S.I. on a certain sum is equal to the same sum. Accordingly what will be the time period, in which rate of interest and time is same number?

Options

A. 8 years
B. `8(1)/(2)` years
C. 7 years
D. 10 years

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI) and the relationship between Principal (P), Rate of Interest (R), and Time (T).

### Step 1: Understand the problem
We are given that the Simple Interest on a certain sum is equal to the same sum (Principal). This means:
\[ \text{SI} = P \]

### Step 2: Write the formula for Simple Interest
The formula for Simple Interest is:
\[ \text{SI} = \frac{P \times R \times T}{100} \]

### Step 3: Substitute the known values
Since we know that SI = P, we can substitute SI in the formula:
\[ P = \frac{P \times R \times T}{100} \]

### Step 4: Simplify the equation
We can cancel P from both sides (assuming P ≠ 0):
\[ 1 = \frac{R \times T}{100} \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ R \times T = 100 \]

### Step 6: Use the condition that Rate and Time are the same
According to the problem, the Rate (R) and Time (T) are the same number. Let's denote this common number as x:
\[ R = T = x \]

### Step 7: Substitute R and T in the equation
Now, substituting R and T in the equation gives us:
\[ x \times x = 100 \]
\[ x^2 = 100 \]

### Step 8: Solve for x
Taking the square root of both sides:
\[ x = 10 \]

### Step 9: Conclusion
Since R = T = x, we find:
- Rate of Interest (R) = 10%
- Time (T) = 10 years

Thus, the time period in which the rate of interest and time is the same number is **10 years**.

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