In what time will the simple interest be `(2)/(5)` of the principal at 8 per cent per annum ?

Options

A. 8 years
B. 7 years
C. 5 years
D. 6 years

Answer

To solve the problem step by step, we will use the formula for Simple Interest (SI):

**Step 1: Understand the relationship between Simple Interest, Principal, Rate, and Time.**
The formula for Simple Interest is given by:
\[ \text{SI} = \frac{P \times R \times T}{100} \]
where:
- \( P \) = Principal
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

**Step 2: Define the variables.**
Let the Principal \( P = 5x \) (we choose this to make calculations easier).
According to the problem, the Simple Interest (SI) is \( \frac{2}{5} \) of the Principal:
\[ \text{SI} = \frac{2}{5} \times P = \frac{2}{5} \times 5x = 2x \]

**Step 3: Substitute the values into the Simple Interest formula.**
Now we substitute \( SI \), \( P \), and \( R \) into the formula:
\[ 2x = \frac{5x \times 8 \times T}{100} \]

**Step 4: Simplify the equation.**
To simplify, we can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 2 = \frac{5 \times 8 \times T}{100} \]

**Step 5: Cross-multiply to solve for \( T \).**
Cross-multiplying gives us:
\[ 2 \times 100 = 5 \times 8 \times T \]
\[ 200 = 40T \]

**Step 6: Solve for \( T \).**
Now, divide both sides by 40:
\[ T = \frac{200}{40} = 5 \]

**Final Answer:**
The time \( T \) is 5 years.

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