A completes 2/5 part of a work in x days. After then he calls B and they together finish the remaining work in 6 days. If B alone takes 100/6 days. Then find the value of x.

(a) 10 days

(b) 50/3 days

(c) 20 days

(d) 40 days

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:

A completes 2/5 part of a work in x days. After then he calls B and they together finish the remaining work in 6 days. If B alone takes 100/6 days. Then find the value of x.

(a) 10 days
(b) 50/3 days
(c) 20 days
(d) 40 days

Solve a work and time problem where A completes 2/5 of a task in x days, and with the help of B, they finish the rest in 6 days. Learn how to calculate the work rates of both A and B to determine the value of x. Suitable for Grades 9-10 students.

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