A sum of money Amounts of Rs. 1016 in 3 years and Rs. 1304 in 7 years. Find the Rate and Principal ?

Answer

To solve the problem step by step, we will use the information given about the amounts after certain years to derive the principal and the rate of interest.

### Step 1: Define the Variables
Let:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum

### Step 2: Set Up the Equations
From the problem, we know:
1. The amount after 3 years is Rs. 1016.
2. The amount after 7 years is Rs. 1304.

Using the formula for the amount in simple interest:
\[ A = P + \frac{P \times R \times T}{100} \]
We can set up two equations based on the amounts given:

**For 3 years:**
\[ 1016 = P + \frac{P \times R \times 3}{100} \quad \text{(Equation 1)} \]

**For 7 years:**
\[ 1304 = P + \frac{P \times R \times 7}{100} \quad \text{(Equation 2)} \]

### Step 3: Simplify the Equations
Rearranging both equations gives us:
1. From Equation 1:
\[ 1016 = P + \frac{3PR}{100} \implies 1016 - P = \frac{3PR}{100} \implies 3PR = 100(1016 - P) \]

2. From Equation 2:
\[ 1304 = P + \frac{7PR}{100} \implies 1304 - P = \frac{7PR}{100} \implies 7PR = 100(1304 - P) \]

### Step 4: Subtract the Two Equations
Now, we can subtract Equation 1 from Equation 2:
\[ (1304 - P) - (1016 - P) = \frac{7PR}{100} - \frac{3PR}{100} \]
This simplifies to:
\[ 1304 - 1016 = \frac{(7PR - 3PR)}{100} \]
\[ 288 = \frac{4PR}{100} \]
Multiplying both sides by 100 gives:
\[ 28800 = 4PR \implies PR = \frac{28800}{4} = 7200 \quad \text{(Equation 3)} \]

### Step 5: Substitute Back to Find Principal
Now substitute \( PR = 7200 \) back into Equation 1:
\[ 1016 = P + \frac{3 \times 7200}{100} \]
Calculating \( \frac{3 \times 7200}{100} \):
\[ \frac{21600}{100} = 216 \]
Now substituting:
\[ 1016 = P + 216 \implies P = 1016 - 216 = 800 \]

### Step 6: Find the Rate of Interest
Now that we have \( P = 800 \), we can find \( R \) using Equation 3:
\[ PR = 7200 \implies 800R = 7200 \implies R = \frac{7200}{800} = 9 \]

### Final Answer
- Principal \( P = 800 \) Rs.
- Rate \( R = 9\% \) per annum.

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