CAT 2022 Slot 2 · Quant · Question 11

One candidate took 30% of the votes cast, so the other three shared the remaining 70%.

They split that 70% in the proportion 1:2:3, which is 6 parts in all.

Each part is \frac{70}{6}%, so the three shares are \frac{70}{6}\%, \frac{140}{6}\% and \frac{210}{6}\%.

In decimals those are about 11.67\%, 23.33\% and 35\%.

So the four candidates took 30\%, 11.67\%, 23.33\% and 35\% of the votes cast.

The winner is the one with 35\%, and the runner-up is the one with 30\%.

Their gap is 35-30=5% of the votes cast.

That gap is 2512 votes, so the votes cast number \frac{2512}{0.05}=50240.

Only 80% of registered voters actually voted, so the registered total is \frac{50240}{0.8}=62800 — the fourth option.

Shortcut. The largest of the three proportional shares is \frac{3}{6}\times 70\%=35\%, which beats the lone 30%, so the gap is a clean 5%. Then 2512 is 5% of the cast votes and 4% of the registered voters, giving \frac{2512}{0.04}=62800 in a single division.