The banker's discount on a sum of money for 1 1/2 years is Rs. 60 and the true discount on the same sum for 2 years is Rs. 75 . The rate per cent is -----

Options

A. `5%`
B. `6%`
C. `6 2/3 %`
D. `3 1/3 %`

Answer

To solve the problem step by step, we need to find the rate of interest based on the given banker's discount and true discount. Here's how to do it:

### Step 1: Understand the Given Information
We know:
- Banker's Discount (BD) for 1.5 years = Rs. 60
- True Discount (TD) for 2 years = Rs. 75

### Step 2: Calculate the Banker's Discount for 2 Years
The Banker's Discount for 2 years can be calculated using the formula:
\[ \text{BD for 2 years} = \text{BD for 1.5 years} \times \frac{2}{1.5} = 60 \times \frac{2}{1.5} = 60 \times \frac{4}{3} = 80 \]
So, the Banker's Discount for 2 years is Rs. 80.

### Step 3: Relate True Discount and Banker's Discount
We know that the True Discount (TD) is given as Rs. 75 for 2 years. The relationship between True Discount, Banker's Discount, and the Sum (Principal) can be expressed as:
\[ \text{TD} = \text{BD} \times \frac{\text{TD}}{\text{BD} - \text{TD}} \]
Substituting the values we have:
\[ 75 = 80 \times \frac{75}{80 - 75} \]
This simplifies to:
\[ 75 = 80 \times \frac{75}{5} \]
\[ 75 = 80 \times 15 \]
This gives us the sum (Principal).

### Step 4: Calculate the Sum (Principal)
From the above equation, we can rearrange to find the sum:
\[ \text{Sum} = \frac{\text{BD} \times \text{TD}}{\text{BD} - \text{TD}} = \frac{80 \times 75}{80 - 75} = \frac{6000}{5} = 1200 \]
So, the Sum (Principal) is Rs. 1200.

### Step 5: Calculate the Rate of Interest
Now we can find the rate of interest using the formula:
\[ \text{Rate} = \frac{\text{SI} \times 100}{\text{Sum} \times \text{Time}} \]
Where SI (Simple Interest) can be considered as the Banker's Discount for 2 years, which is Rs. 80. The time is 2 years.

Substituting the values:
\[ \text{Rate} = \frac{80 \times 100}{1200 \times 2} = \frac{8000}{2400} = \frac{80}{24} = \frac{10}{3} \]
This simplifies to:
\[ \text{Rate} = 3 \frac{1}{3} \% \text{ or } 3.33\% \]

### Final Answer
The rate per cent is **3.33%**.

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