Let's solve the given problems step-by-step:
### Part (a)
Simplify [tex]\(\frac{2}{3} + \frac{7}{3} - \frac{5}{3}\)[/tex]:
1. All the fractions have a common denominator (3), so we can directly add and subtract the numerators.
2. Combining the numerators:
[tex]\[
\frac{2}{3} + \frac{7}{3} - \frac{5}{3} = \frac{2 + 7 - 5}{3} = \frac{4}{3}
\][/tex]
3. Simplifying, we get:
[tex]\[
\frac{4}{3} \approx 1.3333333333333333
\][/tex]
### Part (b)
Simplify [tex]\(2 \frac{3}{5} + 3 \frac{1}{10}\)[/tex]:
1. Convert the mixed fractions to improper fractions:
[tex]\[
2 \frac{3}{5} = \frac{2 \times 5 + 3}{5} = \frac{13}{5}
\][/tex]
[tex]\[
3 \frac{1}{10} = \frac{3 \times 10 + 1}{10} = \frac{31}{10}
\][/tex]
2. Find a common denominator for the fractions [tex]\(\frac{13}{5}\)[/tex] and [tex]\(\frac{31}{10}\)[/tex]. The least common multiple (LCM) of 5 and 10 is 10.
3. Convert the fractions to have the common denominator (10):
[tex]\[
\frac{13}{5} = \frac{13 \times 2}{5 \times 2} = \frac{26}{10}
\][/tex]
[tex]\[
\frac{31}{10} \text{ remains as it is, i.e., } \frac{31}{10}
\][/tex]
4. Add the fractions:
[tex]\[
\frac{26}{10} + \frac{31}{10} = \frac{26 + 31}{10} = \frac{57}{10}
\][/tex]
5. Simplify the result:
[tex]\[
\frac{57}{10} = 5.7
\][/tex]
So, the answers are:
a. [tex]\(\frac{4}{3} \approx 1.3333333333333333\)[/tex]
b. [tex]\(2 \frac{3}{5} + 3 \frac{1}{10} = 5.7\)[/tex]
Note: We also find the numerators to be [tex]\(26\)[/tex] and [tex]\(31\)[/tex] respectively when converted to fractions with a common denominator.
