Given:

1. A completes $\frac{2}{5}$ of the work in $x$ days.
2. A and B together finish the remaining $\frac{3}{5}$ of the work in 6 days.
3. B alone can complete the entire work in $\frac{100}{6}$ days.

We need to find the value of $x$.

Step-by-Step Solution

1. Find the work rates of A and B:

   • Since B can complete the work in $\frac{100}{6}$ days, the work rate of B is: $\text{B's work rate} = \frac{1}{\frac{100}{6}} = \frac{6}{100} = \frac{3}{50} \text{ (work/day)}.$

2. Let A's work rate be $A_{\text{rate}}$.

   • Since A completes $\frac{2}{5}$ of the work in $x$ days: $A_{\text{rate}} = \frac{\frac{2}{5}}{x} = \frac{2}{5x} \text{ (work/day)}.$

3. Combined work rate of A and B:

   • When A and B work together, they complete $\frac{3}{5}$ of the work in 6 days. Therefore, their combined work rate is: $A_{\text{rate}} + B_{\text{rate}} = \frac{\frac{3}{5}}{6} = \frac{3}{30} = \frac{1}{10} \text{ (work/day)}.$

4. Substitute B's work rate into the combined rate to find A's work rate:

   • We know: $A_{\text{rate}} + \frac{3}{50} = \frac{1}{10}.$
   • Isolating $A_{\text{rate}}$: $A_{\text{rate}} = \frac{1}{10} - \frac{3}{50}.$

5. Perform the subtraction:

   • Convert $\frac{1}{10}$ to a fraction with a denominator of 50: $\frac{1}{10} = \frac{5}{50}.$
   • Now subtract: $A_{\text{rate}} = \frac{5}{50} - \frac{3}{50} = \frac{2}{50} = \frac{1}{25} \text{ (work/day)}.$

6. **Find $x$ using A's work rate: