A certain sum of money amounts to 756 in 2 years and to 873 in 3.5 years. Find the sum and the rate of interest.

Options

A. `11%`
B. `13%`
C. `15%`
D. `19%`

Answer

To solve the problem step by step, we will follow the method outlined in the video transcript.

### Step 1: Identify the amounts and time periods
We are given:
- Amount after 2 years (A1) = 756
- Amount after 3.5 years (A2) = 873

### Step 2: Calculate the interest for the additional time period
The interest earned from the end of the second year to the end of the third and a half year is:
\[ \text{Interest} = A2 - A1 = 873 - 756 = 117 \]

### Step 3: Determine the interest for 1 year
Since the interest of 117 is for 1.5 years (from year 2 to year 3.5), we can find the interest for 1 year:
\[ \text{Interest for 1 year} = \frac{117}{1.5} = 78 \]

### Step 4: Calculate the total interest for 2 years
Now, we can find the total interest for 2 years:
\[ \text{Total Interest for 2 years} = 78 \times 2 = 156 \]

### Step 5: Calculate the principal amount
To find the principal amount (P), we use the formula:
\[ \text{Principal} = \text{Amount after 2 years} - \text{Total Interest for 2 years} \]
\[ P = 756 - 156 = 600 \]

### Step 6: Calculate the rate of interest
Using the formula for simple interest:
\[ \text{Simple Interest} = \frac{P \times R \times T}{100} \]
We know:
- Simple Interest for 1 year = 78
- Principal (P) = 600
- Time (T) = 1 year

Plugging in the values:
\[ 78 = \frac{600 \times R \times 1}{100} \]

### Step 7: Solve for R
Rearranging the equation to find R:
\[ R = \frac{78 \times 100}{600} = 13 \]

### Final Answers
- The principal amount (sum) is **600**.
- The rate of interest (R) is **13%**.

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