A earns an interest of rs2880 on a certain sum when it is invested for 3 years offering R% simple interest per annum. B earns an interest of rs4000 on a ceratin sum when it is invested for 5 years offering R% simple interest per annum. Sum invested by A is what per cent more than that invested by B?

Options

A. `(66) 2/3 %`
B. `(33) 1/3 %`
C. `30%`
D. `20%`

Answer

To solve the problem step by step, we will use the formula for simple interest and the information given in the question.

### Step 1: Understand the Simple Interest Formula
The formula for calculating simple interest (SI) is:
\[ SI = \frac{P \times R \times T}{100} \]
Where:
- \(SI\) = Simple Interest
- \(P\) = Principal amount (the initial sum of money)
- \(R\) = Rate of interest per annum
- \(T\) = Time in years

### Step 2: Set Up the Equations for A and B
From the problem, we know:
- A earns an interest of Rs. 2880 over 3 years.
- B earns an interest of Rs. 4000 over 5 years.
- The rate of interest \(R\) is the same for both A and B.

For A:
\[ 2880 = \frac{P_A \times R \times 3}{100} \quad \text{(1)} \]

For B:
\[ 4000 = \frac{P_B \times R \times 5}{100} \quad \text{(2)} \]

### Step 3: Rearrange the Equations to Find Principal Amounts
From equation (1):
\[ P_A = \frac{2880 \times 100}{3R} = \frac{288000}{3R} = \frac{96000}{R} \quad \text{(3)} \]

From equation (2):
\[ P_B = \frac{4000 \times 100}{5R} = \frac{400000}{5R} = \frac{80000}{R} \quad \text{(4)} \]

### Step 4: Find the Ratio of Principal Amounts
Now, we can find the ratio of \(P_A\) to \(P_B\) using equations (3) and (4):
\[ \frac{P_A}{P_B} = \frac{\frac{96000}{R}}{\frac{80000}{R}} = \frac{96000}{80000} = \frac{12}{10} = \frac{6}{5} \]

### Step 5: Express \(P_A\) in Terms of \(P_B\)
From the ratio, we have:
\[ P_A = \frac{6}{5} P_B \]

### Step 6: Calculate the Percentage Difference
To find how much more \(P_A\) is than \(P_B\):
\[ \text{Difference} = P_A - P_B = \frac{6}{5} P_B - P_B = \left(\frac{6}{5} - \frac{5}{5}\right) P_B = \frac{1}{5} P_B \]

Now, to find the percentage more:
\[ \text{Percentage more} = \left(\frac{\text{Difference}}{P_B}\right) \times 100 = \left(\frac{\frac{1}{5} P_B}{P_B}\right) \times 100 = \frac{1}{5} \times 100 = 20\% \]

### Final Answer
The sum invested by A is **20% more** than that invested by B.

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