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Estimation (guesstimates) · 5 marks

Looking at her mathematics book (Part 2), Sheela wonders:

  1. (i) If all the pages of this book were laid out side-by-side on the floor would they cover the entire classroom floor?
  2. (ii) What is the maximum number of such books that can fit in an empty storeroom of dimensions 15 ft × 20 ft × 30 ft?
Answer: (i) No. A page is about 28 cm × 21 cm ≈ 0.059 m2; even 180 separate pages cover only about 10.6 m2, against a classroom floor of about 48 m2. (ii) The storeroom is 9000 ft3 ≈ 25,48,53,000 cm3; a 28 cm × 21 cm × 1 cm book takes 588 cm3, so at most about 4,33,000 books by volume, and about 4,24,000 when stacked neatly.

Step-by-step solution

Idea: (i) Compare the total area of the pages with the floor area. (ii) Compare the storeroom’s volume with one book’s volume, then check how many fit along each wall.

(i) If all the pages of this book were laid out side-by-side on the floor would they cover the entire classroom floor?

  1. Measure: page ≈ 28 cm × 21 cm = 588 cm2 = 0.0588 m2; about 180 pages (90 sheets).½ mark
  2. Even counting every page separately: 180 × 0.0588 ≈ 10.6 m2. (Laying out the 90 sheets: about 5.3 m2.)1 mark
  3. Classroom floor ≈ 8 m × 6 m = 48 m2. 10.6 < 48, so no: the pages cover only about a fifth of the floor.½ mark
No: about 10.6 m2 of pages against about 48 m2 of floor.

(ii) What is the maximum number of such books that can fit in an empty storeroom of dimensions 15 ft × 20 ft × 30 ft?

  1. Storeroom: 15 ft × 20 ft × 30 ft = 457.2 cm × 609.6 cm × 914.4 cm (1 ft = 30.48 cm). Volume = 9000 ft3 ≈ 9000 × 28 317 ≈ 25,48,53,000 cm3.1 mark
  2. Book ≈ 28 cm × 21 cm × 1 cm = 588 cm3. Upper bound = 25,48,53,000 ÷ 588 ≈ 4,33,400.1 mark
  3. Neat stacking: 457.228 → 16, 609.621 → 29, 914.41 → 914 books in a pile: 16 × 29 × 914 = 4,24,096, about 4.2 lakh books.1 mark
About 4,24,000 books (at most about 4,33,000 by volume).
(i) No: about 10.6 m² of pages vs about 48 m² of floor. (ii) About 4,24,000 books stacked neatly (upper bound about 4,33,000 by volume), for a 28 cm × 21 cm × 1 cm book.

Check: 4,24,096 × 588 ≈ 24,93,68,000 cm3, just under the room’s 25,48,53,000 cm3 ✓ (about 98% filled).

Answer to write in the exam

(i)

Area of one page = 28 × 21 = 588 cm2 = 0.0588 m2

180 pages ⇒ 180 × 0.0588 ≈ 10.6 m2

Floor = 8 × 6 = 48 m2

∴ No, the pages cover only about 15 of the floor

(ii)

Room = 457.2 cm × 609.6 cm × 914.4 cm ≈ 25,48,53,000 cm3

Book = 28 × 21 × 1 = 588 cm3 ⇒ at most ≈ 4,33,400

Neat stacking: 16 × 29 × 914 = 4,24,096

∴ About 4.2 lakh books

Common mistakes that cost marks

  • Comparing cm2 with m2 without converting (1 m2 = 10 000 cm2).
  • Using 1 ft = 12 cm or forgetting to convert feet to cm before dividing.
  • Rounding 457.2 ÷ 28 = 16.3 up to 17; only whole books fit.

How this can come in the exam

MCQ (1 mark)

How many sheets of A4 paper (about 0.06 m2 each) cover a floor of 12 m2?

  1. 20
  2. 72
  3. 200
  4. 2000
Show answer

(C) 200
12 ÷ 0.06 = 200.

Try one yourself

How many boxes of 50 cm × 40 cm × 30 cm fit in a room 5 m × 4 m × 3 m?

Show answer

10 × 10 × 10 = 1000 boxes.

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