A sum of Rs. 3170 is divided among X, Y and Z such that if Rs. 13, Rs. 12 and Rs. 18 will be diminished from the shares of X, Y and Z respectively, then their shares will be in the ratio `20 : 18 : 21`. What is the initial share (in Rs.) of Z?

Options

A. 1131
B. 1530
C. 910
D. 1350

Answer

To find the initial share of Z, we will follow these steps:

### Step 1: Set up the equation based on the problem statement
We know that the total amount is Rs. 3170. If Rs. 13, Rs. 12, and Rs. 18 are deducted from the shares of X, Y, and Z respectively, their shares will be in the ratio of 20:18:21.

Let the shares of X, Y, and Z be represented as \( x \), \( y \), and \( z \) respectively. After the deductions, we can express their shares as:
- X's share: \( x - 13 \)
- Y's share: \( y - 12 \)
- Z's share: \( z - 18 \)

According to the problem, these shares are in the ratio:
\[ \frac{x - 13}{20} = \frac{y - 12}{18} = \frac{z - 18}{21} \]

### Step 2: Express the shares in terms of a common variable
Let us denote the common variable as \( k \). Then we can express the shares as:
- \( x - 13 = 20k \) → \( x = 20k + 13 \)
- \( y - 12 = 18k \) → \( y = 18k + 12 \)
- \( z - 18 = 21k \) → \( z = 21k + 18 \)

### Step 3: Write the total amount equation
Now, we know that the total amount is Rs. 3170. Therefore, we can write:
\[ x + y + z = 3170 \]
Substituting the expressions for \( x \), \( y \), and \( z \):
\[ (20k + 13) + (18k + 12) + (21k + 18) = 3170 \]

### Step 4: Simplify the equation
Combining like terms, we get:
\[ 20k + 18k + 21k + 13 + 12 + 18 = 3170 \]
\[ 59k + 43 = 3170 \]

### Step 5: Solve for \( k \)
Subtract 43 from both sides:
\[ 59k = 3170 - 43 \]
\[ 59k = 3127 \]
Now, divide by 59:
\[ k = \frac{3127}{59} = 53 \]

### Step 6: Find the initial share of Z
Now that we have \( k \), we can find the initial share of Z:
\[ z = 21k + 18 \]
Substituting \( k = 53 \):
\[ z = 21(53) + 18 \]
\[ z = 1113 + 18 = 1131 \]

### Final Answer
The initial share of Z is Rs. 1131.

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