What this chapter is about
Proportional reasoning is the ability to think about how two quantities change together in a consistent way. When you say that a car travels 60 kilometres every hour, or that 5 bananas cost ₹20, you are describing a relationship where one quantity depends on another in a fixed pattern. This chapter builds your skill in recognising, setting up and solving problems involving such relationships.
In Class 8, you move beyond simple ratio and proportion problems to think more deeply about direct proportion and how to use it in everyday situations. You learn to identify when two quantities are in proportion, set up the correct relationship, and find unknown values using the unitary method or cross-multiplication.
By the end of this chapter, you should be able to recognise proportional relationships in real life, write them as equations, solve for missing quantities, and check whether your answer makes sense. This reasoning appears everywhere — in recipes, maps, scale drawings, speed calculations, and money problems.
Key ideas
- Two quantities are in direct proportion when increasing one causes the other to increase by the same factor, and their ratio stays constant.
- The unitary method finds the value of one unit first, then multiplies to find the value of any number of units.
- In a proportion statement a : b :: c : d (read as "a is to b as c is to d"), the product of the means (b × c) equals the product of the extremes (a × d). This is the cross-multiplication rule.
- A ratio compares two quantities of the same kind; a proportion states that two ratios are equal.
- If x and y are in direct proportion, then y/x = k (a constant), or equivalently y = kx for some fixed number k.
- Recognising what stays constant in a problem is the key step in proportional reasoning.
- Checking your answer by substituting back into the original relationship helps catch calculation errors.
Formulas and facts to remember
- Formula / Rule: a : b :: c : d means a/b = c/d · Meaning: Two ratios are equal (proportion).
- Formula / Rule: a × d = b × c · Meaning: Cross-multiplication rule for proportions.
- Formula / Rule: Value of 1 unit = Total value ÷ Number of units · Meaning: Unitary method step 1.
- Formula / Rule: Value of n units = (Value of 1 unit) × n · Meaning: Unitary method step 2.
- Formula / Rule: y = kx (k constant) · Meaning: Direct proportion relationship.
- Formula / Rule: k = y/x · Meaning: The constant of proportionality.
Worked examples
Example 1: Cost of notebooks
If 4 notebooks cost ₹120, find the cost of 7 notebooks.
Step 1: Find the cost of 1 notebook. Cost of 1 notebook = 120 ÷ 4 = ₹30.
Step 2: Find the cost of 7 notebooks. Cost of 7 notebooks = 30 × 7 = ₹210.
Answer: 7 notebooks cost ₹210.
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Example 2: Using cross-multiplication
A recipe uses 3 cups of rice for 12 people. How many cups are needed for 20 people?
Let the required cups be x. Set up the proportion: 3/12 = x/20.
Cross-multiply: 3 × 20 = 12 × x 60 = 12x x = 60 ÷ 12 = 5.
Answer: 5 cups of rice are needed for 20 people.
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Example 3: Checking for direct proportion
A bus travels 90 km in 2 hours and 180 km in 4 hours. Are distance and time in direct proportion here?
Find the ratio distance/time in each case: 90/2 = 45 km per hour. 180/4 = 45 km per hour.
Since the ratio is constant (45), distance and time are in direct proportion.
Common mistakes
Writing the proportion upside-down (mixing which quantity goes on top) → always decide which quantity corresponds to which before setting up the ratio.
Forgetting to keep units the same (comparing metres with centimetres) → convert all quantities to the same unit first.
Using cross-multiplication on ratios that are not equal → first confirm the situation is a proportion problem.
Skipping the check step and accepting a clearly unreasonable answer → substitute your answer back to see if the relationship holds.
Confusing direct proportion with other relationships (like inverse proportion) → ask yourself: does doubling one quantity double the other?
Quick revision
- Direct proportion: one quantity doubles, the other doubles too; their ratio is constant.
- Unitary method: find the value of one, then multiply.
- Cross-multiply to solve a/b = c/d: a × d = b × c.
- Always keep units consistent before comparing.
- Check your answer by substituting it back into the original relationship.