The simple interest on a sum of money is `(9)/(35)` of the sum. If the number of years in numerically `(5)/(7)` times of rate percent per annum, then the rate percent per annum is

Options

A. 9
B. 7
C. 6
D. 4

Answer

To solve the problem step-by-step, we will use the formula for Simple Interest (SI):

### Step 1: Understand the given information
We know that the simple interest (SI) on a sum of money is \(\frac{9}{35}\) of the principal amount (P). This can be expressed as:
\[ SI = \frac{9}{35} P \]

### Step 2: Set up the relationship between time (T) and rate (R)
According to the problem, the number of years (T) is \(\frac{5}{7}\) times the rate percent per annum (R). This can be expressed as:
\[ T = \frac{5}{7} R \]

### Step 3: Use the Simple Interest formula
The formula for Simple Interest is given by:
\[ SI = \frac{P \cdot R \cdot T}{100} \]
Substituting the expressions for SI and T into the formula, we have:
\[ \frac{9}{35} P = \frac{P \cdot R \cdot \left(\frac{5}{7} R\right)}{100} \]

### Step 4: Simplify the equation
We can cancel \(P\) from both sides (assuming \(P \neq 0\)):
\[ \frac{9}{35} = \frac{R \cdot \left(\frac{5}{7} R\right)}{100} \]

This simplifies to:
\[ \frac{9}{35} = \frac{5R^2}{700} \]

### Step 5: Cross-multiply to eliminate the fraction
Cross-multiplying gives:
\[ 9 \cdot 700 = 35 \cdot 5R^2 \]
\[ 6300 = 175R^2 \]

### Step 6: Solve for \(R^2\)
Now, divide both sides by 175:
\[ R^2 = \frac{6300}{175} \]
Calculating the right side:
\[ R^2 = 36 \]

### Step 7: Find \(R\)
Taking the square root of both sides:
\[ R = \sqrt{36} = 6 \]

### Conclusion
Thus, the rate percent per annum is:
\[ \boxed{6} \]

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