To solve the addition of the mixed number [tex]\(23 \frac{2}{5}\)[/tex] and the fraction [tex]\(\frac{3}{7}\)[/tex], we will follow these steps:
1. Convert the Mixed Number to an Improper Fraction:
The mixed number [tex]\(23 \frac{2}{5}\)[/tex] needs to be converted to an improper fraction. This involves taking the integer part (23) and converting it along with the fraction part ([tex]\(\frac{2}{5}\)[/tex]) into a single fraction.
[tex]\[
23 \frac{2}{5} = 23 + \frac{2}{5}
\][/tex]
To add these, express 23 as a fraction with the same denominator (5):
[tex]\[
23 = \frac{23 \cdot 5}{5} = \frac{115}{5}
\][/tex]
Now add the two fractions:
[tex]\[
\frac{115}{5} + \frac{2}{5} = \frac{115 + 2}{5} = \frac{117}{5}
\][/tex]
2. Identify the Other Fraction:
The second fraction is [tex]\(\frac{3}{7}\)[/tex].
3. Find a Common Denominator:
The denominators for our fractions are 5 and 7. The common denominator would be their product:
[tex]\[
5 \times 7 = 35
\][/tex]
4. Convert Both Fractions to the Common Denominator:
For [tex]\(\frac{117}{5}\)[/tex]:
[tex]\[
\frac{117 \cdot 7}{5 \cdot 7} = \frac{819}{35}
\][/tex]
For [tex]\(\frac{3}{7}\)[/tex]:
[tex]\[
\frac{3 \cdot 5}{7 \cdot 5} = \frac{15}{35}
\][/tex]
5. Add the Fractions:
With both fractions now having a common denominator, simply add the numerators:
[tex]\[
\frac{819}{35} + \frac{15}{35} = \frac{819 + 15}{35} = \frac{834}{35}
\][/tex]
6. Convert the Result Back to a Mixed Number:
To express [tex]\(\frac{834}{35}\)[/tex] as a mixed number, divide the numerator by the denominator:
[tex]\[
834 \div 35 = 23 \text{ remainder } 29
\][/tex]
So the mixed number is:
[tex]\[
23 \frac{29}{35}
\][/tex]
7. Simplify the Fraction Part if Needed:
Since [tex]\(\frac{29}{35}\)[/tex] is already in its simplest form (the greatest common divisor of 29 and 35 is 1), no further simplification is needed.
Therefore, the final answer to the problem [tex]\(23 \frac{2}{5} + \frac{3}{7}\)[/tex] is:
[tex]\[
23 \frac{29}{35}
\][/tex]
