A certain sum at a certain rate of SI amounts to ? 1125 in 4 yr and ? 1200 in 7 yr. Find the sum and rate of interest.

Answer

To solve the problem step by step, we will use the information provided about the amounts and the time periods to find the principal (sum) and the rate of interest.

### Given:
- Amount after 4 years (A1) = ₹1125
- Amount after 7 years (A2) = ₹1200
- Time for A1 (T1) = 4 years
- Time for A2 (T2) = 7 years

### Step 1: Find the Principal (P)

We can use the formula for the principal based on the amounts and time periods:

\[ P = \frac{A_2 \cdot T_1 - A_1 \cdot T_2}{T_1 - T_2} \]

Substituting the values:

\[ P = \frac{1200 \cdot 4 - 1125 \cdot 7}{4 - 7} \]

### Step 2: Calculate the Numerator

Calculating the numerator:

\[ 1200 \cdot 4 = 4800 \]
\[ 1125 \cdot 7 = 7875 \]

Now substituting these values back into the equation:

\[ P = \frac{4800 - 7875}{4 - 7} \]
\[ P = \frac{-3075}{-3} \]

### Step 3: Calculate the Principal

Now, simplifying the fraction:

\[ P = \frac{3075}{3} = 1025 \]

So, the principal amount (P) is ₹1025.

### Step 4: Find the Rate of Interest (R)

Now we will find the rate of interest using the formula:

\[ R = \frac{A_1 - A_2}{A_2 \cdot T_1 - A_1 \cdot T_2} \cdot 100 \]

Substituting the values:

\[ R = \frac{1125 - 1200}{1200 \cdot 4 - 1125 \cdot 7} \cdot 100 \]

### Step 5: Calculate the Numerator for Rate

Calculating the numerator:

\[ 1125 - 1200 = -75 \]

### Step 6: Calculate the Denominator for Rate

We already calculated the denominator in the previous steps:

\[ 1200 \cdot 4 - 1125 \cdot 7 = 4800 - 7875 = -3075 \]

### Step 7: Substitute Values into Rate Formula

Now substituting these values back into the rate formula:

\[ R = \frac{-75}{-3075} \cdot 100 \]

### Step 8: Calculate the Rate of Interest

Now simplifying:

\[ R = \frac{75}{3075} \cdot 100 \]

Calculating the fraction:

\[ R = 0.0244 \cdot 100 = 2.44\% \]

### Final Answers:
- Principal (P) = ₹1025
- Rate of Interest (R) = 2.44%

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