A certain amount is lent at x% p.a. simple interest for two years. Instead, if the amount was lent at 2x% p.a. simple interest for ‘y’ more years, then the interest would have been five times the earlier interest. What is the value of y?

Options

A. 2 years
B. 3 years
C. 4 years
D. 5 years

Answer

To solve the problem step by step, we need to establish the relationships between the different interests calculated at different rates and for different time periods.

### Step 1: Calculate the interest for the first scenario
The formula for simple interest is given by:

\[ \text{Simple Interest (SI)} = \frac{P \times R \times T}{100} \]

Where:
- \( P \) = Principal amount
- \( R \) = Rate of interest per annum
- \( T \) = Time in years

For the first scenario, the amount is lent at \( x\% \) for 2 years. Therefore, the interest earned (let's call it \( I_1 \)) can be calculated as:

\[ I_1 = \frac{P \times x \times 2}{100} \]

### Step 2: Calculate the interest for the second scenario
In the second scenario, the amount is lent at \( 2x\% \) for \( y + 2 \) years (2 years plus \( y \) more years). The interest earned in this case (let's call it \( I_2 \)) is:

\[ I_2 = \frac{P \times 2x \times (y + 2)}{100} \]

### Step 3: Set up the equation based on the problem statement
According to the problem, the interest from the second scenario is five times the interest from the first scenario. Therefore, we can write the equation:

\[ I_2 = 5 \times I_1 \]

Substituting the expressions for \( I_1 \) and \( I_2 \):

\[ \frac{P \times 2x \times (y + 2)}{100} = 5 \times \frac{P \times x \times 2}{100} \]

### Step 4: Simplify the equation
We can cancel \( P \) and \( 100 \) from both sides (assuming \( P \neq 0 \) and \( 100 \neq 0 \)):

\[ 2x(y + 2) = 10x \]

Now, we can divide both sides by \( 2x \) (assuming \( x \neq 0 \)):

\[ y + 2 = 5 \]

### Step 5: Solve for \( y \)
Now, we can solve for \( y \):

\[ y = 5 - 2 \]
\[ y = 3 \]

### Final Answer
The value of \( y \) is \( 3 \).

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