Time and Work with Inequality Constraints

Let $R, K, G$ be the daily work rates of Rahul, Rakshita, and Gurmeet respectively.

From the actual work done: $6(R + K + G) + 3K = 1 \implies 6(R + G) + 9K = 1$ Let $T$ be the number of days Rakshita takes alone, so $K = 1/T$. Thus, $6(R + G) = 1 - \frac{9}{T}$.

Given conditions:

1. All three together take more than 7 days: $\frac{1}{R + K + G} > 7 \implies R + K + G < \frac{1}{7}$ Substitute $R + G = \frac{1}{6}\left(1 - 9K\right)$: $\frac{1}{6}(1 - 9K) + K < \frac{1}{7} \implies \frac{1}{6} - \frac{1}{2}K < \frac{1}{7} \implies \frac{1}{2}K > \frac{1}{42} \implies K > \frac{1}{21} \implies T < 21$

2. Rahul and Gurmeet together take less than 15 days: $\frac{1}{R + G} < 15 \implies R + G > \frac{1}{15}$ Substitute $R + G = \frac{1}{6}(1 - 9K)$: $\frac{1}{6}(1 - 9K) > \frac{1}{15} \implies 1 - 9K > \frac{2}{5} \implies 9K < \frac{3}{5} \implies K < \frac{1}{15} \implies T > 15$

Thus, $15 < T < 21$.

Hence, $T$ cannot be $21$.