A can complete a work in 20 days and B in 30 days. A worked alone for 4 days and then B completed the remaining work along with C in 18 days . In how many days can C working alone complete the work ?

Options

A. 12
B. 68
C. 72
D. 90

Answer

To solve the problem, we need to determine how much work A, B, and C can do individually and then find out how long C would take to complete the entire work alone.

### Step 1: Determine the work done by A in 4 days.

A can complete the work in 20 days. Therefore, the work done by A in one day is:

\[ \text{Work done by A in 1 day} = \frac{1}{20} \]

In 4 days, A will complete:

\[ \text{Work done by A in 4 days} = 4 \times \frac{1}{20} = \frac{4}{20} = \frac{1}{5} \]

### Step 2: Calculate the remaining work after A's contribution.

The total work can be considered as 1 (the whole work). After A has completed \(\frac{1}{5}\) of the work, the remaining work is:

\[ \text{Remaining work} = 1 - \frac{1}{5} = \frac{4}{5} \]

### Step 3: Determine the combined work rate of B and C.

B can complete the work in 30 days, so the work done by B in one day is:

\[ \text{Work done by B in 1 day} = \frac{1}{30} \]

Let C's work rate be \(\frac{1}{C}\) (where C can complete the work in C days). The combined work done by B and C in one day is:

\[ \text{Work done by B and C in 1 day} = \frac{1}{30} + \frac{1}{C} \]

### Step 4: Calculate the work done by B and C together in 18 days.

In 18 days, B and C together complete the remaining work of \(\frac{4}{5}\):

\[ \text{Work done by B and C in 18 days} = 18 \left( \frac{1}{30} + \frac{1}{C} \right) = \frac{4}{5} \]

### Step 5: Set up the equation.

Now we can set up the equation:

\[ 18 \left( \frac{1}{30} + \frac{1}{C} \right) = \frac{4}{5} \]

### Step 6: Solve for C.

First, simplify the left side:

\[ \frac{18}{30} + \frac{18}{C} = \frac{4}{5} \]

This simplifies to:

\[ \frac{3}{5} + \frac{18}{C} = \frac{4}{5} \]

Subtract \(\frac{3}{5}\) from both sides:

\[ \frac{18}{C} = \frac{4}{5} - \frac{3}{5} = \frac{1}{5} \]

Now, cross-multiply to solve for C:

\[ 18 \cdot 5 = 1 \cdot C \implies C = 90 \]

### Conclusion:

C can complete the work alone in 90 days.