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CBSE Class 8 · Mathematics

Proportional Reasoning-1

Official NCERT chapter from Ganita Prakash Part I (book code hegp1). ExamMaster notes are original teaching at CBSE Class 8 depth.

This lesson follows the official chapter “Proportional Reasoning-1” in Ganita Prakash Part I. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 8
  • Easy level
  • 8 concepts

1Observing Similarity in Change

Similarity in change means two quantities grow (or shrink) in lockstep: when one doubles, the other doubles. A table that stays in the same ratio is the look. A table that drifts is not this similarity.

Two lists that both go up are not automatically proportional.

Figure. Similarity in change means the same multiplier on both parts: 2 becomes 6 and 3 becomes 9, each times 3.

How it works

  1. Write two columnsThis pair of quantities.
  2. See whether each pair simplifies to the same ratioThe lockstep.
  3. If a pair breaks the ratio, the change is not similar in this senseA miss.

2Ratios

A ratio a : b compares two counts of the same kind of unit (or two parts of a whole). 6 : 4 is the same comparison as 3 : 2. The order matters: 6 : 4 is not 4 : 6.

A ratio with no units-story can still be a comparison of two numbers; mixing km with minutes without a rate-word is a different later job.

Figure. A ratio 9 : 18 compares two parts of the same kind. Scaling both by 3 gives 27 : 54; the cross-products 9 times 54 and 18 times 27 both equal 486.

How it works

  1. Write a to b in order6 : 4.
  2. Say what each side countsThis to that.
  3. Keep the orderFirst to second.

3Ratios in their Simplest Form

A ratio in simplest form has no common factor left: 6 : 4 becomes 3 : 2. Divide both sides by the HCF. 3 : 2 is not 2 : 3.

Cancelling only the first number is a house-slide.

Figure. 42 : 39 and 14 : 13 are the same comparison. Divide both parts by 3 to reach the simplest form 14 : 13.

How it works

  1. Find a common factor2 in 6 and 4.
  2. Divide both sides3 : 2.
  3. Stop when HCF is 1Simplest.

Simplify 18 : 24

Write 18 : 24 in simplest form.

  • HCF6
  • 18÷6 : 24÷63 : 4
  • Checkno common factor left

Pro tip. Both sides get the same divide.

4Problem Solving with Proportional Reasoning

A proportion problem is “same ratio”: 3 : 2 = 12 : x means 3x = 24, so x = 8. Cross-multiply or scale both sides by the same number.

Adding 9 to both sides of a ratio is not scaling.

Figure. A proportion problem scales both parts the same way: 10 : 14 becomes 30 : 42 when each part is multiplied by 3. Cross-products both equal 420.

How it works

  1. Write the two ratios as equalThe proportion.
  2. Scale or cross-multiply3x = 2×12.
  3. Solve the unknownx = 8.

5Sharing, but Not Equally!

Sharing not equally means parts in a ratio: 12 sweets in 2 : 1 is 8 and 4, not 6 and 6. The unit is total-parts: 2+1=3, each part is 4.

Halving because “share” sounded equal is the miss of this heading.

Figure. Sharing 24 in the ratio 2 : 1 is not an equal split. The parts are 16 and 8, and they still add to 24.

How it works

  1. Add the ratio parts2 + 1 = 3.
  2. Divide the whole by that sum12 ÷ 3 = 4 per part.
  3. Give each person their parts8 and 4.

6Unit Conversions

A unit conversion is a named trade used as a ratio: 1 m = 100 cm, so 2.4 m = 240 cm. The trade is the given; you scale it.

Inventing 1 m = 10 cm is a broken trade.

Figure. Unit conversion keeps the same length. The stick is 3 m and also 300 cm; the two readings name one span.

How it works

  1. Write the taught trade1 m : 100 cm.
  2. Scale to the given amount2.4 times both sides.
  3. Keep the unit you were asked forcm, or m.

7A definition is a test you can run

A ratio is a test: two numbers in order that you can simplify or scale. If you cannot name both sides, you have a single count, not a ratio.

“6” alone is not 6 : 4.

Figure. A ratio definition is a test you can run: 14 : 13 matches 42 : 39 because both cross-products equal 546.

How it works

  1. Write both sidesThe object.
  2. Simplify or scaleThe test.
  3. If one side is missing, it is not yet a ratioName both.

8Name the given before the unknown

The given are the two quantities or the ratio and the whole. The unknown is the missing term or the share. Copy the order before you swap 2 : 1 into 1 : 2.

Flipping a share because the smaller person “should get more” is a silent given.

Figure. Name the given ratio 10 : 18 and the given 30 before the unknown. Then 10 / 18 = 30 / x gives x = 54.

How it works

  1. Copy the ratio in the given orderThe given.
  2. Name the missing number or the sharesThe unknown.
  3. Then scale or splitGiven first.
12 sweets in the ratio 2 : 1 are
  1. 8 and 4
  2. 6 and 6
  3. 12 and 1

3 parts; each part is 4.

Notes

  • The official chapter title is “Proportional Reasoning-1”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • a : b = (a÷k) : (b÷k)
  • shares: each part = whole / (sum of ratio parts)

Recap

Hold these pegs from the official chapter “Proportional Reasoning-1”. The wording is ExamMaster’s teaching, not a textbook recap.

Observing Similarity in Change
Similarity in change means two quantities grow (or shrink) in lockstep: when one doubles, the other doubles.
Ratios
A ratio a : b compares two counts of the same kind of unit (or two parts of a whole).
Ratios in their Simplest Form
A ratio in simplest form has no common factor left: 6 : 4 becomes 3 : 2.
Problem Solving with Proportional Reasoning
A proportion problem is “same ratio”: 3 : 2 = 12 : x means 3x = 24, so x = 8.
Sharing, but Not Equally!
Sharing not equally means parts in a ratio: 12 sweets in 2 : 1 is 8 and 4, not 6 and 6.
Unit Conversions
A unit conversion is a named trade used as a ratio: 1 m = 100 cm, so 2.4 m = 240 cm.

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