A can do `(7)/(8)` of work in 28 days, B can do `(5)/(6)` of the same work in 20 days. The number of days they will take to complete if they do it together is .

Options

A. `15(3)/(7)` days
B. `17(3)/(5)` days
C. `14(5)/(7)` days
D. `13(5)/(7)` days

Answer

To solve the problem step by step, we need to determine the work rates of A and B, and then find out how long it will take them to complete the work together.

### Step 1: Determine the total work in terms of units
Let's assume the total work is represented by 1 unit.

### Step 2: Calculate A's work rate
A can do \( \frac{7}{8} \) of the work in 28 days. Therefore, the amount of work A can do in one day is:
\[ \text{Work done by A in one day} = \frac{7/8}{28} = \frac{7}{8 \times 28} = \frac{7}{224} = \frac{1}{32} \text{ units per day} \]

### Step 3: Calculate B's work rate
B can do \( \frac{5}{6} \) of the work in 20 days. Therefore, the amount of work B can do in one day is:
\[ \text{Work done by B in one day} = \frac{5/6}{20} = \frac{5}{6 \times 20} = \frac{5}{120} = \frac{1}{24} \text{ units per day} \]

### Step 4: Calculate the combined work rate of A and B
Now, we can find their combined work rate by adding their individual work rates:
\[ \text{Combined work rate} = \text{Work done by A in one day} + \text{Work done by B in one day} = \frac{1}{32} + \frac{1}{24} \]

To add these fractions, we need a common denominator. The least common multiple of 32 and 24 is 96.

Converting the fractions:
\[ \frac{1}{32} = \frac{3}{96} \quad \text{(since } 1 \times 3 = 3 \text{ and } 32 \times 3 = 96\text{)} \]
\[ \frac{1}{24} = \frac{4}{96} \quad \text{(since } 1 \times 4 = 4 \text{ and } 24 \times 4 = 96\text{)} \]

Now, adding these fractions:
\[ \text{Combined work rate} = \frac{3}{96} + \frac{4}{96} = \frac{7}{96} \text{ units per day} \]

### Step 5: Calculate the time taken to complete the work together
If A and B work together, they complete \( \frac{7}{96} \) of the work in one day. To find out how many days it will take to complete 1 unit of work, we take the reciprocal of their combined work rate:
\[ \text{Time taken} = \frac{1 \text{ unit}}{\frac{7}{96} \text{ units per day}} = \frac{96}{7} \text{ days} \]

### Step 6: Final answer
Calculating \( \frac{96}{7} \):
\[ \frac{96}{7} \approx 13.71 \text{ days} \]

Thus, A and B together will take approximately \( 13.71 \) days to complete the work.