Divide Rs1,870 into three parts in such a way that half of the first part, one-third of the second part and one-sixth of the third part are all equal.

Answer

To solve the problem of dividing Rs. 1,870 into three parts such that half of the first part, one-third of the second part, and one-sixth of the third part are all equal, we can follow these steps:

### Step 1: Define the Parts
Let the three parts be:
- First part = \( x \)
- Second part = \( y \)
- Third part = \( z \)

### Step 2: Set Up the Equations
According to the problem, we have:
\[ \frac{x}{2} = \frac{y}{3} = \frac{z}{6} \]
Let’s denote this common value as \( k \). Therefore, we can express \( x \), \( y \), and \( z \) in terms of \( k \):
\[ x = 2k \]
\[ y = 3k \]
\[ z = 6k \]

### Step 3: Set Up the Total Equation
Since the total amount is Rs. 1,870, we can write:
\[ x + y + z = 1870 \]
Substituting the values of \( x \), \( y \), and \( z \):
\[ 2k + 3k + 6k = 1870 \]
This simplifies to:
\[ 11k = 1870 \]

### Step 4: Solve for \( k \)
Now, we can solve for \( k \):
\[ k = \frac{1870}{11} = 170 \]

### Step 5: Find the Values of \( x \), \( y \), and \( z \)
Now that we have \( k \), we can find \( x \), \( y \), and \( z \):
\[ x = 2k = 2 \times 170 = 340 \]
\[ y = 3k = 3 \times 170 = 510 \]
\[ z = 6k = 6 \times 170 = 1020 \]

### Step 6: Conclusion
Thus, the three parts are:
- First part \( x = Rs. 340 \)
- Second part \( y = Rs. 510 \)
- Third part \( z = Rs. 1020 \)

### Final Answer:
The three parts are Rs. 340, Rs. 510, and Rs. 1020.

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