61 is related to 18 following a certain logic. Following the same logic, 213 is related to 38. To which of the following is 509 related, following the same logic? (NOTE : Operations should be performed on the whole numbers, without breaking down the numbers into its constituent digits. E.g. 13 – Operations on 13 such as adding /deleting /multiplying etc., to 13 can be performed. Breaking down 13 into 1 and 3 and then performing mathematical operations on 1 and 3 is not allowed.)
- (a)78
- (b)64
- (c)72
- (d)66
Answer
Why
Correct — D. Rule: add 3 to reach a perfect cube, then square the cube root and add 2.
61 + 3 = 64 = 4³, and 4² + 2 = 18
213 + 3 = 216 = 6³, and 6² + 2 = 38
Now apply it to 509:
509 + 3 = 512 = 8³
8² + 2 = 64 + 2 = 66 → option (d).
Why the others are wrong
- (a)78 — 78 is what you get by continuing the outputs 18, 38 with a doubled gap — +20, then +40. The rule works on 509 itself, not on the two earlier answers.
- (b)64 — 64 is 8² with the +2 dropped. Both worked pairs need it: 4² is 16 but the pair is 18, and 6² is 36 but the pair is 38.
- (c)72 — 72 is 8² + 8 — the number added back is the cube root, not the fixed +2 that turns 16 into 18 and 36 into 38.
Concept
These A-is-related-to-B items hand you two worked pairs and ask for one rule that fits both. Two pairs is enough to kill a coincidence, so test any candidate on both before you use it on the third number.
The way in here is to notice that 61, 213 and 509 all sit 3 below a perfect cube: 64, 216 and 512. Those are the cubes of 4, 6 and 8 — even numbers rising in twos — which is a strong sign the setter built the numbers from cubes and then hid them.
Once the cube root is out, the output is just its square plus 2.
If 512 = 8³ does not come to you, run the rule backwards on the options instead. Subtract 2 from each: only 66 − 2 = 64 is a perfect square, and its root 8 gives 8³ = 512 = 509 + 3.
Key facts
- 61, 213 and 509 are 4³ − 3, 6³ − 3 and 8³ − 3.
- Each output is the cube root squared plus 2, giving 18, 38 and 66.
- The NOTE bars digit-splitting, so 213 is handled as 213 and never as 2, 1 and 3.
Study next
Common traps
- Working on the differences between 61, 213 and 509 instead of finding a rule for each pair on its own.
- Finding the cube root and stopping at its square, which gives 64.
- Testing a candidate rule on the first pair only.
The same A-is-related-to-B format with the whole-number NOTE runs at 9 Sep 2024, 09:00, Reasoning Q.2 (31 to 152, 47 to 168, 66 to 187), where the link turns out to be a flat +121 rather than anything to do with cubes.
The wrapper is identical and the arithmetic is not, which is why the NOTE is worth reading each time.
Related PYQs
No directly related past PYQ was found.