In how much time will the simple interest on a sum of money at `8%` p.a. be `(2)/(5)` of the principal ?

Answer

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

**Step 1: Understand the Problem**
We need to find the time (T) when the Simple Interest (SI) is \( \frac{2}{5} \) of the Principal (P) at a rate of 8% per annum.

**Step 2: Write the Formula for Simple Interest**
The formula for Simple Interest is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \( SI \) = Simple Interest
- \( P \) = Principal
- \( R \) = Rate of Interest (in %)
- \( T \) = Time (in years)

**Step 3: Substitute the Given Values**
We know from the problem that:
- \( SI = \frac{2}{5} P \)
- \( R = 8\% \)

Substituting these values into the formula, we have:
\[ \frac{2}{5} P = \frac{P \times 8 \times T}{100} \]

**Step 4: Cancel the Principal (P)**
Since \( P \) is common on both sides of the equation, we can cancel it out (assuming \( P \neq 0 \)):
\[ \frac{2}{5} = \frac{8 \times T}{100} \]

**Step 5: Rearrange the Equation to Solve for T**
Now, we will rearrange the equation to isolate \( T \):
\[ T = \frac{2}{5} \times \frac{100}{8} \]

**Step 6: Simplify the Right Side**
Calculating the right side:
\[ T = \frac{2 \times 100}{5 \times 8} \]
\[ T = \frac{200}{40} \]
\[ T = 5 \]

**Final Answer:**
The time (T) will be 5 years.

---