M can complete `(3)/(4)` part of a work in 12 days and N can complete `(2)/(7)` part of the same work in 8 days. In how many days will both complete `(11)/(14)` part of the total work?

Options

A. 8
B. 9
C. 7
D. 6

Answer

To solve the problem step-by-step, we will first determine the total work done by M and N in terms of days and then find out how long it will take them to complete \( \frac{11}{14} \) of the total work together.

### Step 1: Calculate the total work done by M
M can complete \( \frac{3}{4} \) of the work in 12 days.

To find out how many days M would take to complete the entire work (W), we can set up the following equation:

\[ \frac{3}{4}W = 12 \text{ days} \]

To find W (the total work), we rearrange the equation:

\[ W = 12 \times \frac{4}{3} = 16 \text{ days} \]

So, M can complete the entire work in 16 days.

### Step 2: Calculate the total work done by N
N can complete \( \frac{2}{7} \) of the work in 8 days.

Using a similar approach, we set up the equation:

\[ \frac{2}{7}W = 8 \text{ days} \]

Rearranging gives us:

\[ W = 8 \times \frac{7}{2} = 28 \text{ days} \]

Thus, N can complete the entire work in 28 days.

### Step 3: Calculate the efficiency of M and N
Now, we can calculate the efficiency (work done per day) of both M and N.

- The efficiency of M:
\[ \text{Efficiency of M} = \frac{W}{\text{Days taken by M}} = \frac{1}{16} \text{ work/day} \]

- The efficiency of N:
\[ \text{Efficiency of N} = \frac{W}{\text{Days taken by N}} = \frac{1}{28} \text{ work/day} \]

### Step 4: Combine the efficiencies
To find the combined efficiency of M and N, we add their efficiencies together:

\[ \text{Combined Efficiency} = \frac{1}{16} + \frac{1}{28} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 16 and 28 is 112.

Converting the fractions:

\[ \frac{1}{16} = \frac{7}{112}, \quad \frac{1}{28} = \frac{4}{112} \]

Now, adding them:

\[ \text{Combined Efficiency} = \frac{7}{112} + \frac{4}{112} = \frac{11}{112} \text{ work/day} \]

### Step 5: Calculate the time taken to complete \( \frac{11}{14} \) of the work
Now, we need to find out how many days it will take for M and N to complete \( \frac{11}{14} \) of the total work.

First, we calculate \( \frac{11}{14} \) of the total work (W):

\[ \frac{11}{14}W = \frac{11}{14} \times 1 = \frac{11}{14} \text{ of the work} \]

Now, we need to find the time taken (T) to complete this work using the combined efficiency:

\[ T = \frac{\text{Work}}{\text{Efficiency}} = \frac{\frac{11}{14}}{\frac{11}{112}} = \frac{11}{14} \times \frac{112}{11} \]

The \( 11 \) cancels out:

\[ T = \frac{112}{14} = 8 \text{ days} \]

### Final Answer
Both M and N will complete \( \frac{11}{14} \) part of the total work in **8 days**.

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