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CBSE Class 8 · Mathematics

Proportional Reasoning-2

Official NCERT chapter from Ganita Prakash Part II (book code hegp2). ExamMaster notes are original teaching at CBSE Class 8 depth.

This lesson follows the official chapter “Proportional Reasoning-2” in Ganita Prakash Part II. 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

1Proportionality — A Quick Recap

A recap: two quantities in proportion keep one ratio. Scale both sides; simplest form cancels a common factor. This chapter adds maps, three-term ratios, splitting a whole, and the inverse case.

Skipping the recap and inventing “whatever looks fair” is a drift.

Figure. A ratio compares two parts of the same kind. Scaling 11 : 19 by 3 gives 33 : 57 and the bars keep the same split. Scale both parts or neither; scaling only one breaks the comparison. Cross-check: 11 x 57 = 19 x 33 = 627.

How it works

  1. Write a ratio and scale it onceThe recap.
  2. See the same comparison after the scaleLockstep.
  3. Then open maps or inverse as askedThis peek’s new jobs.

2Ratios in Maps

A map ratio is a scale: 1 : 50 000 means 1 cm on the map is 50 000 cm on the ground (or the unit-pair you were taught). A measured map-length times the scale gives the ground-length.

A 5 cm road on the map is not a 5 km road without the scale.

Figure. A map ratio is a scale: every length on the page is the same multiple of the matching ground length. The small rectangle is 2 cm; the large one is three times each side, standing for 6 km. Scale both sides or the similar shape breaks.

How it works

  1. Read the scale1 : n, with units as given.
  2. Measure the map lengthThe given cm.
  3. Multiply by n and convert units if neededThe ground length.

Map to ground

Scale 1 : 100 000. A path is 3 cm on the map. How many km on the ground? (1 km = 100 000 cm.)

  • 3 cm on map3 × 100000 cm on ground
  • 300000 cm3 km
  • Check1 cm → 1 km at this scale

Pro tip. The scale is a ratio you multiply.

3Ratios with More than 2 Terms

A ratio with more than two terms is still parts: 2 : 3 : 5 has 10 parts. Order stays. You can compare two of the three, but the whole split uses all three.

Forgetting the third term when you share the whole is a dropped part.

Figure. A three-term ratio is still one whole cut into counted shares. 2 : 3 : 4 means nine equal shares: two, then three, then four. Pairing only two of the three terms throws the third share away.

How it works

  1. Write all terms2 : 3 : 5.
  2. Add them for a whole-split10 parts.
  3. Keep the order if you compare a pairFirst to second to third.

4Dividing a Whole in a Given Ratio

Dividing a whole in a given ratio is the Class 8 share: 40 in 2 : 3 is 16 and 24 because 2+3=5 parts, each part is 8. The whole is the given pie, not a new pie.

Giving 2 and 3 (the ratio numbers as if they were the shares of 40) is the miss.

Figure. To share a whole in a ratio, add the parts first. 9 + 13 = 22, so a whole of 22 splits into 9 and 13 with one unit per share. The two segments are the two people; they must fill the bar with nothing left over.

How it works

  1. Sum the parts2+3=5.
  2. Whole ÷ that sum40÷5=8.
  3. Multiply by each term16 and 24.

5Inverse Proportions

Inverse proportion: when one quantity doubles, the other halves, so the product stays the same. 3 workers × 8 days = 24 worker-days; 6 workers then need 4 days if the job is unchanged.

Treating inverse as “both go up” is the direct-proportion steal.

Figure. Inverse proportion: as one quantity grows, the other shrinks so the product stays put. Two workers need 12 days, three need 8, four need 6 — each pair multiplies to 24. Direct proportion would have made the bars rise together.

How it works

  1. See a job that stays the sameThe constant product.
  2. Write x y = kThe inverse.
  3. Solve the new pair6 × y = 24, y = 4.

6A definition is a test you can run

Proportion (direct or inverse) is a test: constant ratio, or constant product. If you cannot say which is constant, you have two lists, not a proportion-type.

“They both change” is not the test.

Figure. The definition is a test you can run: scale both parts of 10 : 20 by 3 and you still have the same share (30 : 60). Scale only the first part and you get 30 : 20, a different ratio. Equal cross-products decide, not the look of the digits.

How it works

  1. Ask: same ratio, or same product?The type.
  2. Compute that constantThe test.
  3. If neither holds, it is not this chapter’s proportionA different change.

7Name the given before the unknown

The given are the scale or the ratio and the whole (or the worker-days). The unknown is a ground-length or a share or a new time. Copy the type (direct vs inverse) before you scale.

Using 2 : 3 = 40 : x when the job was inverse is a silent swap.

Figure. Name the given ratio and the scale before you write the unknown pair. For 11 : 17 scaled by 3 the unknowns are 33 and 51. Asking for the unknown first is how one part gets scaled and the other forgotten.

How it works

  1. Name direct or inverseThe given type.
  2. Name the missing numberThe unknown.
  3. Then scale or use the productGiven first.

8One worked case is enough at this class

One map-length or one inverse job is enough. A page of the same 1 : 100 000 clone does not add a new scale-rule.

Ten twins of 3 km are still one idea.

Figure. One worked case is enough: 12 : 14 scaled by 3 is 36 : 42. The bars keep the same split, and the cross-products are equal (504). That single check is the school test; a second identical case teaches nothing new.

How it works

  1. Run one map or one inverse caseThe case.
  2. A second only as a checkOptional.
  3. StopThis class.
If 3 workers take 8 days, 6 workers take (same job)
  1. 4 days — inverse; product 24 stays
  2. 16 days
  3. 8 days

Double the workers, half the days.

Notes

  • The official chapter title is “Proportional Reasoning-2”. 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

  • map: ground = map-length × scale-factor (watch units)
  • inverse: x y = k

Recap

Hold these pegs from the official chapter “Proportional Reasoning-2”. The wording is ExamMaster’s teaching, not a textbook recap.

Proportionality — A Quick Recap
A recap: two quantities in proportion keep one ratio.
Ratios in Maps
A map ratio is a scale: 1 : 50 000 means 1 cm on the map is 50 000 cm on the ground (or the unit-pair you were taught).
Ratios with More than 2 Terms
A ratio with more than two terms is still parts: 2 : 3 : 5 has 10 parts.
Dividing a Whole in a Given Ratio
Dividing a whole in a given ratio is the Class 8 share: 40 in 2 : 3 is 16 and 24 because 2+3=5 parts, each part is 8.
Inverse Proportions
Inverse proportion: when one quantity doubles, the other halves, so the product stays the same.
A definition is a test you can run
Proportion (direct or inverse) is a test: constant ratio, or constant product.

Practise Proportional Reasoning-2

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