If a man invests equal sum at the same rate of interest on simple interest for T and T+4 years and the respective ratio of interest gets by man is 1:2 respectively, then find 'T'?

Options

A. 6
B. 2
C. 5
D. 4

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI) and the information given in the question.

### Step 1: Understand the formula for Simple Interest
The formula for Simple Interest is given by:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) = Simple Interest
- \( P \) = Principal amount (the initial sum of money)
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

### Step 2: Set up the equations based on the problem
According to the problem, the man invests the same principal amount \( P \) at the same rate \( R \) for two different time periods: \( T \) years and \( T + 4 \) years. The ratio of the interests earned in these two periods is given as \( 1:2 \).

Let’s denote:
- Interest for \( T \) years:
\[ SI_1 = \frac{P \times R \times T}{100} \]
- Interest for \( T + 4 \) years:
\[ SI_2 = \frac{P \times R \times (T + 4)}{100} \]

### Step 3: Write the ratio of the interests
According to the problem, we have:
\[ \frac{SI_1}{SI_2} = \frac{1}{2} \]
Substituting the expressions for \( SI_1 \) and \( SI_2 \):
\[ \frac{\frac{P \times R \times T}{100}}{\frac{P \times R \times (T + 4)}{100}} = \frac{1}{2} \]

### Step 4: Simplify the equation
Since \( P \) and \( R \) are the same in both cases, they cancel out:
\[ \frac{T}{T + 4} = \frac{1}{2} \]

### Step 5: Cross-multiply to solve for \( T \)
Cross-multiplying gives:
\[ 2T = T + 4 \]

### Step 6: Solve for \( T \)
Now, subtract \( T \) from both sides:
\[ 2T - T = 4 \]
\[ T = 4 \]

### Conclusion
Thus, the value of \( T \) is \( 4 \).