If A alone can do a job in 40 days then, in how many days can B alone do the job if together they can do the job in 8 days?

Options

A. 15
B. 10
C. 20
D. 25

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Work Done by A
A can complete the job in 40 days. This means that A's work rate (efficiency) is:
\[ \text{Efficiency of A} = \frac{1 \text{ job}}{40 \text{ days}} = \frac{1}{40} \text{ jobs per day} \]

### Step 2: Understand the Work Done by A and B Together
A and B together can complete the job in 8 days. Therefore, their combined work rate is:
\[ \text{Efficiency of A and B} = \frac{1 \text{ job}}{8 \text{ days}} = \frac{1}{8} \text{ jobs per day} \]

### Step 3: Calculate the Efficiency of B
Now, we can find the efficiency of B. We know the efficiency of A and the combined efficiency of A and B:
\[ \text{Efficiency of B} = \text{Efficiency of A and B} - \text{Efficiency of A} \]
Substituting the values we have:
\[ \text{Efficiency of B} = \frac{1}{8} - \frac{1}{40} \]

### Step 4: Find a Common Denominator
To subtract these fractions, we need a common denominator. The least common multiple (LCM) of 8 and 40 is 40. We can convert \(\frac{1}{8}\) to have a denominator of 40:
\[ \frac{1}{8} = \frac{5}{40} \]
Now we can perform the subtraction:
\[ \text{Efficiency of B} = \frac{5}{40} - \frac{1}{40} = \frac{4}{40} = \frac{1}{10} \text{ jobs per day} \]

### Step 5: Calculate the Time Taken by B Alone
To find out how many days B alone would take to complete the job, we take the reciprocal of B's efficiency:
\[ \text{Time taken by B} = \frac{1 \text{ job}}{\text{Efficiency of B}} = \frac{1}{\frac{1}{10}} = 10 \text{ days} \]

### Final Answer
B alone can complete the job in **10 days**.

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