CAT 2023 Slot 3 · Quant · Question 8

Let the daily rates be r for Rahul, k for Rakshita and g for Gurmeet.

The job was finished by all three working 6 days and Rakshita a further 3 days, so 6\left( r+k+g \right)+3k=1.

Expanding, 6\left( r+g \right)+6k+3k=1, that is 6\left( r+g \right)+9k=1.

So r+g=\frac{1-9k}{6}.

All three together take more than 7 days, so their combined rate is less than \frac{1}{7}: r+k+g<\frac{1}{7}.

Substituting, \frac{1-9k}{6}+k<\frac{1}{7}, that is \frac{1-3k}{6}<\frac{1}{7}.

Cross-multiplying, 7-21k<6, so 21k>1 and k>\frac{1}{21}.

A rate above \frac{1}{21} means Rakshita alone would take fewer than 21 days.

Rahul and Gurmeet together take less than 15 days, so r+g>\frac{1}{15}.

Substituting, \frac{1-9k}{6}>\frac{1}{15}, so 15-135k>6 and k<\frac{1}{15}.

That means Rakshita alone would take more than 15 days.

So her solo time lies strictly between 15 and 21 days: 16, 17 and 20 are all possible, but 21 is not — the second option.

Shortcut. The question asks what cannot be, so look for the boundary. The strict inequality "more than 7 days" turns into "fewer than 21 days" for Rakshita, and 21 sits exactly on the excluded edge — a classic endpoint trap.