Adding and Subtracting Fractions: Additional Practice (Set 2)

The least common denominator of $2$, $3$, and $6$ is $6$. Rewrite each fraction over $6$.

$\frac{1}{2} = \frac{3}{6}, \qquad \frac{1}{3} = \frac{2}{6}, \qquad \frac{1}{6} = \frac{1}{6}$

Now add all three over the common denominator.

$\frac{3}{6} + \frac{2}{6} + \frac{1}{6} = \frac{6}{6} = 1$

Adding all the tops and bottoms separately gives $\frac{3}{11}$, which is wrong.

The least common denominator of $10$ and $6$ is $30$. Rewrite each fraction over $30$.

$\frac{7}{10} = \frac{21}{30}, \qquad \frac{1}{6} = \frac{5}{30}$

Now subtract and simplify.

$\frac{21}{30} - \frac{5}{30} = \frac{16}{30} = \frac{8}{15}$

The greatest common factor of $16$ and $30$ is $2$, so $\frac{16}{30} = \frac{8}{15}$.

The least common denominator of $3$ and $4$ is $12$. Rewrite each fraction over $12$.

$\frac{2}{3} = \frac{8}{12}, \qquad \frac{3}{4} = \frac{9}{12}$

Now add over the common denominator.

$\frac{8}{12} + \frac{9}{12} = \frac{17}{12}$

This improper fraction is the final answer. Adding the tops and bottoms separately gives $\frac{5}{7}$, which is wrong.

The least common denominator of $10$, $5$, and $10$ is $10$. Rewrite $\frac{2}{5}$ over $10$.

$\frac{2}{5} = \frac{4}{10}$

Then subtract left to right and simplify.

$\frac{9}{10} - \frac{4}{10} - \frac{1}{10} = \frac{4}{10} = \frac{2}{5}$

The greatest common factor of $4$ and $10$ is $2$, so $\frac{4}{10} = \frac{2}{5}$.

Write everything over the common denominator $12$.

$1 = \frac{12}{12}, \qquad \frac{1}{3} = \frac{4}{12}, \qquad \frac{1}{4} = \frac{3}{12}$

Then subtract left to right.

$\frac{12}{12} - \frac{4}{12} - \frac{3}{12} = \frac{5}{12}$

The least common denominator of $9$ and $6$ is $18$. Rewrite each fraction over $18$.

$\frac{7}{9} = \frac{14}{18}, \qquad \frac{1}{6} = \frac{3}{18}$

Now subtract over the common denominator.

$\frac{14}{18} - \frac{3}{18} = \frac{11}{18}$

Since $11$ and $18$ share no factor above $1$, $\frac{11}{18}$ is in lowest terms.

The least common denominator of $10$ and $15$ is $30$. Rewrite each fraction over $30$.

$\frac{3}{10} = \frac{9}{30}, \qquad \frac{2}{15} = \frac{4}{30}$

Now add over the common denominator.

$\frac{9}{30} + \frac{4}{30} = \frac{13}{30}$

Since $13$ and $30$ share no factor above $1$, $\frac{13}{30}$ is in lowest terms.

Write the whole number $3$ as a fraction over $5$: $3 = \frac{15}{5}$.

$3 - \frac{4}{5} = \frac{15}{5} - \frac{4}{5} = \frac{11}{5}$

This improper fraction is the final answer.

The missing piece is $\frac{5}{6} - \frac{1}{3}$.

$\frac{5}{6} - \frac{1}{3} = \frac{5}{6} - \frac{2}{6} = \frac{3}{6} = \frac{1}{2}$

Check: $\frac{1}{2} + \frac{1}{3} = \frac{3}{6} + \frac{2}{6} = \frac{5}{6}$, so the missing number is $\frac{1}{2}$.

The least common denominator of $12$ and $18$ is $36$. Rewrite each fraction over $36$.

$\frac{5}{12} = \frac{15}{36}, \qquad \frac{7}{18} = \frac{14}{36}$

Now add over the common denominator.

$\frac{15}{36} + \frac{14}{36} = \frac{29}{36}$

Since $29$ and $36$ share no factor above $1$, $\frac{29}{36}$ is in lowest terms.

The least common denominator of $6$ and $10$ is $30$. Rewrite each fraction over $30$.

$\frac{1}{6} = \frac{5}{30}, \qquad \frac{3}{10} = \frac{9}{30}$

Now add and simplify.

$\frac{5}{30} + \frac{9}{30} = \frac{14}{30} = \frac{7}{15}$

The greatest common factor of $14$ and $30$ is $2$, so $\frac{14}{30} = \frac{7}{15}$.

The least common denominator of $8$ and $6$ is $24$. Rewrite each fraction over $24$.

$\frac{7}{8} = \frac{21}{24}, \qquad \frac{5}{6} = \frac{20}{24}$

Now subtract over the common denominator.

$\frac{21}{24} - \frac{20}{24} = \frac{1}{24}$

Since $1$ and $24$ share no factor above $1$, $\frac{1}{24}$ is in lowest terms.