A and B together can complete a work in 35 days while A alone can complete the same work in 60 days. In how many days, B alone will be able to complete the same work?

Options

A. 80 days
B. 84 days
C. 84 days
D. 92 days

Answer

To find out how many days B alone will take to complete the work, we can follow these steps:

### Step 1: Determine the work rate of A
A can complete the work in 60 days. Therefore, the work rate of A is:
\[ \text{Work rate of A} = \frac{1 \text{ work}}{60 \text{ days}} = \frac{1}{60} \text{ work per day} \]

### Step 2: Determine the work rate of A and B together
A and B together can complete the work in 35 days. Therefore, their combined work rate is:
\[ \text{Work rate of A and B} = \frac{1 \text{ work}}{35 \text{ days}} = \frac{1}{35} \text{ work per day} \]

### Step 3: Set up the equation for B's work rate
Let B alone take X days to complete the work. Thus, the work rate of B is:
\[ \text{Work rate of B} = \frac{1 \text{ work}}{X \text{ days}} = \frac{1}{X} \text{ work per day} \]

### Step 4: Combine the work rates
Since the work rate of A and B together is the sum of their individual work rates, we can write:
\[ \text{Work rate of A} + \text{Work rate of B} = \text{Work rate of A and B} \]
Substituting the values we have:
\[ \frac{1}{60} + \frac{1}{X} = \frac{1}{35} \]

### Step 5: Solve for X
To solve for X, we first find a common denominator for the fractions. The common denominator for 60, X, and 35 is 420X. We can rewrite the equation:
\[ \frac{7X}{420X} + \frac{420}{420X} = \frac{12}{420} \]
This simplifies to:
\[ 7X + 420 = 12X \]
Now, rearranging gives:
\[ 420 = 12X - 7X \]
\[ 420 = 5X \]
Dividing both sides by 5:
\[ X = \frac{420}{5} = 84 \]

### Conclusion
B alone will be able to complete the same work in **84 days**.

---