A is thrice as good a workman as B and therefore is able to finish a job in 30 days less than B. How many days will they take to finish the job working together?

Options

A. `10 (1)/(4)`
B. `11 (3)/(4)`
C. `11 (1)/(4)`
D. `11 (1)/(3)`

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Define the variables
Let the number of days A takes to complete the job be \( x \). Since A is thrice as good a workman as B, B will take \( 3x \) days to complete the same job.

### Step 2: Set up the equation
According to the problem, A finishes the job in 30 days less than B. Therefore, we can set up the equation:
\[ 3x - x = 30 \]

### Step 3: Simplify the equation
This simplifies to:
\[ 2x = 30 \]

### Step 4: Solve for \( x \)
Now, divide both sides by 2:
\[ x = \frac{30}{2} = 15 \]
So, A takes 15 days to complete the job.

### Step 5: Calculate B's time
Since B takes \( 3x \) days:
\[ B = 3 \times 15 = 45 \text{ days} \]

### Step 6: Calculate the work done by A and B together
To find out how many days A and B will take to finish the job together, we use the formula:
\[ \text{Time taken together} = \frac{A \times B}{A + B} \]
Substituting the values we found:
\[ \text{Time taken together} = \frac{15 \times 45}{15 + 45} \]

### Step 7: Calculate the numerator and denominator
Calculating the numerator:
\[ 15 \times 45 = 675 \]
Calculating the denominator:
\[ 15 + 45 = 60 \]

### Step 8: Divide to find the time taken together
Now, we divide:
\[ \text{Time taken together} = \frac{675}{60} = 11.25 \text{ days} \]
This can also be expressed as:
\[ 11 \frac{1}{4} \text{ days} \]

### Final Answer
Thus, A and B together will take \( 11 \frac{1}{4} \) days to finish the job.

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