To complete a work, A takes 50% more time than B. If together they take 18 days to complete the work, how much time shall B take to do it ?

Options

A. 30 days
B. 35 days
C. 40 days
D. 45 days

Answer

To solve the problem step by step, let's denote the time taken by B to complete the work alone as \( x \) days. According to the problem, A takes 50% more time than B, which means A takes \( x + 0.5x = 1.5x \) days to complete the work alone.

Now, we know that when A and B work together, they can complete the work in 18 days. We can express their work rates as follows:

- The work rate of A is \( \frac{1}{1.5x} = \frac{2}{3x} \) (since work done is the reciprocal of time taken).
- The work rate of B is \( \frac{1}{x} \).

When A and B work together, their combined work rate is:

\[ \frac{2}{3x} + \frac{1}{x} = \frac{2 + 3}{3x} = \frac{5}{3x} \]

Since they complete the work together in 18 days, their combined work rate can also be expressed as:

\[ \frac{1}{18} \]

Setting the two expressions for the combined work rate equal to each other gives us:

\[ \frac{5}{3x} = \frac{1}{18} \]

Now, we can cross-multiply to solve for \( x \):

\[ 5 \cdot 18 = 3x \]

\[ 90 = 3x \]

Dividing both sides by 3:

\[ x = 30 \]

Thus, B takes 30 days to complete the work alone.

### Summary of Steps:
1. Denote the time taken by B as \( x \).
2. A takes \( 1.5x \) days.
3. Calculate the work rates of A and B.
4. Set the combined work rate equal to \( \frac{1}{18} \).
5. Solve for \( x \).