A can do `1/2` of the work in 5 days and B can do `2/5` of the weak in 10 days .in how many days both A and B together can do the work ?

Options

A. `7 (2)/(7) days `
B. `7(1)/(7) days `
C. `7 (3)/(7) days `
D. `7(5)/(7) days `

Answer

To solve the problem, we need to determine how long it will take for A and B to complete the work together. Let's break it down step by step:

### Step 1: Determine the Work Done by A
A can do \( \frac{1}{2} \) of the work in 5 days.
To find A's efficiency (work done per day), we calculate:
\[ \text{Efficiency of A} = \frac{\text{Work}}{\text{Time}} = \frac{\frac{1}{2}}{5} = \frac{1}{10} \]
This means A can complete \( \frac{1}{10} \) of the work in one day.

### Step 2: Determine the Work Done by B
B can do \( \frac{2}{5} \) of the work in 10 days.
To find B's efficiency, we calculate:
\[ \text{Efficiency of B} = \frac{\text{Work}}{\text{Time}} = \frac{\frac{2}{5}}{10} = \frac{2}{50} = \frac{1}{25} \]
This means B can complete \( \frac{1}{25} \) of the work in one day.

### Step 3: Combine the Efficiencies of A and B
Now, we add the efficiencies of A and B to find their combined efficiency:
\[ \text{Combined Efficiency} = \text{Efficiency of A} + \text{Efficiency of B} = \frac{1}{10} + \frac{1}{25} \]
To add these fractions, we need a common denominator. The least common multiple of 10 and 25 is 50.
\[ \frac{1}{10} = \frac{5}{50} \quad \text{and} \quad \frac{1}{25} = \frac{2}{50} \]
So,
\[ \text{Combined Efficiency} = \frac{5}{50} + \frac{2}{50} = \frac{7}{50} \]

### Step 4: Calculate the Time Taken to Complete the Work Together
The time taken to complete the entire work when A and B work together is the inverse of their combined efficiency:
\[ \text{Time} = \frac{1}{\text{Combined Efficiency}} = \frac{1}{\frac{7}{50}} = \frac{50}{7} \]
This can be expressed as:
\[ \frac{50}{7} \text{ days} \approx 7 \frac{1}{7} \text{ days} \]

### Final Answer
Thus, A and B together can complete the work in \( \frac{50}{7} \) days or approximately \( 7 \frac{1}{7} \) days.

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