To solve the given addition of mixed numbers [tex]\(1 \frac{3}{8} + 3 \frac{7}{8}\)[/tex], let's break it down into step-by-step reasoning:
### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert each mixed number to an improper fraction.
For [tex]\(1 \frac{3}{8}\)[/tex]:
[tex]\[1 \frac{3}{8} = 1 + \frac{3}{8} = \frac{8}{8} + \frac{3}{8} = \frac{11}{8}\][/tex]
For [tex]\(3 \frac{7}{8}\)[/tex]:
[tex]\[3 \frac{7}{8} = 3 + \frac{7}{8} = \frac{24}{8} + \frac{7}{8} = \frac{31}{8}\][/tex]
### Step 2: Add the Improper Fractions
Now we add these two improper fractions.
[tex]\[\frac{11}{8} + \frac{31}{8} = \frac{11 + 31}{8} = \frac{42}{8}\][/tex]
### Step 3: Simplify the Fraction (if necessary)
Next, we simplify the fraction [tex]\(\frac{42}{8}\)[/tex] by converting it to a mixed number.
To convert [tex]\(\frac{42}{8}\)[/tex] to a mixed number, we divide the numerator by the denominator to get the whole part and the remainder.
[tex]\[42 \div 8 = 5 \qquad\text{with a remainder of } 2\][/tex]
Thus, we can write:
[tex]\[\frac{42}{8} = 5 \frac{2}{8}\][/tex]
### Step 4: Simplify the Fractional Part (if necessary)
If needed, simplify the fractional part. However, [tex]\(\frac{2}{8}\)[/tex] can be further simplified by dividing the numerator and the denominator by their greatest common divisor, which is 2.
[tex]\[\frac{2}{8} = \frac{2 \div 2}{8 \div 2} = \frac{1}{4}\][/tex]
So we have:
[tex]\[5 \frac{2}{8} = 5 \frac{1}{4}\][/tex]
In this case, the fraction [tex]\(\frac{2}{8}\)[/tex] remains as it is, but if we simplify it, we get [tex]\(\frac{1}{4}\)[/tex], which is the same thing in fractional part representation without changing the whole number.
Thus, we can conclude that:
[tex]\[1 \frac{3}{8} + 3 \frac{7}{8} = 5 \frac{2}{8}\][/tex]
Or equivalently:
[tex]\[1 \frac{3}{8} + 3 \frac{7}{8} = 5 \frac{1}{4}\][/tex]
This detailed step-by-step approach confirms our initial result:
[tex]\[
1 \frac{3}{8} + 3 \frac{7}{8} = 5 \frac{2}{8}
\][/tex]
