If simple interest on some amount for 3 years is 600 and compound interest on same amount for 2 years is 450 then find the rate of interest?

Options

A. A.`25%`
B. B.`20%`
C. C. `22% `
D. D.`22.5%`

Answer

To solve the problem, we need to find the rate of interest given the simple interest (SI) for 3 years and the compound interest (CI) for 2 years.

### Step 1: Calculate the Simple Interest per Year
The total simple interest for 3 years is given as 600. To find the simple interest for one year, we can use the formula:

\[ \text{SI per year} = \frac{\text{Total SI}}{\text{Number of years}} = \frac{600}{3} = 200 \]

### Step 2: Relate Simple Interest to Compound Interest
The simple interest for one year is 200. In the case of compound interest, the first year's interest is equal to the simple interest for that year. Therefore, the first year of compound interest (CI) is also 200.

### Step 3: Calculate the Total Compound Interest for 2 Years
The total compound interest for 2 years is given as 450. We know that the first year of CI is 200.

Let the second year of CI be represented as \( x \). Therefore, we can write:

\[ \text{Total CI for 2 years} = \text{First year CI} + \text{Second year CI} \]
\[ 450 = 200 + x \]

### Step 4: Solve for the Second Year of CI
Now, we can solve for \( x \):

\[ x = 450 - 200 = 250 \]

### Step 5: Calculate the Rate of Interest
The second year of CI is the interest earned on the principal plus the first year's interest. The amount at the end of the first year is:

\[ \text{Principal} + \text{First year CI} = P + 200 \]

The second year's interest (which is 250) can be expressed as:

\[ \text{Second year CI} = \text{Amount at end of first year} \times \frac{R}{100} \]
\[ 250 = (P + 200) \times \frac{R}{100} \]

### Step 6: Use the Simple Interest Formula to Find Principal
We can also express the principal in terms of the total simple interest:

\[ 600 = P \times \frac{R}{100} \times 3 \]
\[ P = \frac{600 \times 100}{3R} = \frac{20000}{3R} \]

### Step 7: Substitute Principal into the CI Equation
Now substitute \( P \) into the second year CI equation:

\[ 250 = \left(\frac{20000}{3R} + 200\right) \times \frac{R}{100} \]

### Step 8: Solve for R
Now we can solve for \( R \):

1. Multiply both sides by 100:
\[ 25000 = \left(\frac{20000}{3R} + 200\right)R \]

2. Distributing \( R \):
\[ 25000 = \frac{20000}{3} + 200R \]

3. Rearranging gives:
\[ 200R = 25000 - \frac{20000}{3} \]

4. Convert \( 25000 \) to a fraction with a common denominator:
\[ 25000 = \frac{75000}{3} \]

5. Now subtract:
\[ 200R = \frac{75000 - 20000}{3} = \frac{55000}{3} \]

6. Finally, solve for \( R \):
\[ R = \frac{55000}{600} = \frac{55}{6} \approx 9.17\% \]

### Final Answer
The rate of interest is approximately \( 9.17\% \).

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