Amartya wants to invest ₹45,000 in Indira Vikas Patra (IVP) & in Public Provident fund (PPF). He wants to invest at least ₹ 10,000 in PPF & at least ₹5000 in IVP. If the rate of interest on PPF is 8% per annum & that on IVP is 7% per annum. Formulate the above problem as LPP to determine maximum yearly income.

Answer

To solve the problem of Amartya's investment in Indira Vikas Patra (IVP) and Public Provident Fund (PPF), we will formulate it as a Linear Programming Problem (LPP) and determine the maximum yearly income.

### Step 1: Define the Variables
Let:
- \( x \) = amount invested in PPF
- \( y \) = amount invested in IVP

### Step 2: Set Up the Objective Function
The objective is to maximize the yearly income from the investments. The income from PPF at an interest rate of 8% is \( 0.08x \) and from IVP at an interest rate of 7% is \( 0.07y \). Therefore, the total income \( Z \) can be expressed as:
\[ Z = 0.08x + 0.07y \]

### Step 3: Set Up the Constraints
Based on the problem statement, we have the following constraints:
1. The total investment should be ₹45,000:
\[ x + y = 45000 \quad \text{(Constraint 1)} \]
2. The minimum investment in PPF should be ₹10,000:
\[ x \geq 10000 \quad \text{(Constraint 2)} \]
3. The minimum investment in IVP should be ₹5,000:
\[ y \geq 5000 \quad \text{(Constraint 3)} \]

### Step 4: Rewrite the Constraints
We can rewrite the first constraint to express \( y \) in terms of \( x \):
\[ y = 45000 - x \]

### Step 5: Combine Constraints
Now, we have:
- From Constraint 2: \( x \geq 10000 \)
- From Constraint 3: \( y \geq 5000 \) implies \( 45000 - x \geq 5000 \) or \( x \leq 40000 \)

### Step 6: Summary of Constraints
Thus, the constraints can be summarized as:
1. \( 10000 \leq x \leq 40000 \)
2. \( y \geq 5000 \)

### Step 7: Determine the Feasible Region
The feasible region is defined by the constraints. We can find the vertices of the feasible region by substituting the limits of \( x \):
1. If \( x = 10000 \):
\[ y = 45000 - 10000 = 35000 \]
Point: \( (10000, 35000) \)

2. If \( x = 40000 \):
\[ y = 45000 - 40000 = 5000 \]
Point: \( (40000, 5000) \)

### Step 8: Calculate the Income at Each Vertex
Now, we calculate the income \( Z \) at each vertex:
1. At \( (10000, 35000) \):
\[ Z = 0.08(10000) + 0.07(35000) = 800 + 2450 = 3250 \]

2. At \( (40000, 5000) \):
\[ Z = 0.08(40000) + 0.07(5000) = 3200 + 350 = 3550 \]

### Step 9: Determine the Maximum Income
Comparing the incomes:
- At \( (10000, 35000) \): Income = ₹3250
- At \( (40000, 5000) \): Income = ₹3550

The maximum yearly income is ₹3550.

### Conclusion
Amartya should invest ₹40,000 in PPF and ₹5,000 in IVP to achieve the maximum yearly income of ₹3550.

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