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?

Options

A. `a^2 = bc`
B. `b^2 = ca`
C. `c^2 = ab`
D. abc = 1

Answer

To solve the problem step by step, we will use the formula for simple interest and analyze the relationships given in the question.

### Step 1: Understand the Simple Interest Formula
The formula for simple interest (SI) is given by:
\[ SI = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal amount
- \( R \) = Rate of interest
- \( T \) = Time period

### Step 2: Set Up the First Case
In the first case, we have:
- \( A \) is the simple interest on \( B \).
Using the formula, we can express this as:
\[ A = \frac{B \times R \times T}{100} \]
Since the rate of interest and time are the same for both cases, we can denote \( R \) and \( T \) as constants.

### Step 3: Rearranging the First Case
Rearranging the equation from Step 2 gives us:
\[ A = \frac{BRT}{100} \]

### Step 4: Set Up the Second Case
In the second case, we have:
- \( B \) is the simple interest on \( C \).
Using the formula again, we can express this as:
\[ B = \frac{C \times R \times T}{100} \]

### Step 5: Rearranging the Second Case
Rearranging the equation from Step 4 gives us:
\[ B = \frac{CRT}{100} \]

### Step 6: Equate the Rates of Interest
Since the rates of interest are the same for both cases, we can set the two expressions for \( R \) equal to each other:
From the first case, we can express \( R \) as:
\[ R = \frac{100A}{BT} \]
From the second case, we can express \( R \) as:
\[ R = \frac{100B}{CT} \]

### Step 7: Set the Two Expressions for \( R \) Equal
Now we can set the two expressions for \( R \) equal to each other:
\[ \frac{100A}{BT} = \frac{100B}{CT} \]

### Step 8: Simplifying the Equation
Cancelling out \( 100 \) and \( T \) from both sides gives:
\[ \frac{A}{B} = \frac{B}{C} \]

### Step 9: Cross Multiplying
Cross multiplying gives us:
\[ A \cdot C = B \cdot B \]
or
\[ AC = B^2 \]

### Conclusion
Thus, the correct relationship is:
\[ B^2 = AC \]