Find the missing number: 6 : 42 :: 9 : ?

Correct — D.
Rule: n : n × (n + 1).

First pair: 6 × 7 = 42.
Second pair: 9 × 10 = 90.

The same rule written as n² + n gives 36 + 6 = 42 and 81 + 9 = 90 → option (d).

Why the others are wrong

• (a)
  92 — 92 is 9 × 10 + 2. The first pair adds nothing after the product, since 6 × 7 lands exactly on 42, so the extra 2 has no source.
• (b)
  88 — 88 is 9 × 10 − 2, and its digit sum of 16 shows it is not a multiple of 9. The rule multiplies 9 by 10, so the answer must be one.
• (c)
  86 — 86 is 9 × 10 − 4, and its digit sum of 14 rules out a multiple of 9. Nothing is subtracted in the first pair, where 6 × 7 = 42 exactly.

A number analogy hides one operation that turns the first term into the second. Test the common forms: a multiple, a square plus or minus something, a product of consecutive numbers.

42 = 6 × 7 points to n × (n + 1). A quick filter follows: the answer must be a multiple of 9, and of 92, 88, 86 and 90, only 90 passes the digit-sum test.

Key facts

• n × (n + 1) equals n² + n: 6 × 7 = 36 + 6 = 42.
• Products of consecutive numbers: 7 × 8 = 56, 9 × 10 = 90, 10 × 11 = 110.
• A number is a multiple of 9 exactly when its digit sum is a multiple of 9.

Common traps

• Settling on the first rule that fits one pair without testing it against the options
• Reading 42 as 6 × 7 and then multiplying 9 by 7

No directly related past PYQ was found.