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

Options

A. 0.04
B. 0.06
C. 0.05
D. 0.07

Answer

To solve the problem step by step, let's denote the principal as \( P \) and the amount as \( A \).

### Step 1: Set up the initial ratio
From the problem, we know that the ratio of the principal and amount for a certain period is given as:
\[ \frac{P}{A} = \frac{4}{5} \]
This implies that:
\[ A = \frac{5}{4}P \]

### Step 2: Set up the ratio after 3 years
After 3 years, the ratio of the principal and amount changes to:
\[ \frac{P}{A'} = \frac{5}{7} \]
Where \( A' \) is the amount after 3 years. This implies that:
\[ A' = \frac{7}{5}P \]

### Step 3: Relate the two amounts
Since the amount after 3 years can also be expressed in terms of the initial amount and the interest earned, we can write:
\[ A' = A + \text{SI} \]
Where SI is the simple interest earned in 3 years. The simple interest can be calculated using the formula:
\[ \text{SI} = \frac{P \times r \times t}{100} \]
Where \( r \) is the rate of interest and \( t \) is the time in years.

### Step 4: Substitute the values
From Step 1, we know:
\[ A = \frac{5}{4}P \]
Now substituting \( A \) into the expression for \( A' \):
\[ A' = \frac{5}{4}P + \frac{P \times r \times 3}{100} \]
Setting this equal to the expression from Step 2:
\[ \frac{7}{5}P = \frac{5}{4}P + \frac{P \times r \times 3}{100} \]

### Step 5: Simplify the equation
To eliminate \( P \) from the equation, we can divide through by \( P \) (assuming \( P \neq 0 \)):
\[ \frac{7}{5} = \frac{5}{4} + \frac{3r}{100} \]

### Step 6: Solve for \( r \)
Now, we need to solve for \( r \). First, we can convert the fractions to a common denominator:
\[ \frac{7}{5} = \frac{28}{20}, \quad \frac{5}{4} = \frac{25}{20} \]
So, we rewrite the equation:
\[ \frac{28}{20} = \frac{25}{20} + \frac{3r}{100} \]
Subtract \( \frac{25}{20} \) from both sides:
\[ \frac{28}{20} - \frac{25}{20} = \frac{3r}{100} \]
This simplifies to:
\[ \frac{3}{20} = \frac{3r}{100} \]

### Step 7: Cross-multiply to find \( r \)
Cross-multiplying gives:
\[ 3 \times 100 = 3r \times 20 \]
\[ 300 = 60r \]
Now, divide both sides by 60:
\[ r = \frac{300}{60} = 5 \]

### Conclusion
The rate of interest \( r \) is \( 5\% \).