CSAT Solved Papers/ 2025/Q16

2025 CSAT — Q16

Quant Logical & quantitative reasoning 2.5 marks Medium

If 7∗24=257 * 24 = 25 and 12∗16=2012 * 16 = 20, then what is 16∗6316 * 63 equal to?

  1. A 70
  2. B 66
  3. C 65 Answer
  4. D 64

Worked rationale

Read the two given results as a hidden rule. Test the sum of squares under a root:

7∗24=72+242=49+576=625=25 ✓7 * 24 = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \ \checkmark 12∗16=122+162=144+256=400=20 ✓12 * 16 = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \ \checkmark

The rule is a∗b=a2+b2a * b = \sqrt{a^2 + b^2} — the hypotenuse of a right triangle with legs a,ba,b. Apply it:

16∗63=162+632=256+3969=4225=65.16 * 63 = \sqrt{16^2 + 63^2} = \sqrt{256 + 3969} = \sqrt{4225} = 65.

(Recognising the Pythagorean triple (16,63,65)(16, 63, 65) confirms it instantly.)

Answer: (c) 65.

Why the other options miss

  • A
    reached for the wrong rule: guesses an additive/average rule (e.g. 16+63=7916 + 63 = 79 then adjusted, or 16+63?\tfrac{16+63}{?}) instead of the root-of-sum-of-squares.
  • B
    an arithmetic slip: takes 4225\sqrt{4225} as ≈65\approx 65 but rounds or miscomputes the square root to 6666 (662=4356≠422566^2 = 4356 \ne 4225).
  • D
    an arithmetic slip: drops a unit (63→6463 \to 64 confusion) or reads 4096=64\sqrt{4096}=64, mis-squaring 6363 as ≈602\approx 60^2.

Specialist insight

Redefined-operator items are decoded by testing structural rules against the given data points, not by inventing one. The fingerprint here is that 2525 and 2020 are exact integers and (7,24,25)(7,24,25), (12,16,20)=(3,4,5)×4(12,16,20)=(3,4,5)\times4 are Pythagorean — that pattern screams “hypotenuse.” Once the rule is fixed by both anchors (one data point can fit many rules; two pin it down), the answer is mechanical. Knowing the common triples (7,24,25),(8,15,17),(9,40,41),(16,63,65),(20,21,29)(7,24,25),(8,15,17),(9,40,41),(16,63,65),(20,21,29) turns the final square root into instant recall — exactly the speed edge CSAT rewards.

The trap, in one line

The operator is a∗b=a2+b2a*b = \sqrt{a^2+b^2} (a Pythagorean hypotenuse) — verify it on both given pairs before applying; 16∗63=6516*63 = 65.

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