If the discount on Rs. 3050 be equal to the simple interest on Rs. 3000 for the same time. Find the time, the rate on interest being `5%` per annum-

Options

A. 4 months
B. 6 months
C. 3 months
D. None of these

Answer

To solve the problem step by step, we will follow the mathematical approach to find the time when the discount on Rs. 3050 is equal to the simple interest on Rs. 3000 at a rate of 5% per annum.

### Step 1: Understand the Problem
We need to find the time (T) when the discount on Rs. 3050 is equal to the simple interest on Rs. 3000 for the same time period at a rate of 5% per annum.

### Step 2: Write the Formula for True Discount
The formula for True Discount (TD) is given by:
\[ TD = \frac{P \times R \times T}{100 + R \times T} \]
Where:
- P = Present Value (or the amount on which discount is calculated)
- R = Rate of interest
- T = Time

### Step 3: Set Up the Equation
Using the formula for True Discount for Rs. 3050:
\[ TD = \frac{3050 \times 5 \times T}{100 + 5T} \]

For the Simple Interest (SI) on Rs. 3000:
\[ SI = \frac{P \times R \times T}{100} = \frac{3000 \times 5 \times T}{100} \]

### Step 4: Equate the Two Expressions
According to the problem, the True Discount is equal to the Simple Interest:
\[ \frac{3050 \times 5 \times T}{100 + 5T} = \frac{3000 \times 5 \times T}{100} \]

### Step 5: Simplify the Equation
We can cancel out the common terms (5T) from both sides:
\[ \frac{3050}{100 + 5T} = \frac{3000}{100} \]

### Step 6: Cross Multiply
Cross multiplying gives us:
\[ 3050 \times 100 = 3000 \times (100 + 5T) \]

### Step 7: Expand and Rearrange
Expanding the equation:
\[ 305000 = 300000 + 15000T \]
Rearranging gives:
\[ 305000 - 300000 = 15000T \]
\[ 5000 = 15000T \]

### Step 8: Solve for T
Now, divide both sides by 15000:
\[ T = \frac{5000}{15000} = \frac{1}{3} \text{ years} \]

### Step 9: Convert Time to Months
To convert years into months, multiply by 12:
\[ T = \frac{1}{3} \times 12 = 4 \text{ months} \]

### Final Answer
The time (T) is **4 months**.

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