Answer :
Let's solve each of these equations involving fractions step-by-step.
### a) [tex]\(\frac{1}{4} + \frac{2}{4}\)[/tex]
The denominators are the same, so we can add the numerators:
[tex]\[ \frac{1}{4} + \frac{2}{4} = \frac{1+2}{4} = \frac{3}{4} \][/tex]
### c) [tex]\(\frac{1}{4} + \frac{7}{8}\)[/tex]
To add these fractions, we need a common denominator. The least common denominator (LCD) of 4 and 8 is 8:
[tex]\[ \frac{1}{4} = \frac{2}{8} \][/tex]
Now, add the fractions:
[tex]\[ \frac{2}{8} + \frac{7}{8} = \frac{2+7}{8} = \frac{9}{8} = 1 \frac{1}{8} \][/tex]
### e) [tex]\(\frac{2}{10} + \frac{3}{14}\)[/tex]
Find the least common denominator (LCD) of 10 and 14, which is 70:
[tex]\[ \frac{2}{10} = \frac{2 \times 7}{10 \times 7} = \frac{14}{70} \][/tex]
[tex]\[ \frac{3}{14} = \frac{3 \times 5}{14 \times 5} = \frac{15}{70} \][/tex]
Now, add the fractions:
[tex]\[ \frac{14}{70} + \frac{15}{70} = \frac{14+15}{70} = \frac{29}{70} \][/tex]
### f) [tex]\(\frac{21}{22} + \frac{3}{5}\)[/tex]
Find the least common denominator (LCD) of 22 and 5, which is 110:
[tex]\[ \frac{21}{22} = \frac{21 \times 5}{22 \times 5} = \frac{105}{110} \][/tex]
[tex]\[ \frac{3}{5} = \frac{3 \times 22}{5 \times 22} = \frac{66}{110} \][/tex]
Now, add the fractions:
[tex]\[ \frac{105}{110} + \frac{66}{110} = \frac{105+66}{110} = \frac{171}{110} \][/tex]
### g) [tex]\(\frac{4}{5} - \frac{2}{5}\)[/tex]
The denominators are the same, so we can subtract the numerators:
[tex]\[ \frac{4}{5} - \frac{2}{5} = \frac{4-2}{5} = \frac{2}{5} \][/tex]
### h) [tex]\(10 - \frac{3}{11}\)[/tex]
Convert 10 to a fraction with a denominator of 11:
[tex]\[ 10 = \frac{110}{11} \][/tex]
Now, subtract the fractions:
[tex]\[ \frac{110}{11} - \frac{3}{11} = \frac{110-3}{11} = \frac{107}{11} \][/tex]
### i) [tex]\(\frac{3}{4} - \frac{1}{6}\)[/tex]
Find the least common denominator (LCD) of 4 and 6, which is 12:
[tex]\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \][/tex]
[tex]\[ \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \][/tex]
Now, subtract the fractions:
[tex]\[ \frac{9}{12} - \frac{2}{12} = \frac{9-2}{12} = \frac{7}{12} \][/tex]
### j) [tex]\(\frac{9}{11} - \frac{3}{22}\)[/tex]
Find the least common denominator (LCD) of 11 and 22, which is 22:
[tex]\[ \frac{9}{11} = \frac{9 \times 2}{11 \times 2} = \frac{18}{22} \][/tex]
[tex]\[ \frac{3}{22} = \frac{3}{22} \][/tex]
Now, subtract the fractions:
[tex]\[ \frac{18}{22} - \frac{3}{22} = \frac{18-3}{22} = \frac{15}{22} \][/tex]
### Summary of Results
a) [tex]\(\frac{3}{4}\)[/tex]
c) [tex]\(\frac{9}{8} = 1 \frac{1}{8}\)[/tex]
e) [tex]\(\frac{29}{70}\)[/tex]
f) [tex]\(\frac{171}{110}\)[/tex]
g) [tex]\(\frac{2}{5}\)[/tex]
h) [tex]\(\frac{107}{11}\)[/tex]
i) [tex]\(\frac{7}{12}\)[/tex]
j) [tex]\(\frac{15}{22}\)[/tex]
### a) [tex]\(\frac{1}{4} + \frac{2}{4}\)[/tex]
The denominators are the same, so we can add the numerators:
[tex]\[ \frac{1}{4} + \frac{2}{4} = \frac{1+2}{4} = \frac{3}{4} \][/tex]
### c) [tex]\(\frac{1}{4} + \frac{7}{8}\)[/tex]
To add these fractions, we need a common denominator. The least common denominator (LCD) of 4 and 8 is 8:
[tex]\[ \frac{1}{4} = \frac{2}{8} \][/tex]
Now, add the fractions:
[tex]\[ \frac{2}{8} + \frac{7}{8} = \frac{2+7}{8} = \frac{9}{8} = 1 \frac{1}{8} \][/tex]
### e) [tex]\(\frac{2}{10} + \frac{3}{14}\)[/tex]
Find the least common denominator (LCD) of 10 and 14, which is 70:
[tex]\[ \frac{2}{10} = \frac{2 \times 7}{10 \times 7} = \frac{14}{70} \][/tex]
[tex]\[ \frac{3}{14} = \frac{3 \times 5}{14 \times 5} = \frac{15}{70} \][/tex]
Now, add the fractions:
[tex]\[ \frac{14}{70} + \frac{15}{70} = \frac{14+15}{70} = \frac{29}{70} \][/tex]
### f) [tex]\(\frac{21}{22} + \frac{3}{5}\)[/tex]
Find the least common denominator (LCD) of 22 and 5, which is 110:
[tex]\[ \frac{21}{22} = \frac{21 \times 5}{22 \times 5} = \frac{105}{110} \][/tex]
[tex]\[ \frac{3}{5} = \frac{3 \times 22}{5 \times 22} = \frac{66}{110} \][/tex]
Now, add the fractions:
[tex]\[ \frac{105}{110} + \frac{66}{110} = \frac{105+66}{110} = \frac{171}{110} \][/tex]
### g) [tex]\(\frac{4}{5} - \frac{2}{5}\)[/tex]
The denominators are the same, so we can subtract the numerators:
[tex]\[ \frac{4}{5} - \frac{2}{5} = \frac{4-2}{5} = \frac{2}{5} \][/tex]
### h) [tex]\(10 - \frac{3}{11}\)[/tex]
Convert 10 to a fraction with a denominator of 11:
[tex]\[ 10 = \frac{110}{11} \][/tex]
Now, subtract the fractions:
[tex]\[ \frac{110}{11} - \frac{3}{11} = \frac{110-3}{11} = \frac{107}{11} \][/tex]
### i) [tex]\(\frac{3}{4} - \frac{1}{6}\)[/tex]
Find the least common denominator (LCD) of 4 and 6, which is 12:
[tex]\[ \frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12} \][/tex]
[tex]\[ \frac{1}{6} = \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \][/tex]
Now, subtract the fractions:
[tex]\[ \frac{9}{12} - \frac{2}{12} = \frac{9-2}{12} = \frac{7}{12} \][/tex]
### j) [tex]\(\frac{9}{11} - \frac{3}{22}\)[/tex]
Find the least common denominator (LCD) of 11 and 22, which is 22:
[tex]\[ \frac{9}{11} = \frac{9 \times 2}{11 \times 2} = \frac{18}{22} \][/tex]
[tex]\[ \frac{3}{22} = \frac{3}{22} \][/tex]
Now, subtract the fractions:
[tex]\[ \frac{18}{22} - \frac{3}{22} = \frac{18-3}{22} = \frac{15}{22} \][/tex]
### Summary of Results
a) [tex]\(\frac{3}{4}\)[/tex]
c) [tex]\(\frac{9}{8} = 1 \frac{1}{8}\)[/tex]
e) [tex]\(\frac{29}{70}\)[/tex]
f) [tex]\(\frac{171}{110}\)[/tex]
g) [tex]\(\frac{2}{5}\)[/tex]
h) [tex]\(\frac{107}{11}\)[/tex]
i) [tex]\(\frac{7}{12}\)[/tex]
j) [tex]\(\frac{15}{22}\)[/tex]
