹50 × 7 = ₹350.
Common mistakes to avoid
- Reversing a ratio — writing b : a when a : b is meant.
- Comparing quantities in different units without converting first.
- Forgetting to simplify a ratio to lowest terms.
- Multiplying the wrong pairs when checking a proportion (use a×d and b×c).
Proportional Reasoning 1 — MCQ Quiz
10 questions with instant feedback. Use number keys 1–4 to answer.
The ratio 10 : 15 in simplest form is:
Practice questions
Short answer
Write 25 : 35 in simplest form.
Divide both by 5: 5 : 7.
How do you check if four numbers are in proportion?
See whether the cross products are equal: a×d = b×c.
Define direct proportion.
Two quantities are in direct proportion if they change together so that their ratio stays constant.
Long answer
Explain the unitary method using the cost of notebooks, with a worked example.
The unitary method first finds the value of a single unit, then scales up. For example, if 6 notebooks cost ₹90, one notebook costs ₹90 ÷ 6 = ₹15. To find the cost of 10 notebooks, multiply: ₹15 × 10 = ₹150. By going through the value of one unit, we can find the cost of any number of notebooks reliably.
What does it mean for two quantities to be in direct proportion? Give the rule and an example.
Two quantities are in direct proportion when an increase in one causes a proportional increase in the other, keeping their ratio constant: x₁/y₁ = x₂/y₂. For instance, if a car uses 4 litres of petrol for 60 km, then for 90 km the petrol needed satisfies 4/60 = x/90, giving x = 6 litres. Doubling the distance would double the petrol used.
HOTS (Higher Order Thinking)
A map uses 1 cm to represent 5 km. How far apart in reality are two towns 7 cm apart on the map?
Using direct proportion, 1 cm → 5 km, so 7 cm → 7 × 5 = 35 km.
If the ratio of boys to girls in a class is 3 : 2 and there are 18 boys, how many girls are there?
3 parts = 18, so 1 part = 6, and girls = 2 parts = 12.
Quick revision
Revision notes
- Ratio a : b compares like quantities; simplify like fractions; order matters.
- Proportion a : b :: c : d means a×d = b×c.
- Unitary method: find one unit, then multiply.
- Direct proportion: quantities rise/fall together, ratio constant.
Key takeaways
- Always compare quantities in the same units.
- Cross products test a proportion quickly.
- The unitary method scales any quantity reliably.
Frequently asked questions
What is a ratio?
A comparison of two quantities of the same kind by division, written a : b.
How do I find a missing term in a proportion?
Use the cross-product rule a×d = b×c and solve for the unknown.
What is the unitary method?
A way to solve problems by first finding the value of one unit, then multiplying.