A sum of Rs. 3690 is divided among P, Q, R and S such that `("P's share")/("Q's share") = ("Q's share")/("R's share") = ("R's share")/("S' share") = (4)/(5)`.
Then, Q's share is _______ .

Options

A. Rs.1250
B. Rs. 1000
C. Rs. 640
D. Rs. 800

Answer

To solve the problem of dividing Rs. 3690 among P, Q, R, and S based on the given ratios, we can follow these steps:

### Step 1: Understand the Ratios
We are given that:
\[ \frac{P}{Q} = \frac{Q}{R} = \frac{R}{S} = \frac{4}{5} \]
This means that we can express the shares of P, Q, R, and S in terms of a common variable.

### Step 2: Express Shares in Terms of a Common Variable
Let’s denote Q's share as \( 5x \) (since Q is the middle term in the ratio). Then we can express the other shares as:
- \( P = \frac{4}{5} \times Q = \frac{4}{5} \times 5x = 4x \)
- \( R = \frac{5}{4} \times Q = \frac{5}{4} \times 5x = 6.25x \)
- \( S = \frac{5}{4} \times R = \frac{5}{4} \times 6.25x = 7.8125x \)

However, to avoid fractions, we can multiply all shares by 20 to make calculations easier:
- \( P = 4x \times 20 = 80x \)
- \( Q = 5x \times 20 = 100x \)
- \( R = 6.25x \times 20 = 125x \)
- \( S = 7.8125x \times 20 = 156.25x \)

### Step 3: Calculate Total Shares
Now, we can add all the shares together:
\[ P + Q + R + S = 80x + 100x + 125x + 156.25x = 461.25x \]

### Step 4: Set Up the Equation
We know that the total amount is Rs. 3690, so we can set up the equation:
\[ 461.25x = 3690 \]

### Step 5: Solve for x
Now, we solve for \( x \):
\[ x = \frac{3690}{461.25} = 8 \]

### Step 6: Find Q's Share
Now that we have \( x \), we can find Q's share:
\[ Q = 100x = 100 \times 8 = 800 \]

### Final Answer
Thus, Q's share is **Rs. 800**.

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