Express 0.121212... as a fraction.

Correct — A. Let x = 0.121212…
The repeating block 12 has two digits, so multiply by 100 to move one full block past the point.

100x = 12.121212…
Subtract x: 100x − x = 12.1212… − 0.1212…
99x = 12
x = 12⁄99

HCF of 12 and 99 is 3, so divide both: x = 4⁄33 → option (a)

Why the others are wrong

• (b)
  5⁄33 — 5⁄33 = 15⁄99 = 0.151515…, repeating 15, not 12. A fraction n⁄33 repeats the two digits of 3n, and 3 × 5 = 15.
• (c)
  7⁄33 — 7⁄33 = 21⁄99 = 0.212121…, the digits of 12 in the wrong order. The block 12 needs 12⁄99, which is 4⁄33.
• (d)
  12⁄33 — 12⁄33 keeps the block 12 but puts it over 33 instead of 99. It equals 36⁄99 = 0.363636…, three times the given decimal.

In a pure recurring decimal the repeating block starts right after the point. Multiplying by 10ⁿ, where n is the length of the block, moves exactly one block to the left of the point, so the endless tails line up.

Subtracting the original then cancels the tail. That is why the result is the block over n nines: 0.1212… = 12⁄99.

Key facts

• A pure recurring decimal equals its repeating block over as many 9s as the block has digits: 0.1212… = 12⁄99.
• 12⁄99 reduces by the HCF 3 to 4⁄33.
• For n from 1 to 32, n⁄33 = 3n⁄99, so its repeating block is 3n written as two digits: 1⁄33 = 0.0303…
• In a mixed recurring decimal such as 0.1222…, subtract the non-repeating part and divide by 90: (12 − 1)⁄90 = 11⁄90.

Common traps

• Multiplying by 10 instead of 100: 10x = 1.2121… does not line up with x = 0.1212…, so the tail does not cancel.
• Seeing 12 in the block and picking 12⁄33, which puts the block over 33 instead of 99.

No directly related past PYQ was found.