A person borrows some money for 8 years at a rate of simple interest. If the ratio of principal and total interest is 5 : 8, then what is the rate (in percentage) of interest?

Options

A. 10
B. 20
C. 25
D. 30

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Understand the given information
We know that the ratio of the principal (P) to the total interest (I) is given as 5:8. This means:
\[ \frac{P}{I} = \frac{5}{8} \]

### Step 2: Express the principal and interest in terms of a variable
Let’s assume the principal amount (P) is \(5x\) and the total interest (I) is \(8x\), where \(x\) is a common multiplier.

### Step 3: Use the formula for simple interest
The formula for simple interest is:
\[ I = \frac{P \times R \times T}{100} \]
where:
- \(I\) is the total interest,
- \(P\) is the principal,
- \(R\) is the rate of interest,
- \(T\) is the time in years.

### Step 4: Substitute the known values into the formula
From our assumption:
- \(I = 8x\)
- \(P = 5x\)
- \(T = 8\) years

Substituting these values into the simple interest formula gives:
\[ 8x = \frac{5x \times R \times 8}{100} \]

### Step 5: Simplify the equation
Now, simplify the equation:
\[ 8x = \frac{40xR}{100} \]
\[ 8x = \frac{2xR}{5} \]

### Step 6: Eliminate \(x\) from both sides
Assuming \(x \neq 0\), we can divide both sides by \(x\):
\[ 8 = \frac{2R}{5} \]

### Step 7: Solve for \(R\)
To isolate \(R\), multiply both sides by 5:
\[ 40 = 2R \]
Now, divide both sides by 2:
\[ R = 20 \]

### Step 8: Conclusion
The rate of interest \(R\) is 20%.

### Final Answer
The rate of interest is **20%**.

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