Find the value of m and n respectively
`5(1)/(m)xxn(3)/(4)=20`

Options

A. 3,1
B. 3,3
C. 4,1
D. 5,3

Answer

To solve the equation \( 5 \frac{1}{m} \times n \frac{3}{4} = 20 \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
The mixed number \( 5 \frac{1}{m} \) can be converted to an improper fraction:
\[ 5 \frac{1}{m} = \frac{5m + 1}{m} \]
Similarly, \( n \frac{3}{4} \) can be expressed as:
\[ n \frac{3}{4} = \frac{4n + 3}{4} \]

### Step 2: Rewrite the Equation
Now, we can rewrite the equation using the improper fractions:
\[ \frac{5m + 1}{m} \times \frac{4n + 3}{4} = 20 \]

### Step 3: Simplify the Equation
To eliminate the fraction, we can multiply both sides by \( 4m \):
\[ (5m + 1)(4n + 3) = 80m \]

### Step 4: Expand the Left Side
Expanding the left side gives:
\[ 20mn + 15m + 4n + 3 = 80m \]

### Step 5: Rearrange the Equation
Rearranging the equation to bring all terms to one side results in:
\[ 20mn + 4n + 15m + 3 - 80m = 0 \]
which simplifies to:
\[ 20mn + 4n - 65m + 3 = 0 \]

### Step 6: Solve for Values of m and n
At this point, we can try different integer values for \( m \) and \( n \) to find a solution.

Let's try \( m = 3 \) and \( n = 1 \):
\[ 20(3)(1) + 4(1) - 65(3) + 3 = 60 + 4 - 195 + 3 = -128 \quad (\text{not a solution}) \]

Next, let's try \( m = 3 \) and \( n = 3 \):
\[ 20(3)(3) + 4(3) - 65(3) + 3 = 180 + 12 - 195 + 3 = 0 \quad (\text{this works!}) \]

Thus, the values of \( m \) and \( n \) are:
\[ m = 3, \quad n = 3 \]

### Final Answer
The values of \( m \) and \( n \) are \( 3 \) and \( 3 \) respectively.

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