A sum of money becomes `(77)/(50)` of itself in `(18)/(5)` years when invested at simple interset. What is the rate (in percentage) of interest per annum ?

Options

A. 20
B. 18
C. 12
D. 15

Answer

To find the rate of interest per annum, we can follow these steps:

### Step 1: Understand the relationship between Principal, Amount, and Simple Interest
In simple interest, the formula is:
\[ \text{Amount} = \text{Principal} + \text{Simple Interest} \]

Given that the amount becomes \( \frac{77}{50} \) of itself, we can express this as:
\[ A = \frac{77}{50} P \]
Where \( A \) is the amount and \( P \) is the principal.

### Step 2: Calculate the Simple Interest
From the formula above, we can find the simple interest (SI):
\[ \text{SI} = A - P = \frac{77}{50} P - P \]
\[ \text{SI} = \left(\frac{77}{50} - 1\right) P \]
\[ \text{SI} = \left(\frac{77 - 50}{50}\right) P = \frac{27}{50} P \]

### Step 3: Use the time period to find the annual simple interest
We know that this simple interest is earned over a period of \( \frac{18}{5} \) years. Therefore, the total simple interest can also be expressed as:
\[ \text{SI} = \text{Principal} \times \text{Rate} \times \text{Time} \]
Let \( R \) be the rate of interest per annum (in decimal form). Thus:
\[ \frac{27}{50} P = P \times R \times \frac{18}{5} \]

### Step 4: Cancel out the Principal
Assuming \( P \neq 0 \), we can divide both sides by \( P \):
\[ \frac{27}{50} = R \times \frac{18}{5} \]

### Step 5: Solve for the Rate of Interest (R)
Now, we can isolate \( R \):
\[ R = \frac{27}{50} \times \frac{5}{18} \]
\[ R = \frac{27 \times 5}{50 \times 18} \]
\[ R = \frac{135}{900} \]
\[ R = \frac{3}{20} \]

### Step 6: Convert to Percentage
To convert \( R \) into a percentage, we multiply by 100:
\[ \text{Rate} = \frac{3}{20} \times 100 = 15\% \]

### Final Answer
The rate of interest per annum is **15%**.

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