Simplify: `10(2)/5xx8(4)/5` divided by `4(2)/5`

Options

A. `20(4)/5`
B. `5/(104)`
C. 64
D. 21

Answer

To simplify the expression \( \frac{10 \frac{2}{5} \times 8 \frac{4}{5}}{4 \frac{2}{5}} \), we will follow these steps:

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

1. For \( 10 \frac{2}{5} \):
\[ 10 \times 5 + 2 = 50 + 2 = 52 \quad \Rightarrow \quad \frac{52}{5} \]

2. For \( 8 \frac{4}{5} \):
\[ 8 \times 5 + 4 = 40 + 4 = 44 \quad \Rightarrow \quad \frac{44}{5} \]

3. For \( 4 \frac{2}{5} \):
\[ 4 \times 5 + 2 = 20 + 2 = 22 \quad \Rightarrow \quad \frac{22}{5} \]

### Step 2: Rewrite the Expression
Now we can rewrite the original expression using improper fractions:
\[ \frac{\frac{52}{5} \times \frac{44}{5}}{\frac{22}{5}} \]

### Step 3: Simplify the Division of Fractions
To divide by a fraction, we multiply by its reciprocal:
\[ \frac{52}{5} \times \frac{44}{5} \times \frac{5}{22} \]

### Step 4: Cancel Common Factors
Now, we can cancel the common factors:
- The \( 5 \) in the numerator and denominator cancels out.
- \( 44 \) and \( 22 \) can be simplified: \( 44 \div 22 = 2 \).

So we have:
\[ \frac{52 \times 2}{5} \]

### Step 5: Multiply the Numerator
Now multiply the numbers in the numerator:
\[ 52 \times 2 = 104 \]
Thus, we have:
\[ \frac{104}{5} \]

### Step 6: Convert to Mixed Number
Finally, we convert \( \frac{104}{5} \) back to a mixed number:
- Divide \( 104 \) by \( 5 \):
- \( 5 \times 20 = 100 \) (which is the largest multiple of \( 5 \) less than \( 104 \))
- Remainder: \( 104 - 100 = 4 \)

So, \( \frac{104}{5} = 20 \frac{4}{5} \).

### Final Answer
The simplified form of the expression is:
\[ 20 \frac{4}{5} \]