To solve the problem [tex]\(\frac{19}{6} + \frac{10}{3}\)[/tex], we need to follow these steps:
1. Identify the fractions and their denominators: We have the fractions [tex]\(\frac{19}{6}\)[/tex] and [tex]\(\frac{10}{3}\)[/tex]. The denominators are 6 and 3, respectively.
2. Find a common denominator: To add the fractions, we need a common denominator. The least common multiple (LCM) of 6 and 3 is 6.
3. Convert the fractions to have the common denominator:
- The first fraction, [tex]\(\frac{19}{6}\)[/tex], already has the denominator 6.
- The second fraction, [tex]\(\frac{10}{3}\)[/tex], needs to be converted to have the denominator 6. We do this by multiplying both the numerator and the denominator by 2:
[tex]\[
\frac{10}{3} = \frac{10 \times 2}{3 \times 2} = \frac{20}{6}
\][/tex]
4. Add the fractions now that they have the same denominator:
[tex]\[
\frac{19}{6} + \frac{20}{6} = \frac{19 + 20}{6} = \frac{39}{6}
\][/tex]
5. Simplify the resulting fraction: Finally, we simplify [tex]\(\frac{39}{6}\)[/tex] by finding the greatest common divisor (GCD) of 39 and 6, which is 3. We divide both the numerator and the denominator by 3:
[tex]\[
\frac{39}{6} = \frac{39 \div 3}{6 \div 3} = \frac{13}{2}
\][/tex]
So, [tex]\(\frac{19}{6} + \frac{10}{3} = \frac{13}{2}\)[/tex].
