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Grade 8/ Mathematics/ Proportional Reasoning – 1
Chapter 7 · NCERT Class 8 Ganita Prakash

Proportional Reasoning – 1

Double the recipe, scale the map, work out the discount — proportional reasoning is the maths of "if this, then how much of that?". Ratios, the unitary method and percentages are three faces of the same idea. Tap each one to see how it works.

📐 3 topics⏱ ~25 min📝 12-question quiz
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The six ideas of proportion

These tools all answer the same kind of question — scaling one quantity with another. Tap each term to see what it means and a quick example.

Explore · Ratio & proportiontap a term

Learn

The three big ideas

  • A ratio compares two quantities of the same kind by division, written a : b. The order matters: 3 : 2 is not the same as 2 : 3.
  • Simplify a ratio by dividing both terms by their HCF: 20 : 30 = 2 : 3. Equivalent ratios come from multiplying or dividing both terms by the same number: 3 : 4 = 9 : 12.
  • A proportion says two ratios are equal: a : b :: c : d. Then the product of the extremes equals the product of the means: a × d = b × c.

Worked example. Are 2 : 3 and 8 : 12 in proportion?

1. Check the cross products: extremes 2 × 12 = 24; means 3 × 8 = 24.

2. They are equal, so yes — 2 : 3 :: 8 : 12 is a true proportion.

Common mistake: forming a ratio from different units. To compare 50 cm and 2 m, first make the units the same (2 m = 200 cm), giving 50 : 200 = 1 : 4.
  • Unitary method: first find the value of one unit, then multiply for the number you need. 5 pens cost ₹60 → 1 pen = ₹12 → 8 pens = ₹96.
  • Direct proportion: two quantities that rise and fall together so their ratio stays constant (y ÷ x = k). More petrol → more distance; more kg → more cost.
  • You can solve direct-proportion problems by the unitary method or by setting two equal ratios.

Worked example. A car uses 4 litres of petrol for 60 km. How far will 10 litres take it?

1. Distance per litre = 60 ÷ 4 = 15 km (the value of one unit).

2. For 10 litres: 10 × 15 = 150 km.

3. Check by ratio: 60 : 4 = 150 : 10, since 60 × 10 = 4 × 150. ✓

  • Percent means "per hundred": x% = x ÷ 100. It is simply a ratio with denominator 100, which makes different quantities easy to compare.
  • Convert a fraction or decimal to a percentage by writing it out of 100: 3/4 = 75%, 0.6 = 60%.
  • Percentage of a quantity: 20% of 250 = (20 ÷ 100) × 250 = 50.
  • Discounts, marks, GST and interest are all everyday percentages.

Worked example. A bag marked ₹800 is sold at a 15% discount. What is the sale price?

1. Discount = 15% of 800 = (15 ÷ 100) × 800 = ₹120.

2. Sale price = 800 − 120 = ₹680.

Where you'll meet it

Proportion at work

Cooking & recipes

A recipe for 4 people becomes a recipe for 10 by keeping every ingredient in the same ratio. The unitary method tells you exactly how much of each to add.

Maps & scale drawings

A map scale like 1 cm : 5 km is a ratio. Measuring a distance on the map and scaling up by that ratio gives the real distance on the ground.

Shopping & money

Discounts, GST, exam marks and interest are all percentages. Knowing how to take a percentage of an amount lets you check a bill or compare two offers in seconds.

Check yourself

Competency quiz

Modelled on the competency-based pattern — MCQ, assertion–reason and a case study, testing whether you can use the ideas, not just recall them.

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Interactive built to the OpenMAIC approach (THU-MAIC, MIT). Content from the NCERT Class 8 Ganita Prakash textbook (ncert.nic.in).

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