If A and B can do a piece of work in 20 days, and A alone can do the same work in 30 days, then in how many days can B alone complete the same work?

Options

A. 60
B. 40
C. 75
D. 50

Answer

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

### Step 1: Determine the work done by A and B together
If A and B can complete the work in 20 days, we can calculate their combined work rate. The work rate is the reciprocal of the time taken to complete the work.

\[ \text{Work rate of A and B together} = \frac{1}{20} \text{ (work per day)} \]

### Step 2: Determine the work done by A alone
A can complete the work alone in 30 days. Therefore, A's work rate is:

\[ \text{Work rate of A} = \frac{1}{30} \text{ (work per day)} \]

### Step 3: Calculate the work done by B alone
To find B's work rate, we can use the equation for combined work rates. The combined work rate of A and B is equal to the sum of their individual work rates:

\[ \text{Work rate of A and B} = \text{Work rate of A} + \text{Work rate of B} \]

Substituting the known values:

\[ \frac{1}{20} = \frac{1}{30} + \text{Work rate of B} \]

### Step 4: Solve for B's work rate
Now, we need to isolate B's work rate:

\[ \text{Work rate of B} = \frac{1}{20} - \frac{1}{30} \]

To perform this subtraction, we need a common denominator. The least common multiple of 20 and 30 is 60. Thus, we convert the fractions:

\[ \frac{1}{20} = \frac{3}{60} \quad \text{and} \quad \frac{1}{30} = \frac{2}{60} \]

Now substituting back:

\[ \text{Work rate of B} = \frac{3}{60} - \frac{2}{60} = \frac{1}{60} \text{ (work per day)} \]

### Step 5: Calculate the time taken by B to complete the work alone
Since B's work rate is \(\frac{1}{60}\), it means B can complete the work in:

\[ \text{Time taken by B} = \frac{1}{\text{Work rate of B}} = \frac{1}{\frac{1}{60}} = 60 \text{ days} \]

### Final Answer
B alone can complete the work in **60 days**.

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