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.)?
2000
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 \(x = 1000\):
| Individual | Ratio Expression | Calculated Share (in Rs.) |
|---|---|---|
| P | \(5x\) | \(5 \times 1000 = 5000\) |
| Q | \(2x\) | \(2 \times 1000 = 2000\) |
| R | \(4x\) | \(4 \times 1000 = 4000\) |
| S | \(3x\) | \(3 \times 1000 = 3000\) |
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.
Important Questions from Ratio and Proportion
In a mixture of 156 litres, the ratio of milk and water is 7 : 6. How much water should be added to make the ratio 14 : 13?
If the ratio of the first to second number is 3 : 4 and that of the second to the third number is 8 : 5, and sum of three numbers is 190 then the third number is:
The third proportional to 9 and 15 is:
The average age of 3 persons is 30 years.If their ages are in the ratio of 3 : 5 : 7 respectively, then the age of the eldest person is:
The income of A and B are in the ratio 5 : 3. The expenses of A, B and C are in the ratio of 8 : 5 : 2. If C spends 2000 and B saves ₹ 700, then A saves:
