Rs 7750 is divided among X, Y and Z such that 3 times of X’s share is equal to 5 times of Y’s share which is equal to 2 times of Z’s share. What is the share (in Rs) of Z?

Options

A. 4250
B. 3750
C. 3875
D. 4475

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Set up the equations based on the given information.
We know that:
- \(3 \times \text{X's share} = 5 \times \text{Y's share} = 2 \times \text{Z's share}\)

Let's denote:
- X's share = \(x\)
- Y's share = \(y\)
- Z's share = \(z\)

From the information given, we can express the shares in terms of a common variable \(k\):
- \(3x = k\) → \(x = \frac{k}{3}\)
- \(5y = k\) → \(y = \frac{k}{5}\)
- \(2z = k\) → \(z = \frac{k}{2}\)

### Step 2: Write the equation for the total amount.
The total amount shared among X, Y, and Z is Rs 7750. Therefore, we can write:
\[ x + y + z = 7750 \]

Substituting the values of \(x\), \(y\), and \(z\) in terms of \(k\):
\[ \frac{k}{3} + \frac{k}{5} + \frac{k}{2} = 7750 \]

### Step 3: Find a common denominator and simplify.
The least common multiple (LCM) of 3, 5, and 2 is 30. We can rewrite the equation as:
\[ \frac{10k}{30} + \frac{6k}{30} + \frac{15k}{30} = 7750 \]

Combining the fractions gives:
\[ \frac{31k}{30} = 7750 \]

### Step 4: Solve for \(k\).
To isolate \(k\), we multiply both sides by 30:
\[ 31k = 7750 \times 30 \]
Calculating the right side:
\[ 31k = 232500 \]
Now, divide by 31:
\[ k = \frac{232500}{31} = 7500 \]

### Step 5: Find Z's share.
Now that we have \(k\), we can find Z's share using the equation \(z = \frac{k}{2}\):
\[ z = \frac{7500}{2} = 3750 \]

### Final Answer:
The share of Z is Rs 3750.

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