If a is simple interest on b and b is simple interest on c. Rate of interest and time is the same in both the cases, then which of the following is correct ?

Question

Accepted Answer

To understand why Option 2 ($b^2 = ca$) is the correct answer, let's break down the problem using the principles of simple interest.

### Explanation for Option 2:

1. **Simple Interest Formula**: The formula for simple interest is given by:
   $$
   \text{Simple Interest (SI)} = \frac{P \times R \times T}{100}
   $$
   where $ P $ is the principal amount, $ R $ is the rate of interest per annum, and $ T $ is the time in years.

2. **Given Conditions**:
   - $ a $ is the simple interest on $ b $.
   - $ b $ is the simple interest on $ c $.
   - The rate of interest ($ R $) and time ($ T $) are the same in both cases.

3. **Expressing $ a $ and $ b $ in terms of $ b $ and $ c $**:
   - For $ a $ as simple interest on $ b $:
     $$
     a = \frac{b \times R \times T}{100}
     $$
   - For $ b $ as simple interest on $ c $:
     $$
     b = \frac{c \times R \times T}{100}
     $$

4. **Substituting the expression for $ b $ into the equation for $ a $**:
   - Substitute $ b = \frac{c \times R \times T}{100} $ into the equation for $ a $:
     $$
     a = \frac{\left(\frac{c \times R \times T}{100}\right) \times R \times T}{100}
     $$
   - Simplifying, we get:
     $$
     a = \frac{c \times (R \times T)^2}{10000}
     $$

5. **Relating $ a $, $ b $, and $ c $**:
   - From the above expressions, we can derive:
     $$
     b^2 = \left(\frac{c \times R \times T}{100}\right)^2 = \frac{c^2 \times (R \times T)^2}{10000}
     $$
   - Since $ a = \frac{c \times (R \times T)^2}{10000} $, we can equate:
     $$
     b^2 = c \times a
     $$

Thus, Option 2 ($b^2 = ca$) is correct.

### Explanation for Why Other Options are Incorrect:

1. **Option 1 ($a^2 = bc$)**:
   - From our derivation, $ a $ is not directly related to $ b $ and $ c $ in a squared relationship. The correct relationship involves $ b^2 $, not $ a^2 $.

2. **Option 3 ($c^2 = ab$)**:
   - The derived relationship shows $ b^2 = ca $, not $ c^2 = ab $. The roles of $ b $ and $ c $ are not interchangeable in this context.

3. **Option 4 (abc = 1)**:
   - There is no basis in the simple interest calculations that would lead to the product of $ a $, $ b $, and $ c $ equating to 1. This option does not align with the derived relationships.

In conclusion, the correct relationship derived from the given conditions is $ b^2 = ca $, making Option 2 the correct answer.