The present worth of bill due 7 months, hence is Rs. 1200. if the bill were due at the end of 2.5 years, its present worth would be Rs1016. Find the rate of interest of the bill.

Options

A. 8%
B. 9%
C. 10%
D. 11%

Answer

To find the rate of interest for the given problem, we can follow these steps:

### Step 1: Understand the Problem
We have two scenarios for the present worth of a bill:
1. Present worth (PW) due in 7 months is Rs. 1200.
2. Present worth (PW) due in 2.5 years is Rs. 1016.

We need to find the rate of interest (R).

### Step 2: Set Up the Formula for Present Worth
The formula for present worth (PW) is given by:
\[ PW = \frac{A}{(1 + rt)} \]
Where:
- \( A \) = Amount due in the future
- \( r \) = Rate of interest per year (as a decimal)
- \( t \) = Time in years

### Step 3: Write Equations for Both Scenarios
1. For the first scenario (due in 7 months):
\[ 1200 = \frac{A}{(1 + r \cdot \frac{7}{12})} \]
(Here, \( t = \frac{7}{12} \) years)

2. For the second scenario (due in 2.5 years):
\[ 1016 = \frac{A}{(1 + r \cdot 2.5)} \]

### Step 4: Express Amount \( A \) from Both Equations
From the first equation, we can express \( A \):
\[ A = 1200 \cdot (1 + r \cdot \frac{7}{12}) \]

From the second equation, we can express \( A \) as well:
\[ A = 1016 \cdot (1 + r \cdot 2.5) \]

### Step 5: Set the Two Expressions for \( A \) Equal
Since both expressions represent the same amount \( A \), we can set them equal to each other:
\[ 1200 \cdot (1 + r \cdot \frac{7}{12}) = 1016 \cdot (1 + r \cdot 2.5) \]

### Step 6: Expand and Rearrange the Equation
Expanding both sides:
\[ 1200 + 1200 \cdot r \cdot \frac{7}{12} = 1016 + 1016 \cdot r \cdot 2.5 \]

This simplifies to:
\[ 1200 + 700r = 1016 + 2540r \]

### Step 7: Collect Like Terms
Rearranging gives us:
\[ 1200 - 1016 = 2540r - 700r \]
\[ 184 = 1840r \]

### Step 8: Solve for \( r \)
Dividing both sides by 1840:
\[ r = \frac{184}{1840} = 0.1 \]

### Step 9: Convert to Percentage
To express \( r \) as a percentage, multiply by 100:
\[ R = 0.1 \times 100 = 10\% \]

### Final Answer
The rate of interest (R) is **10%**.

---