Sure, let's add the mixed numbers [tex]\( 5 \frac{4}{5} \)[/tex] and [tex]\( 3 \frac{6}{7} \)[/tex] step-by-step:
### Step 1: Convert Mixed Numbers to Improper Fractions
First, convert each mixed number to an improper fraction.
For [tex]\( 5 \frac{4}{5} \)[/tex]:
- The whole number part is 5.
- The fraction part is [tex]\( \frac{4}{5} \)[/tex].
To convert [tex]\( 5 \frac{4}{5} \)[/tex] to an improper fraction:
[tex]\[
5 \frac{4}{5} = \frac{(5 \times 5) + 4}{5} = \frac{25 + 4}{5} = \frac{29}{5}
\][/tex]
For [tex]\( 3 \frac{6}{7} \)[/tex]:
- The whole number part is 3.
- The fraction part is [tex]\( \frac{6}{7} \)[/tex].
To convert [tex]\( 3 \frac{6}{7} \)[/tex] to an improper fraction:
[tex]\[
3 \frac{6}{7} = \frac{(3 \times 7) + 6}{7} = \frac{21 + 6}{7} = \frac{27}{7}
\][/tex]
### Step 2: Find a Common Denominator
Next, find a common denominator to add these fractions. The denominators are 5 and 7. The common denominator is:
[tex]\[
5 \times 7 = 35
\][/tex]
### Step 3: Adjust Numerators for the Common Denominator
Convert each fraction to have the common denominator of 35.
For [tex]\( \frac{29}{5} \)[/tex]:
[tex]\[
\frac{29}{5} = \frac{29 \times 7}{5 \times 7} = \frac{203}{35}
\][/tex]
For [tex]\( \frac{27}{7} \)[/tex]:
[tex]\[
\frac{27}{7} = \frac{27 \times 5}{7 \times 5} = \frac{135}{35}
\][/tex]
### Step 4: Add the Adjusted Fractions
Now add the numerators, keeping the common denominator:
[tex]\[
\frac{203}{35} + \frac{135}{35} = \frac{203 + 135}{35} = \frac{338}{35}
\][/tex]
### Step 5: Simplify the Result
Now simplify the improper fraction [tex]\( \frac{338}{35} \)[/tex] if possible. Find the greatest common divisor (GCD) of 338 and 35, which is 1. Since their GCD is 1, the fraction is already in its simplest form.
### Step 6: Convert the Improper Fraction to a Mixed Number
Finally, convert [tex]\( \frac{338}{35} \)[/tex] to a mixed number:
Divide 338 by 35:
[tex]\[
338 \div 35 = 9 \text{ (quotient)} \quad \text{remainder:} \quad 338 - (35 \times 9) = 23
\][/tex]
Thus,
[tex]\[
\frac{338}{35} = 9 \frac{23}{35}
\][/tex]
So, the sum of [tex]\( 5 \frac{4}{5} \)[/tex] and [tex]\( 3 \frac{6}{7} \)[/tex] is:
[tex]\[
9 \frac{23}{35}
\][/tex]
