Select the set in which the numbers are related in the same way as are the numbers of the following sets. (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.) (19, 456, 6) (16, 320, 5)

Correct — A. Rule: middle number = first × third × 4.

Test both given sets.
19 × 6 = 114, and 114 × 4 = 456.
16 × 5 = 80, and 80 × 4 = 320.

Now the keyed set.
18 × 9 = 162, and 162 × 4 = 648.

That is (18, 648, 9), option (a). Put the other way round, middle ÷ (first × third) has to equal 4.

Why the others are wrong

• (b)
  (14, 210, 5) — 14 × 5 × 4 = 280, not 210. This set runs on a multiplier of 3, because 210 ÷ 70 = 3.
• (c)
  (21, 504, 8) — 21 × 8 × 4 = 672, not 504. The multiplier here is 3 as well, since 504 ÷ 168 = 3.
• (d)
  (18, 378, 7) — 18 × 7 × 4 = 504, not 378. Once again the multiplier is 3, as 378 ÷ 126 = 3.

A three-number set is usually held together by one arithmetic sentence linking all three, not by two separate rules. Look for a product or a quotient before anything else.

Here 456 ÷ (19 × 6) = 4 and 320 ÷ (16 × 5) = 4. One division on each given set fixes the constant, and then a single multiplication tests each option.

The design of this item is worth noticing. All three rejected sets obey middle = first × third × 3, so anyone who checks only one given set can be pulled towards any of them.

Key facts

• 456 = 19 × 6 × 4 and 320 = 16 × 5 × 4.
• The keyed set satisfies 648 = 18 × 9 × 4.
• All three rejected sets satisfy middle = first × third × 3.

Common traps

• Deriving the constant from one given set and never testing it on the second.
• Breaking the numbers into digits, which the question's note explicitly forbids.

No directly related past PYQ was found.