To solve the problem of dividing Rs. 3,105 among K, L, and M in the proportions of \( \frac{2}{3} : \frac{3}{4} : \frac{1}{2} \), we will follow these steps:
### Step 1: Find a common denominator for the proportions
The proportions given are \( \frac{2}{3} \), \( \frac{3}{4} \), and \( \frac{1}{2} \). To add these fractions, we need a common denominator. The least common multiple (LCM) of the denominators (3, 4, and 2) is 12.
### Step 2: Convert each fraction to have the common denominator
Now, we convert each fraction:
- \( \frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12} \)
- \( \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \)
- \( \frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12} \)
### Step 3: Add the converted fractions
Now we add the fractions:
\[
\frac{8}{12} + \frac{9}{12} + \frac{6}{12} = \frac{8 + 9 + 6}{12} = \frac{23}{12}
\]
### Step 4: Calculate the value of one part
The total amount to be divided is Rs. 3,105. To find the value of one part, we set up the equation:
\[
\frac{23}{12} \text{ parts} = 3105
\]
To find the value of one part, we calculate:
\[
1 \text{ part} = \frac{3105 \times 12}{23}
\]
### Step 5: Simplify the calculation
Calculating \( 3105 \times 12 \):
\[
3105 \times 12 = 37260
\]
Now divide by 23:
\[
1 \text{ part} = \frac{37260}{23} = 1620
\]
### Step 6: Calculate the individual shares
Now we can calculate the shares for K, L, and M:
- K's share:
\[
K = \frac{8}{12} \times 1620 = \frac{8 \times 1620}{12} = 1080
\]
- L's share:
\[
L = \frac{9}{12} \times 1620 = \frac{9 \times 1620}{12} = 1215
\]
- M's share:
\[
M = \frac{6}{12} \times 1620 = \frac{6 \times 1620}{12} = 810
\]
### Step 7: Identify the least amount
From the calculated shares:
- K gets Rs. 1,080
- L gets Rs. 1,215
- M gets Rs. 810
The least amount is Rs. 810, which is M's share.
### Final Answer
Thus, the least amount received is:
\[
\text{M gets Rs. } 810
\]
