The different in compound interest and simple interest for 2 years on a sum of money is Rs. 160. the simple interest for 2 years be Rs.2880, the rate percent is -

Options

A. `(5)5/9%`
B. `(12)1/9%`
C. `(11)1/9` %
D. `9%`

Answer

To solve the problem step by step, we will use the information provided in the question and the video transcript.

### Step 1: Understand the given information
We know that:
- The difference between Compound Interest (CI) and Simple Interest (SI) for 2 years is Rs. 160.
- The Simple Interest for 2 years is Rs. 2880.

### Step 2: Use the formula for Simple Interest
The formula for Simple Interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

Given that \( SI = 2880 \) and \( T = 2 \):
\[ 2880 = \frac{P \times R \times 2}{100} \]

### Step 3: Rearranging the equation to find \( P \times R \)
From the equation:
\[ P \times R = \frac{2880 \times 100}{2} = 144000 \]

### Step 4: Use the formula for Compound Interest
The formula for Compound Interest (CI) for 2 years is:
\[ CI = P \left(1 + \frac{R}{100}\right)^2 - P \]
This can be simplified to:
\[ CI = P \left[\left(1 + \frac{R}{100}\right)^2 - 1\right] \]

### Step 5: Expand the Compound Interest formula
Using the binomial expansion for \( \left(1 + \frac{R}{100}\right)^2 \):
\[ \left(1 + \frac{R}{100}\right)^2 = 1 + 2\frac{R}{100} + \left(\frac{R}{100}\right)^2 \]
Thus,
\[ CI = P \left[2\frac{R}{100} + \left(\frac{R}{100}\right)^2\right] \]

### Step 6: Substitute the value of \( P \)
Now substituting \( P \) from the earlier equation \( P = \frac{144000}{R} \):
\[ CI = \frac{144000}{R} \left[2\frac{R}{100} + \frac{R^2}{10000}\right] \]
This simplifies to:
\[ CI = \frac{144000 \times 2}{100} + \frac{144000 \times R}{10000} \]
\[ CI = 2880 + \frac{1440R}{100} \]

### Step 7: Find the difference between CI and SI
We know that:
\[ CI - SI = 160 \]
Substituting the values we have:
\[ \left(2880 + \frac{1440R}{100}\right) - 2880 = 160 \]
This simplifies to:
\[ \frac{1440R}{100} = 160 \]

### Step 8: Solve for \( R \)
Now, solving for \( R \):
\[ 1440R = 16000 \]
\[ R = \frac{16000}{1440} = \frac{100}{9} \]
This gives:
\[ R \approx 11.11\% \]

### Conclusion
The rate percent is approximately \( 11.11\% \).

---