Which fraction will be multiplied by fraction 2 `5/7`. so that its product will be 4 `13/21` ?

Options

A. `57/97`
B. `21/97`
C. `7/19`
D. `97/57`

Answer

To find the fraction that needs to be multiplied by \( 2 \frac{5}{7} \) to yield \( 4 \frac{13}{21} \), we can follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert both mixed numbers into improper fractions.

For \( 2 \frac{5}{7} \):
\[ 2 \frac{5}{7} = \frac{(2 \times 7) + 5}{7} = \frac{14 + 5}{7} = \frac{19}{7} \]

For \( 4 \frac{13}{21} \):
\[ 4 \frac{13}{21} = \frac{(4 \times 21) + 13}{21} = \frac{84 + 13}{21} = \frac{97}{21} \]

### Step 2: Set Up the Equation
Let the unknown fraction be \( \frac{x}{y} \). We set up the equation based on the problem statement:
\[ \frac{x}{y} \times \frac{19}{7} = \frac{97}{21} \]

### Step 3: Isolate the Unknown Fraction
To isolate \( \frac{x}{y} \), we can multiply both sides of the equation by the reciprocal of \( \frac{19}{7} \):
\[ \frac{x}{y} = \frac{97}{21} \times \frac{7}{19} \]

### Step 4: Perform the Multiplication
Now we perform the multiplication:
\[ \frac{x}{y} = \frac{97 \times 7}{21 \times 19} \]

Calculating the numerator:
\[ 97 \times 7 = 679 \]

Calculating the denominator:
\[ 21 \times 19 = 399 \]

Thus, we have:
\[ \frac{x}{y} = \frac{679}{399} \]

### Step 5: Simplify the Fraction (if necessary)
We check if \( \frac{679}{399} \) can be simplified. Since 679 and 399 have no common factors, this fraction is already in its simplest form.

### Conclusion
The fraction that needs to be multiplied by \( 2 \frac{5}{7} \) to get \( 4 \frac{13}{21} \) is:
\[ \frac{679}{399} \]