Alright, let's walk through this problem step-by-step to multiply the mixed numbers [tex]\(5 \frac{3}{4}\)[/tex] and [tex]\(1 \frac{3}{4}\)[/tex] and simplify the answer.
### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers to improper fractions.
For [tex]\(5 \frac{3}{4}\)[/tex]:
[tex]\[ 5 \frac{3}{4} = \frac{(5 \times 4) + 3}{4} = \frac{20 + 3}{4} = \frac{23}{4} \][/tex]
For [tex]\(1 \frac{3}{4}\)[/tex]:
[tex]\[ 1 \frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4} \][/tex]
### Step 2: Multiply the Improper Fractions
Next, we multiply these two improper fractions:
[tex]\[ \frac{23}{4} \times \frac{7}{4} = \frac{23 \times 7}{4 \times 4} = \frac{161}{16} \][/tex]
### Step 3: Simplify the Fraction (if needed)
In our case, [tex]\(\frac{161}{16}\)[/tex] is already in its simplest form as the numerator and the denominator have no common factors other than 1.
### Step 4: Convert the Improper Fraction to a Mixed Number
Now, we convert the improper fraction [tex]\(\frac{161}{16}\)[/tex] back to a mixed number.
To do this, we divide the numerator by the denominator:
[tex]\[ 161 \div 16 = 10 \quad \text{remainder} \quad 1 \][/tex]
This means:
[tex]\[ \frac{161}{16} = 10 \frac{1}{16} \][/tex]
### Summary
We multiplied [tex]\(5 \frac{3}{4}\)[/tex] and [tex]\(1 \frac{3}{4}\)[/tex], which simplifies to:
[tex]\[ 10 \frac{1}{16} \][/tex]
Therefore, the correct answer is:
[tex]\[ 10 \frac{1}{16} \][/tex]
So the third option [tex]\(10 \frac{1}{16}\)[/tex] is the correct choice.
