CSAT Solved Papers/ 2022/Q59

2022 CSAT — Q59

Quant Statement validity 2.5 marks Medium

Consider the following statements in respect of a rectangular sheet of length 2020 cm and breadth 88 cm:

  1. It is possible to cut the sheet exactly into 44 square sheets.

  2. It is possible to cut the sheet into 1010 triangular sheets of equal area.

Which of the above statements is/are correct?

  1. A 1 only
  2. B 2 only
  3. C Both 1 and 2 Answer
  4. D Neither 1 nor 2

Worked rationale

Statement 1 — cut 20×820 \times 8 into 44 squares (no waste). Peel off squares greedily:

20×8  →  (8×8)+(8×8)+(4×8)⏟=(4×4)+(4×4).20\times 8 \;\to\; (8\times 8) + (8\times 8) + \underbrace{(4\times 8)}_{= (4\times 4)+(4\times 4)}.

That is four squares — two of side 88 and two of side 44 — exactly tiling the sheet with no leftover. Correct.

Statement 2 — cut into 1010 equal-area triangles. Area =160= 160, so each triangle must be 1616. Split the sheet into 55 equal sub-rectangles (4×84\times 8 each, area 3232); each sub-rectangle’s diagonal cuts it into 22 triangles of area 1616. That gives 5×2=105 \times 2 = 10 triangles, all of equal area. Correct.

Answer: (c) Both 1 and 2.

Visual solution

The same solve, worked by hand — read it, then trace it.

Hand-drawn worked solution for UPSC 2022 CSAT Q59 — Statement validity
Tap the drawing to open it full size for the fine detail.

Why the other options miss

  • A
    missed a case: doubts the equal-area triangulation, not seeing the 55-strips-times-diagonal construction.
  • B
    missed a case: insists the 44 squares must be equal and rejects (1), missing the 8,8,4,48,8,4,4 dissection.
  • D
    solved the wrong question: treats both dissections as impossible without attempting a construction.

Specialist insight

Both claims are constructive existence statements — produce one valid cut and the claim is true. Square dissection: peel the largest square (8×88\times 8) twice, then halve the 4×84\times 8 remainder into two 4×44\times 4. Equal-area triangulation: any rectangle splits into nn equal triangles for even nn (here 55 strips ×\times diagonal). The trap is reading ”44 square sheets” as ”44 equal squares,” which is not stated.

The trap, in one line

Squares 8,8,4,48,8,4,4 tile it; 55 strips ×\times diagonal give 1010 equal triangles — both hold ⇒\Rightarrow (c).

← All 2022 CSAT questions