Schooling · Class 8 · Mathematics · Proportional Reasoning-2 · Chapter on a page
CBSE Class 8 · Mathematics · NCERT Ganita Prakash Part-II
Proportional Reasoning-2: 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.
This note's current text has no applied fact-check on record.
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
- inverse proportion
- To check direct proportion
- To check inverse proportion
- unitary method
- In direct proportion, if x₁/y₁ =…
- In inverse proportion, x₁ × y₁ = x₂…
- Formulas and facts
- Direct proportion
- Inverse proportion
- Unitary method for direct proportion
- Unitary method for inverse proportion
- Scale on maps
- Watch out for
- Assuming every problem is direct…
- Forgetting to keep units the same…
- In inverse proportion, dividing…
- Setting up the proportion upside down
- Using cross-multiplication wrongly…
What this chapter is about
This chapter takes your understanding of ratios and proportions further. In earlier classes, you learnt what a ratio is and how to check if two ratios are equal (proportion). Now you will explore deeper ideas: direct proportion, inverse proportion, and how to solve real-life problems using these concepts.
Key ideas
- Two quantities are in direct proportion if their ratio stays constant. When one doubles, the other also doubles. Example: cost and number of items bought at a fixed price per item.
- Two quantities are in inverse proportion if their product stays constant. When one doubles, the other becomes half. Example: more workers finish a job in fewer days.
- To check direct proportion: divide corresponding values — if all answers are equal, it is direct proportion.
- To check inverse proportion: multiply corresponding values — if all products are equal, it is inverse proportion.
- The unitary method finds the value for one unit first, then scales up or down. It works for both types of proportion.
- In direct proportion, if x₁/y₁ = x₂/y₂, you can cross-multiply to find an unknown.
- In inverse proportion, x₁ × y₁ = x₂ × y₂ helps you find an unknown.
- Real-life clues: phrases like "at the same rate" or "per item" suggest direct proportion; phrases like "working together" or "sharing equally" often suggest inverse proportion.
Formulas and facts to remember
- Direct proportion: x₁/y₁ = x₂/y₂ (ratio is constant)
- Inverse proportion: x₁ × y₁ = x₂ × y₂ (product is constant)
- Unitary method for direct proportion: Value for 1 unit = Total value ÷ Number of units
- Unitary method for inverse proportion: If 1 person takes T days, then n persons take T/n days (when work stays the same)
- Scale on maps: Map distance and actual distance are in direct proportion when using the same scale.
Common mistakes
- Assuming every problem is direct proportion → first check whether increasing one quantity increases or decreases the other.
- Forgetting to keep units the same before comparing → always convert to the same unit (all in metres, all in hours, etc.).
- In inverse proportion, dividing instead of multiplying → remember, for inverse proportion you multiply corresponding values.
- Setting up the proportion upside down → write both ratios in the same order (first quantity on top, second on bottom, or vice versa, but be consistent).
- Using cross-multiplication wrongly in inverse proportion → cross-multiplication works for equal ratios (direct), not for inverse; use the product formula instead.
Quick revision
- Direct proportion: ratio stays the same; when one goes up, the other goes up.
- Inverse proportion: product stays the same; when one goes up, the other goes down.
- Unitary method: find the value for 1 unit, then multiply or divide as needed.
- Direct → x₁/y₁ = x₂/y₂; Inverse → x₁ × y₁ = x₂ × y₂.
- Always identify the type of proportion before solving.