Suman divided a certain sum between her three daughters in the ratio 2 : 3 : 4. Had she divided the sum in the ratio `1/2:1/3:1/4`, the daughter who got the least share earlier, would have got ₹3,500 more. The sum was:

Options

A. ₹14,000
B. ₹14,500
C. ₹13,750
D. ₹14,625

Answer

To solve the problem step by step, we will follow the given ratios and the information provided about the shares of Suman's daughters.

### Step 1: Understand the Ratios
Suman divides a certain sum among her three daughters in the ratio of 2:3:4. Let's denote the shares of the daughters as:
- Daughter 1: 2x
- Daughter 2: 3x
- Daughter 3: 4x

### Step 2: Calculate the Total Ratio
The total parts in the ratio 2:3:4 is:
\[ 2 + 3 + 4 = 9 \]
So, the total sum can be expressed as:
\[ \text{Total Sum} = 9x \]

### Step 3: Determine the Shares in the Second Ratio
If the sum were divided in the ratio \( \frac{1}{2} : \frac{1}{3} : \frac{1}{4} \), we first need to convert these fractions into a common ratio. The least common multiple (LCM) of the denominators (2, 3, and 4) is 12. Thus, we can express the new ratio as:
- \( \frac{1}{2} = \frac{6}{12} \)
- \( \frac{1}{3} = \frac{4}{12} \)
- \( \frac{1}{4} = \frac{3}{12} \)

This gives us the new ratio:
\[ 6:4:3 \]

### Step 4: Calculate the Total Parts of the New Ratio
The total parts in the new ratio 6:4:3 is:
\[ 6 + 4 + 3 = 13 \]
So, the total sum in this case can be expressed as:
\[ \text{Total Sum} = 13y \]

### Step 5: Identify the Least Share in Both Ratios
From the first ratio (2:3:4), the least share is:
\[ 2x \]

From the second ratio (6:4:3), the least share is:
\[ 3y \]

### Step 6: Set Up the Equation Based on the Given Information
According to the problem, the daughter who got the least share earlier (2x) would have received ₹3,500 more in the second division (3y). Therefore, we can set up the equation:
\[ 3y = 2x + 3500 \]

### Step 7: Relate the Two Expressions for the Total Sum
Since both expressions represent the same total sum, we can equate them:
\[ 9x = 13y \]

### Step 8: Solve the Equations
From the equation \( 9x = 13y \), we can express \( y \) in terms of \( x \):
\[ y = \frac{9x}{13} \]

Now substitute \( y \) in the earlier equation:
\[ 3\left(\frac{9x}{13}\right) = 2x + 3500 \]
\[ \frac{27x}{13} = 2x + 3500 \]

### Step 9: Clear the Fraction
Multiply the entire equation by 13 to eliminate the fraction:
\[ 27x = 26x + 45500 \]
\[ 27x - 26x = 45500 \]
\[ x = 45500 \]

### Step 10: Calculate the Total Sum
Now substitute \( x \) back into the total sum:
\[ \text{Total Sum} = 9x = 9 \times 45500 = 409500 \]

### Final Answer
The total sum Suman divided among her daughters is ₹409,500.