A sum of `₹ 221` is divided among, X,Y and Z such taht X gets `₹ 52` more than X.Y gets `₹ 26` more than Z. The ratio of the shares of X,Y and Z respectively is :

Options

A. `9:5:3`
B. `9:3:5`
C. ` 5:9:3`
D. `10:6:5`

Explanation

To solve the problem, we need to find the shares of X, Y, and Z based on the conditions provided. Let's break it down step by step.

### Step 1: Define the variables
Let:
- The amount received by Z be \( z \).
- The amount received by Y be \( y \).
- The amount received by X be \( x \).

### Step 2: Set up the equations based on the problem statement
According to the problem:
1. \( x = y + 52 \) (X gets ₹52 more than Y)
2. \( y = z + 26 \) (Y gets ₹26 more than Z)
3. The total amount is \( x + y + z = 221 \).

### Step 3: Substitute the values
From the second equation, we can express \( z \) in terms of \( y \):
\[ z = y - 26 \]

Now, substitute \( z \) in the total amount equation:
\[ x + y + (y - 26) = 221 \]

### Step 4: Substitute \( x \) in the total amount equation
Now, substitute \( x \) from the first equation:
\[ (y + 52) + y + (y - 26) = 221 \]

### Step 5: Simplify the equation
Combine like terms:
\[ 3y + 26 = 221 \]

### Step 6: Solve for \( y \)
Subtract 26 from both sides:
\[ 3y = 221 - 26 \]
\[ 3y = 195 \]

Now, divide by 3:
\[ y = \frac{195}{3} = 65 \]

### Step 7: Find \( x \) and \( z \)
Now that we have \( y \):
- To find \( x \):
\[ x = y + 52 = 65 + 52 = 117 \]

- To find \( z \):
\[ z = y - 26 = 65 - 26 = 39 \]

### Step 8: Write the shares of X, Y, and Z
Now we have:
- \( x = 117 \)
- \( y = 65 \)
- \( z = 39 \)

### Step 9: Find the ratio of X, Y, and Z
The ratio of the shares of X, Y, and Z is:
\[ x : y : z = 117 : 65 : 39 \]

### Step 10: Simplify the ratio
To simplify the ratio, we can divide each term by their greatest common divisor (GCD). The GCD of 117, 65, and 39 is 13.
- \( \frac{117}{13} = 9 \)
- \( \frac{65}{13} = 5 \)
- \( \frac{39}{13} = 3 \)

Thus, the simplified ratio is:
\[ 9 : 5 : 3 \]

### Final Answer
The ratio of the shares of X, Y, and Z is \( 9 : 5 : 3 \).

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