Schooling · Class 8 · Mathematics · Proportional Reasoning-1 · Chapter on a page
CBSE Class 8 · Mathematics · NCERT Ganita Prakash Part-I
Proportional Reasoning-1: chapter on a page
A cut of Shishya's notes for this chapter: the first paragraph of the introduction and the key ideas, formulas and facts, common mistakes and quick revision lists, word for word. Nothing is added or reworded; the worked examples are left out (they are on the chapter page).
About the notes: Written by Shishya's AI on 26 Sept 2026 from the chapter's title and class level, in Shishya's own words — not a copy or summary of the textbook. Read the official chapter for the book's own text, activities and exercises.
Fact-checked on 21 Oct 2026: an AI-assisted check of this note against cited sources; the errors it found were corrected that day. Some errors may remain.
Mind map
Drawn from the same notes by a fixed rule (no AI): each label is the item's own words — its bold term, its label before a colon, a short formula as written, or its first words.
The mind map as a list
- Key ideas
- direct proportion
- unitary method
- In a proportion statement a
- ratio
- If x and y are in direct proportion…
- Recognising what stays constant in a…
- Checking your answer by substituting…
- Formulas and facts
- a : b :: c : d means a/b = c/d
- a × d = b × c
- Formula / Rule
- Value of n units = (Value of 1 unit)…
- y = kx (k constant)
- k = y/x
- Watch out for
- Writing the proportion upside-down…
- Forgetting to keep units the same…
- Using cross-multiplication on ratios…
- Skipping the check step and…
- Confusing direct proportion with…
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.
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, which can even be of different kinds (such as number of students and kilograms of rice); 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.
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.