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Add the following mixed numbers:

[tex]\[ 6 \frac{38}{45} + 2 \frac{5}{9} \][/tex]

Enter your answer in the box as a mixed number in simplest form:

[tex]\[ \square \][/tex]

Answer :

Let's solve the problem of adding the mixed numbers [tex]\(6 \frac{38}{45}\)[/tex] and [tex]\(2 \frac{5}{9}\)[/tex].

### Step 1: Convert Mixed Numbers to Improper Fractions

1. Convert [tex]\(6 \frac{38}{45}\)[/tex] to an improper fraction:

Multiply the whole number by the denominator and add the numerator:
[tex]\[
6 \times 45 + 38 = 270 + 38 = 308
\][/tex]
So, [tex]\(6 \frac{38}{45} = \frac{308}{45}\)[/tex].

2. Convert [tex]\(2 \frac{5}{9}\)[/tex] to an improper fraction:

Multiply the whole number by the denominator and add the numerator:
[tex]\[
2 \times 9 + 5 = 18 + 5 = 23
\][/tex]
So, [tex]\(2 \frac{5}{9} = \frac{23}{9}\)[/tex].

### Step 2: Find a Common Denominator

The common denominator between 45 and 9 is 45.

Convert [tex]\(\frac{23}{9}\)[/tex] to an equivalent fraction with a denominator of 45:
[tex]\[
\frac{23}{9} \times \frac{5}{5} = \frac{115}{45}
\][/tex]

### Step 3: Add the Fractions

Now that both fractions have the same denominator, we add them:
[tex]\[
\frac{308}{45} + \frac{115}{45} = \frac{308 + 115}{45} = \frac{423}{45}
\][/tex]

### Step 4: Convert the Sum to a Mixed Number

1. Find the Whole Number Part:

Divide the numerator by the denominator:
[tex]\[
423 \div 45 = 9 \quad \text{(whole number)}
\][/tex]

2. Find the Remainder for the Fractional Part:

The remainder is:
[tex]\[
423 \mod 45 = 18
\][/tex]

So, the fractional part is [tex]\(\frac{18}{45}\)[/tex].

### Step 5: Simplify the Fraction

To simplify [tex]\(\frac{18}{45}\)[/tex], find the greatest common divisor (GCD) of 18 and 45, which is 9:

[tex]\[
\frac{18}{45} = \frac{18 \div 9}{45 \div 9} = \frac{2}{5}
\][/tex]

### Solution

Putting it all together, the mixed number result is:
[tex]\[
9 \frac{2}{5}
\][/tex]

So, the answer is [tex]\(9 \frac{2}{5}\)[/tex].

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