A sum of Rs.18000 is divided between P, Q, R, and S such that the ratio of shares of P and Q is 8:9, that of Q and R is `2:3` and that of R and S is 9:13. Find the difference between the shares of Q and R?

Options

A. Rs.2520
B. Rs.1120
C. Rs.3780
D. Rs.1620

Answer

To solve the problem step by step, we will follow the ratios given and find the shares of P, Q, R, and S, and then calculate the difference between the shares of Q and R.

### Step 1: Understand the Ratios
We have the following ratios:
- P:Q = 8:9
- Q:R = 2:3
- R:S = 9:13

### Step 2: Express All Shares in Terms of a Common Variable
Let us denote the shares of P, Q, R, and S in terms of a common variable \( k \).

1. From the ratio P:Q = 8:9, we can express:
- P = 8k
- Q = 9k

2. From the ratio Q:R = 2:3, we can express:
- Q = 2m
- R = 3m

Since Q is already expressed as 9k, we can set:
- 9k = 2m
- Therefore, \( m = \frac{9k}{2} \)

Now substituting for R:
- R = 3m = 3 * \( \frac{9k}{2} \) = \( \frac{27k}{2} \)

3. From the ratio R:S = 9:13, we can express:
- R = 9n
- S = 13n

Since R is already expressed as \( \frac{27k}{2} \), we can set:
- \( \frac{27k}{2} = 9n \)
- Therefore, \( n = \frac{27k}{18} = \frac{3k}{2} \)

Now substituting for S:
- S = 13n = 13 * \( \frac{3k}{2} \) = \( \frac{39k}{2} \)

### Step 3: Combine All Shares
Now we have:
- P = 8k
- Q = 9k
- R = \( \frac{27k}{2} \)
- S = \( \frac{39k}{2} \)

### Step 4: Set Up the Equation
The total sum of the shares is Rs. 18000:
\[ 8k + 9k + \frac{27k}{2} + \frac{39k}{2} = 18000 \]

### Step 5: Simplify the Equation
Combine the terms:
\[ 8k + 9k + \frac{27k + 39k}{2} = 18000 \]
\[ 17k + \frac{66k}{2} = 18000 \]
\[ 17k + 33k = 18000 \]
\[ 50k = 18000 \]

### Step 6: Solve for k
\[ k = \frac{18000}{50} = 360 \]

### Step 7: Find the Shares
Now we can find the individual shares:
- P = 8k = 8 * 360 = 2880
- Q = 9k = 9 * 360 = 3240
- R = \( \frac{27k}{2} = \frac{27 * 360}{2} = 4860 \)
- S = \( \frac{39k}{2} = \frac{39 * 360}{2} = 7020 \)

### Step 8: Calculate the Difference Between Q and R
The difference between the shares of Q and R is:
\[ R - Q = 4860 - 3240 = 1620 \]

### Final Answer
The difference between the shares of Q and R is Rs. 1620.

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