CAT 2024 Slot 3 · Quant · Question 19

The two farms together cover 20+30=50 hectares.

Across the whole 50 hectares, wheat and mustard are in the ratio 11:9, a total of 20 parts.

So the total wheat is \frac{11}{20}\times 50=27.5 hectares and the total mustard is \frac{9}{20}\times 50=22.5 hectares.

On Vimal's 30 hectares, wheat and mustard are in the ratio 5:3, a total of 8 parts.

So Vimal's wheat is \frac{5}{8}\times 30=18.75 hectares and his mustard is \frac{3}{8}\times 30=11.25 hectares.

Rajesh's wheat is what remains: 27.5-18.75=8.75 hectares.

Rajesh's mustard is 22.5-11.25=11.25 hectares.

As a check, 8.75+11.25=20, which is exactly Rajesh's holding.

The ratio is 8.75:11.25, and multiplying both by 4 gives 35:45.

Dividing by 5, the ratio is 7:9 — the second option.

Shortcut. This is alligation in disguise: Vimal's wheat share is \frac{5}{8}=62.5\% and the blended share is \frac{11}{20}=55\%. Vimal pulls the average up, so Rajesh must sit below 55%, which rules out options 1 and 3 at once. Of the remaining two, \frac{7}{16}=43.75\% gives the right weighted average over 20 and 30 hectares.