In an election between R and S, R gets 110% of the votes of S and beats S by 770 votes. If there were no invalid votes, then the total number of votes polled is:

Correct — A. R gets 110% of S's votes, so write S = x and R = 1.1x.

R − S = 770
1.1x − x = 0.1x = 770
x = 7,700 votes for S

R = 1.1 × 7,700 = 8,470
Total polled = 7,700 + 8,470 = 16,170 → option (a)

Shortcut: the margin is 10% of S and the poll is 210% of S, so the total is 21 × 770 = 16,170.

Why the others are wrong

• (b)
  17160 — 17,160 is the right digits in the wrong order. The total has to be 21 × 770, and 770 × 22 = 16,940 — no whole number of margins reaches 17,160.
• (c)
  17610 — 17,610 is the same five digits reshuffled again. Between 770 × 22 = 16,940 and 770 × 23 = 17,710 there is nothing at 17,610.
• (d)
  16710 — 16,710 transposes the 1 and the 7 of the correct total. It is what a candidate writes after computing 16,170 and then copying it out of order.

The phrase to decode is 110% of the votes of S. It fixes R against S, not against the poll: R = 1.1S, so the margin R − S is 0.1S, one tenth of S's votes.

With no invalid votes the two candidates hold the entire poll, so the total is S + R = 2.1S.

That gives the ratio that does all the work: the margin is 1 part while the total is 21 parts. Multiply the 770-vote margin by 21 and the answer appears without solving for either candidate.

Key facts

• R gets 110% of S's votes, so R beats S by 10% of S's votes, not by 10% of the poll.
• With no invalid votes the poll equals S + 1.1S = 2.1S.
• The total is 21 times the margin, giving 21 × 770 = 16,170 votes.

Common traps

• Treating the 770-vote margin as 10% of the total poll instead of 10% of S
• Reading 110% of S as 110% more than S, which would make R = 2.1S
• Subtracting an invalid-vote share that the stem explicitly rules out

No directly related past PYQ was found.