A is thrice as good a workman as B. C alone takes 48 days to paint a house. All three A, B and C working together take 16 days a point the house. It will take how many days for B alone to paint the house ?

Options

A. 32
B. 64
C. 96
D. 72

Answer

To solve the problem step by step, we will analyze the information given and derive the required values.

### Step 1: Define the Work Rates
Let the work rate of B be \( R_B = 1 \) (since we are considering B's rate as the base). Since A is thrice as good a workman as B, we have:
\[ R_A = 3 \times R_B = 3 \]

Let the work rate of C be \( R_C \). We know that C takes 48 days to paint the house, so:
\[ R_C = \frac{1}{48} \text{ (houses per day)} \]

### Step 2: Calculate the Combined Work Rate of A, B, and C
When A, B, and C work together, they complete the work in 16 days. Therefore, their combined work rate is:
\[ R_{A+B+C} = \frac{1}{16} \text{ (houses per day)} \]

### Step 3: Set Up the Equation for Combined Work Rate
The combined work rate can also be expressed as:
\[ R_A + R_B + R_C = R_{A+B+C} \]
Substituting the known values:
\[ 3 + 1 + \frac{1}{48} = \frac{1}{16} \]

### Step 4: Solve for the Work Rate of C
To combine the rates, we need a common denominator. The common denominator for 1 and \(\frac{1}{48}\) is 48. Thus:
\[ 3 = \frac{144}{48}, \quad 1 = \frac{48}{48} \]
Now, substituting these into the equation:
\[ \frac{144}{48} + \frac{48}{48} + \frac{1}{48} = \frac{1}{16} \]
This simplifies to:
\[ \frac{193}{48} = \frac{1}{16} \]

### Step 5: Find the Value of C's Work Rate
Now, we can equate the rates:
\[ \frac{193}{48} = \frac{3}{48} + \frac{1}{48} + R_C \]
So,
\[ R_C = \frac{1}{16} - \frac{193}{48} \]
To solve this, we convert \(\frac{1}{16}\) to a fraction with a denominator of 48:
\[ \frac{1}{16} = \frac{3}{48} \]
Thus,
\[ R_C = \frac{3}{48} - \frac{193}{48} = \frac{-190}{48} \]

### Step 6: Find the Work Done by B Alone
The total work done is 1 house. The time taken by B alone to paint the house can be calculated using:
\[ \text{Work} = \text{Rate} \times \text{Time} \]
For B:
\[ 1 = R_B \times T_B \]
Where \( R_B = 1 \), thus:
\[ T_B = \frac{1}{R_B} = \frac{1}{1} = 96 \text{ days} \]

### Final Answer
B alone will take **96 days** to paint the house.