Solve Complex Division: 27½ ÷ (2⅕ ÷ ⅘) Step-by-Step

Let's solve the problem step-by-step.

• Step 1: Convert the mixed numbers to improper fractions.

Convert $27\frac{1}{2}$ to an improper fraction:
$27\frac{1}{2} = \frac{2 \times 27 + 1}{2} = \frac{55}{2}$.

Convert $2\frac{1}{5}$ to an improper fraction:
$2\frac{1}{5} = \frac{5 \times 2 + 1}{5} = \frac{11}{5}$.

• Step 2: Simplify $2\frac{1}{5} : \frac{4}{5}$ using reciprocal multiplication.

$\frac{11}{5} : \frac{4}{5} = \frac{11}{5} \times \frac{5}{4} = \frac{11 \times 5}{5 \times 4} = \frac{11}{4}$.

• Step 3: Divide $27\frac{1}{2}$ by the result from Step 2.

$\frac{55}{2} : \frac{11}{4} = \frac{55}{2} \times \frac{4}{11} = \frac{55 \times 4}{2 \times 11}$.

Simplify the expression:
The $55$ in the numerator and $11$ in the denominator simplify as follows: $55 \div 11 = 5$, leaving us with:
$\frac{5 \times 4}{2} = \frac{20}{2} = 10$.

Therefore, the solution to the problem is $10$, matching the correct choice.