To solve the problem [tex]\(10 \frac{6}{7} \div 7 \frac{1}{2}\)[/tex], we need to follow several steps to divide these mixed numbers.
### Step 1: Convert mixed numbers to improper fractions
First, let's convert both mixed numbers to improper fractions.
1. For [tex]\(10 \frac{6}{7}\)[/tex]:
[tex]\[10\frac{6}{7} = 10 + \frac{6}{7}\][/tex]
To convert the integer part (10) to a fraction with denominator 7:
[tex]\[10 = \frac{10 \times 7}{7} = \frac{70}{7}\][/tex]
Now, add [tex]\(\frac{6}{7}\)[/tex] to [tex]\(\frac{70}{7}\)[/tex]:
[tex]\[10 \frac{6}{7} = \frac{70}{7} + \frac{6}{7} = \frac{70 + 6}{7} = \frac{76}{7}\][/tex]
2. For [tex]\(7 \frac{1}{2}\)[/tex]:
[tex]\[7 \frac{1}{2} = 7 + \frac{1}{2}\][/tex]
To convert the integer part (7) to a fraction with denominator 2:
[tex]\[7 = \frac{7 \times 2}{2} = \frac{14}{2}\][/tex]
Now, add [tex]\(\frac{1}{2}\)[/tex] to [tex]\(\frac{14}{2}\)[/tex]:
[tex]\[7 \frac{1}{2} = \frac{14}{2} + \frac{1}{2} = \frac{14 + 1}{2} = \frac{15}{2}\][/tex]
### Step 2: Divide the improper fractions
Now, we need to divide the two improper fractions:
[tex]\[\frac{76}{7} \div \frac{15}{2}\][/tex]
Dividing by a fraction is the same as multiplying by its reciprocal:
[tex]\[\frac{76}{7} \times \frac{2}{15}\][/tex]
### Step 3: Multiply the fractions
To multiply fractions, you multiply the numerators and the denominators:
[tex]\[
\frac{76 \times 2}{7 \times 15} = \frac{152}{105}
\][/tex]
### Step 4: Simplify the fraction
Next, we simplify the resulting fraction, [tex]\(\frac{152}{105}\)[/tex]. However, in this case, [tex]\(\frac{152}{105}\)[/tex] is already in its simplest form as their greatest common divisor is 1.
### Step 5: Convert the improper fraction to a mixed number
Now, let's convert [tex]\(\frac{152}{105}\)[/tex] back to a mixed number. We do this by dividing the numerator by the denominator:
[tex]\[ 152 \div 105 = 1 \text{ (quotient) with a remainder of } 47 \][/tex]
So, the integer part is 1, and the fractional part is:
[tex]\[\frac{47}{105}\][/tex]
### Conclusion
Therefore,
[tex]\[10 \frac{6}{7} \div 7 \frac{1}{2} = 1 \frac{47}{105}\][/tex]
