8 1/4 : c = 13 3/4 : 2 1/3

To solve the equation \( \frac{8 \frac{1}{4}}{c} = \frac{13 \frac{3}{4}}{2 \frac{1}{3}} \), we first need to rewrite the mixed numbers as improper fractions.

1. Convert \( 8 \frac{1}{4} \) to an improper fraction: \[ 8 \frac{1}{4} = 8 \times 4 + 1 = 32 + 1 = \frac{33}{4} \]

2. Convert \( 13 \frac{3}{4} \) to an improper fraction: \[ 13 \frac{3}{4} = 13 \times 4 + 3 = 52 + 3 = \frac{55}{4} \]

3. Convert \( 2 \frac{1}{3} \) to an improper fraction: \[ 2 \frac{1}{3} = 2 \times 3 + 1 = 6 + 1 = \frac{7}{3} \]

Now the equation becomes: \[ \frac{\frac{33}{4}}{c} = \frac{\frac{55}{4}}{\frac{7}{3}} \]

Next, we simplify the right side: \[ \frac{\frac{55}{4}}{\frac{7}{3}} = \frac{55}{4} \times \frac{3}{7} = \frac{55 \times 3}{4 \times 7} = \frac{165}{28} \]

Now we have: \[ \frac{\frac{33}{4}}{c} = \frac{165}{28} \]

Cross-multiply to solve for \( c \): \[ 33 \times 28 = 165 \times c \]

Calculating \( 33 \times 28 \): \[ 33 \times 28 = 924 \]

So we have: \[ 924 = 165c \]

Now, solve for \( c \): \[ c = \frac{924}{165} \]

Now we can simplify \( \frac{924}{165} \) by finding the GCD of 924 and 165, which is 33:

1. Divide both the numerator and the denominator by 33: \[ c = \frac{924 \div 33}{165 \div 33} = \frac{28}{5} \]

Thus, the value of \( c \) is: \[ c = 5.6 \quad \text{or} \quad c = \frac{28}{5} \]