A sum of Rs 16,200 was divided among A, B and C. Had A received Rs 1,200 less, the ratio of the shares would have been 3 : 4 : 5. What was A's share, initially?

Options

A. Rs 4,950
B. Rs 6,250
C. Rs 3,750
D. Rs 6,050

Answer

To solve the problem step by step, let's denote A's initial share as \( x \), B's share as \( y \), and C's share as \( z \).

### Step 1: Set up the equation for the total sum
We know that the total amount of money distributed among A, B, and C is Rs 16,200. Therefore, we can write the equation:
\[ x + y + z = 16200 \]

### Step 2: Adjust A's share and set up the ratio
According to the problem, if A had received Rs 1,200 less, his new share would be \( x - 1200 \). The new ratio of their shares would then be 3:4:5. This can be expressed as:
\[ \frac{x - 1200}{3} = \frac{y}{4} = \frac{z}{5} \]

### Step 3: Express y and z in terms of x
Let’s denote the common multiple of the ratios as \( k \). Then we can express y and z in terms of x:
\[ x - 1200 = 3k \quad (1) \]
\[ y = 4k \quad (2) \]
\[ z = 5k \quad (3) \]

### Step 4: Substitute y and z into the total sum equation
Now substitute equations (2) and (3) into the total sum equation:
\[ x + 4k + 5k = 16200 \]
This simplifies to:
\[ x + 9k = 16200 \quad (4) \]

### Step 5: Substitute x from equation (1) into equation (4)
From equation (1), we can express \( k \) in terms of \( x \):
\[ k = \frac{x - 1200}{3} \]
Now substitute \( k \) back into equation (4):
\[ x + 9\left(\frac{x - 1200}{3}\right) = 16200 \]
Multiply through by 3 to eliminate the fraction:
\[ 3x + 9(x - 1200) = 48600 \]
This simplifies to:
\[ 3x + 9x - 10800 = 48600 \]
Combine like terms:
\[ 12x - 10800 = 48600 \]

### Step 6: Solve for x
Add 10800 to both sides:
\[ 12x = 59400 \]
Now divide by 12:
\[ x = 4950 \]

### Step 7: Find A's initial share
Thus, A's initial share is Rs 4,950.

### Final Answer
A's initial share is **Rs 4,950**.