Sure, we can solve each part step by step. Let's start with the given fractions and mixed numbers.
### (a) [tex]\(1 \frac{1}{3} + 3 \frac{2}{3}\)[/tex]
First, we need to convert the mixed numbers into improper fractions:
- [tex]\( 1 \frac{1}{3} \)[/tex] can be written as [tex]\( 1 + \frac{1}{3} \)[/tex], which is [tex]\( \frac{4}{3} \)[/tex].
- [tex]\( 3 \frac{2}{3} \)[/tex] can be written as [tex]\( 3 + \frac{2}{3} \)[/tex], which is [tex]\( \frac{11}{3} \)[/tex].
Now, add the fractions:
[tex]\[ \frac{4}{3} + \frac{11}{3} = \frac{4 + 11}{3} = \frac{15}{3} = 5 \][/tex]
### (b) [tex]\( \frac{2}{3} + \frac{3}{4} + \frac{1}{2} \)[/tex]
To add these fractions, we need a common denominator. The least common denominator (LCD) for 3, 4, and 2 is 12.
- Convert each fraction to have the denominator 12:
[tex]\[
\frac{2}{3} = \frac{2 \times 4}{3 \times 4} = \frac{8}{12}
\][/tex]
[tex]\[
\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}
\][/tex]
[tex]\[
\frac{1}{2} = \frac{1 \times 6}{2 \times 6} = \frac{6}{12}
\][/tex]
Now, add them:
[tex]\[ \frac{8}{12} + \frac{9}{12} + \frac{6}{12} = \frac{8 + 9 + 6}{12} = \frac{23}{12} \approx 1.9166666666666665 \][/tex]
### (c) [tex]\( 4 \frac{2}{3} + 3 \frac{1}{4} \)[/tex]
Convert the mixed numbers to improper fractions:
- [tex]\( 4 \frac{2}{3} \)[/tex] can be written as [tex]\( 4 + \frac{2}{3} \)[/tex], which is [tex]\( \frac{14}{3} \)[/tex].
- [tex]\( 3 \frac{1}{4} \)[/tex] can be written as [tex]\( 3 + \frac{1}{4} \)[/tex], which is [tex]\( \frac{13}{4} \)[/tex].
To add these fractions, find a common denominator. The LCD for 3 and 4 is 12.
- Convert the fractions:
[tex]\[
\frac{14}{3} = \frac{14 \times 4}{3 \times 4} = \frac{56}{12}
\][/tex]
[tex]\[
\frac{13}{4} = \frac{13 \times 3}{4 \times 3} = \frac{39}{12}
\][/tex]
Now, add them:
[tex]\[ \frac{56}{12} + \frac{39}{12} = \frac{56 + 39}{12} = \frac{95}{12} \approx 7.916666666666667 \][/tex]
### (d) [tex]\( \frac{5}{6} - \frac{1}{3} \)[/tex]
First, convert [tex]\(\frac{1}{3}\)[/tex] to a fraction with the same denominator as [tex]\(\frac{5}{6}\)[/tex], which is 6.
[tex]\[ \frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6} \][/tex]
Now, subtract the fractions:
[tex]\[ \frac{5}{6} - \frac{2}{6} = \frac{5 - 2}{6} = \frac{3}{6} = \frac{1}{2} = 0.5 \][/tex]
So the solutions are:
(a) [tex]\(5\)[/tex]
(b) [tex]\(1.9166666666666665\)[/tex]
(c) [tex]\(7.916666666666667\)[/tex]
(d) [tex]\(0.5\)[/tex]
