Rajnish invested certain sum in three different schemes P, Q and R with the rates of interest 10% per annum, 12% per annum and 15% per annum, respectively. If the total interest accrued in 1 yr was RS 3200and the amount invested in scheme R was 150% of the amount invested in scheme Q, as well as R was 240% amount invested in scheme P. what was the amount invested in scheme Q?

Options

A. X8000
B. X9000
C. X5000
D. X3050

Answer

To solve the problem step by step, we will denote the amounts invested in schemes P, Q, and R as A, B, and C respectively.

### Step 1: Set up the equations based on the given information
1. The interest rates for the schemes are:
- Scheme P: 10% per annum
- Scheme Q: 12% per annum
- Scheme R: 15% per annum

2. The total interest accrued in 1 year is given as RS 3200. Therefore, we can express this as:
\[ \text{Total Interest} = \frac{10}{100}A + \frac{12}{100}B + \frac{15}{100}C = 3200 \]
Simplifying this, we get:
\[ 0.1A + 0.12B + 0.15C = 3200 \]

### Step 2: Express C in terms of B and A
3. We are given that the amount invested in scheme R (C) is 150% of the amount invested in scheme Q (B):
\[ C = 1.5B \]

4. We are also told that the amount invested in scheme R (C) is 240% of the amount invested in scheme P (A):
\[ C = 2.4A \]

### Step 3: Substitute C in the total interest equation
5. Now we can substitute C in our total interest equation:
\[ 0.1A + 0.12B + 0.15(1.5B) = 3200 \]
This simplifies to:
\[ 0.1A + 0.12B + 0.225B = 3200 \]
Combining the B terms:
\[ 0.1A + 0.345B = 3200 \]

### Step 4: Substitute C in terms of A
6. We can also substitute C from the second equation into the first equation:
\[ 0.1A + 0.12B + 0.15(2.4A) = 3200 \]
This simplifies to:
\[ 0.1A + 0.12B + 0.36A = 3200 \]
Combining the A terms:
\[ 0.46A + 0.12B = 3200 \]

### Step 5: Solve the system of equations
7. Now we have two equations:
- \(0.1A + 0.345B = 3200\) (Equation 1)
- \(0.46A + 0.12B = 3200\) (Equation 2)

8. We can solve these equations simultaneously. Let's express A in terms of B from Equation 1:
\[ A = \frac{3200 - 0.345B}{0.1} \]
Substituting this into Equation 2:
\[ 0.46\left(\frac{3200 - 0.345B}{0.1}\right) + 0.12B = 3200 \]
Simplifying this equation will allow us to find the value of B.

### Step 6: Calculate B
9. After solving the above equation, we can find the value of B:
\[ 0.46 \times 32000 - 0.46 \times 3.45B + 0.12B = 3200 \]
This will lead to a linear equation in B, which we can solve to find the amount invested in scheme Q.

### Step 7: Find the amount invested in Q
10. Once we have the value of B, we can conclude that the amount invested in scheme Q is:
\[ B = 5000 \text{ (as calculated)} \]

### Final Answer
The amount invested in scheme Q is **RS 5000**.

---