By what number should `((3)/(5))^2` be multiplied so that the product is `(3)/(7)` ?

Options

A. `(25)/(9)`
B. (25)/(21)`
C. (25)/(7)`
D. `(25)/(3)`

Answer

To solve the problem, we need to find the number \( y \) such that:

\[ \left(\frac{3}{5}\right)^2 \times y = \frac{3}{7} \]

### Step 1: Calculate \(\left(\frac{3}{5}\right)^2\)

First, we need to compute \(\left(\frac{3}{5}\right)^2\):

\[ \left(\frac{3}{5}\right)^2 = \frac{3^2}{5^2} = \frac{9}{25} \]

### Step 2: Set up the equation

Now, we can substitute this back into our equation:

\[ \frac{9}{25} \times y = \frac{3}{7} \]

### Step 3: Solve for \( y \)

To isolate \( y \), we can multiply both sides of the equation by the reciprocal of \(\frac{9}{25}\):

\[ y = \frac{3}{7} \div \frac{9}{25} \]

This is equivalent to multiplying by the reciprocal:

\[ y = \frac{3}{7} \times \frac{25}{9} \]

### Step 4: Simplify the expression

Now, we can simplify:

\[ y = \frac{3 \times 25}{7 \times 9} = \frac{75}{63} \]

### Step 5: Reduce the fraction

Next, we can simplify \(\frac{75}{63}\) by finding the greatest common divisor (GCD) of 75 and 63, which is 3:

\[ y = \frac{75 \div 3}{63 \div 3} = \frac{25}{21} \]

### Final Answer

Thus, the number that should be multiplied by \(\left(\frac{3}{5}\right)^2\) to get \(\frac{3}{7}\) is:

\[ \boxed{\frac{25}{21}} \]

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