A sum of money is to be distributed among P, Q, R, and S in the proportion 5 ∶ 2 ∶ 4 ∶ 3, respectively.

If R gets Rs. 1000 more than S, what is the share of Q (in Rs.)?

Money Distribution Problem: Understanding Ratios

This problem focuses on the distribution of a sum of money among four individuals—P, Q, R, and S—according to a specified ratio. We are given a condition about the difference in shares between R and S, which is key to finding the value of each part of the ratio and subsequently, the specific share of Q.

Ratio of Money Distribution

The shares of P, Q, R, and S are in the proportion 5 ∶ 2 ∶ 4 ∶ 3, respectively. This means that if the total money is divided into parts, P gets 5 parts, Q gets 2 parts, R gets 4 parts, and S gets 3 parts.

- P's share ∶ Q's share ∶ R's share ∶ S's share = 5 ∶ 2 ∶ 4 ∶ 3

Representing Individual Shares Algebraically

To calculate the actual money distributed, we can introduce a common multiplier, often denoted as $x$, for each part of the ratio. This allows us to express each person's share as an algebraic term:

- Share of P $=5x$
- Share of Q $=2x$
- Share of R $=4x$
- Share of S $=3x$

**Calculating the Value of Each Share Part ($x$)**

The problem states a critical piece of information: "R gets Rs. 1000 more than S". We can translate this into an equation using our algebraic representations of the shares:

R's share $-$ S's share $=$ Rs. 1000

Substitute the expressions for R's and S's shares into the equation:

$4x-3x=1000$

Simplifying the equation gives us the value of $x$:

$x=1000$

This means that each 'part' in our ratio corresponds to Rs. 1000.

Determining Q's Share

Now that we have found the value of $x$, we can easily calculate Q's share. From our initial representation:

Q's share $=2x$

Substitute the value of $x=1000$:

Q's share $=2\times 1000$

Q's share $=2000$

Therefore, the share of Q is Rs. 2000.

Verification of Money Distribution

Let's quickly verify the shares for all individuals using $x=1000$:We can see that R's share (Rs. 4000) is indeed Rs. 1000 more than S's share (Rs. 3000), which matches the condition given in the problem statement.