With a given rate of simple inerest, the ratio of principal and amount for a certain period of time is 4 : 5. After 3 years, with the same rate of interest, the ratio of the principal and amount becomes 5 : 7. The rate of interest is

Options

A. `4%`
B. `6%`
C. `5%`
D. `7%`

Answer

To solve the problem step by step, let's break it down clearly:

### Step 1: Understand the Ratios
We are given two ratios of Principal (P) and Amount (A):
1. For the first case, the ratio of P to A is 4:5.
2. For the second case (after 3 years), the ratio of P to A is 5:7.

### Step 2: Set Up the Equations
From the first ratio (4:5), we can express:
- Principal (P) = 4x
- Amount (A) = 5x

From the second ratio (5:7), we can express:
- Principal (P) = 5y
- Amount (A) = 7y

### Step 3: Calculate Simple Interest for Both Cases
The formula for Amount (A) in terms of Principal (P), Rate (R), and Time (T) for Simple Interest is:
\[ A = P + \text{SI} \]
Where:
\[ \text{SI} = \frac{P \times R \times T}{100} \]

#### For the First Case:
Using the first case:
\[ A = P + \text{SI} \]
\[ 5x = 4x + \frac{4x \times R \times t_1}{100} \]
Where \( t_1 \) is the time period for the first case.

Rearranging gives:
\[ 5x - 4x = \frac{4x \times R \times t_1}{100} \]
\[ x = \frac{4x \times R \times t_1}{100} \]
Dividing both sides by \( x \) (assuming \( x \neq 0 \)):
\[ 1 = \frac{4R \times t_1}{100} \]
Thus,
\[ 100 = 4R \times t_1 \]
\[ t_1 = \frac{100}{4R} = \frac{25}{R} \]

#### For the Second Case:
Using the second case:
\[ 7y = 5y + \frac{5y \times R \times t_2}{100} \]
Where \( t_2 = t_1 + 3 \) (since it is after 3 years).

Rearranging gives:
\[ 7y - 5y = \frac{5y \times R \times t_2}{100} \]
\[ 2y = \frac{5y \times R \times t_2}{100} \]
Dividing both sides by \( y \) (assuming \( y \neq 0 \)):
\[ 2 = \frac{5R \times t_2}{100} \]
Thus,
\[ 200 = 5R \times t_2 \]
\[ t_2 = \frac{200}{5R} = \frac{40}{R} \]

### Step 4: Relate the Times
Since \( t_2 = t_1 + 3 \):
\[ \frac{40}{R} = \frac{25}{R} + 3 \]

### Step 5: Solve for R
Multiply through by \( R \) to eliminate the denominator:
\[ 40 = 25 + 3R \]
Rearranging gives:
\[ 40 - 25 = 3R \]
\[ 15 = 3R \]
Thus,
\[ R = \frac{15}{3} = 5 \]

### Conclusion
The rate of interest (R) is **5%**.

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