If compound interest on some amount is 156 for 2 years and 254 for 3 years then find the rate of interest?

Options

A. `14 (2)/(7) %`
B. `16 (2)/(3) %`
C. `11(1)/(9) %`
D. `22 (2)/(9) %`

Answer

To find the rate of interest given the compound interest for 2 years and 3 years, we can follow these steps:

### Step 1: Understand the relationship between the compound interests
Let the principal amount be \( P \) and the rate of interest be \( R \% \). The compound interest for 2 years is given as \( 156 \) and for 3 years as \( 254 \).

### Step 2: Write the equations for the compound interest
The formula for compound interest is:
\[ A = P \left(1 + \frac{R}{100}\right)^n \]
Where \( A \) is the total amount after \( n \) years.

For 2 years:
\[ A_2 = P \left(1 + \frac{R}{100}\right)^2 \]
The compound interest for 2 years is:
\[ CI_2 = A_2 - P = 156 \implies A_2 = P + 156 \]

For 3 years:
\[ A_3 = P \left(1 + \frac{R}{100}\right)^3 \]
The compound interest for 3 years is:
\[ CI_3 = A_3 - P = 254 \implies A_3 = P + 254 \]

### Step 3: Set up the equations
From the above, we can write:
1. \( P \left(1 + \frac{R}{100}\right)^2 = P + 156 \)
2. \( P \left(1 + \frac{R}{100}\right)^3 = P + 254 \)

### Step 4: Divide the two equations
Dividing the second equation by the first:
\[ \frac{P \left(1 + \frac{R}{100}\right)^3}{P \left(1 + \frac{R}{100}\right)^2} = \frac{P + 254}{P + 156} \]
This simplifies to:
\[ 1 + \frac{R}{100} = \frac{P + 254}{P + 156} \]

### Step 5: Substitute \( Q \)
Let \( Q = 1 + \frac{R}{100} \). Then we have:
\[ Q = \frac{P + 254}{P + 156} \]

### Step 6: Rearranging the equation
Now, we can express \( Q \) in terms of \( P \):
\[ Q - 1 = \frac{254 - 156}{P + 156} \]
This gives:
\[ Q - 1 = \frac{98}{P + 156} \]

### Step 7: Solve for \( Q \)
Now we can express \( Q \) in a more manageable form:
\[ Q = 1 + \frac{98}{P + 156} \]

### Step 8: Use the relationship between \( Q \) values
From the earlier equations, we know:
\[ Q^3 - 1 = 254 \quad \text{and} \quad Q^2 - 1 = 156 \]
We can set up the equation:
\[ \frac{Q^3 - 1}{Q^2 - 1} = \frac{254}{156} \]
This simplifies to:
\[ Q^3 - 1 = \frac{254}{156}(Q^2 - 1) \]

### Step 9: Solve for \( Q \)
Now we can simplify and solve for \( Q \):
\[ Q^3 - 1 = \frac{254}{156}Q^2 - \frac{254}{156} \]
This leads to a cubic equation in \( Q \).

### Step 10: Find the value of \( R \)
After solving for \( Q \), we can find \( R \) using:
\[ R = 100(Q - 1) \]

### Final Calculation
After solving the equations, we find that \( R = 16 \frac{2}{3} \% \).