Subtracting Fractions

1. We look for a common denominator.

2. For each fraction, we create an equivalent fraction.

3. We subtract only the numerators.

Subtract the following: ||\frac{1}{2}-\frac{1}{4}||

1. We look for a common denominator.
   
   Multiples of |2=\{2,\underbrace{\color{red}{4}}_{\color{blue}{2^{nd} \ \text{multiple}}},6,8,...\}|
   Multiples of |4=\{\underbrace{\color{red}{4}}_{\color{green}{1^{st} \ \text{multiple}}}, 8, 12, 16,...\}|
   Thus, the common denominator is |\color{red}{4}.|

2. For each fraction, we create an equivalent fraction. ||\frac{1}{2}^{\color{blue}{\times 2}}_{\color{blue}{\times 2}}=\frac{2}{\color{red}{4}} \\\\ \frac{1}{4}^{\color{green}{\times 1}}_{\color{green}{\times 1}}=\frac{1}{\color{red}{4}}||

3. We subtract only the numerators.||\begin{align} \frac{1}{2}-\frac{1}{4} &= \frac{2}{\color{red}{4}}-\frac{1}{\color{red}{4}}\\\\ &=\frac{2-1}{\color{red}{4}}\\\\ &=\frac{1}{\color{red}{4}}\end{align}||

Subtract the following: ||\frac{5}{\color{blue}{6}}-\frac{4}{\color{green}{5}}||

1. We look for a common denominator.
   
   By multiplying the denominators, we find the common denominator: ||\color{blue}{6} \times \color{green}{5} = \color{red}{30}||

2. For each fraction, we create an equivalent fraction. ||\frac{5}{\color{blue}{6}}^{\color{green}{\times 5}}_{\color{green}{\times 5}} =\frac{25}{\color{red}{30}} \\\\ \frac{4}{\color{green}{5}}^{\color{blue}{\times 6}}_{\color{blue}{\times 6}} = \frac{24}{\color{red}{30}}||

3. We subtract only the numerators. ||\begin{align} \frac{5}{\color{blue}{6}} - \frac{4}{\color{green}{5}} &= \frac{25}{\color{red}{30}} - \frac{24}{\color{red}{30}} \\\\ &= \frac{25-24}{\color{red}{30}} \\\\ &= \frac{1}{\color{red}{30}} \end{align}||

For example, take the fraction |\dfrac{3}{4}.| The 4^th line from |0| represents a whole, or the fraction |\dfrac{4}{4}.| 

1. We look for the common denominator of the fractions.

2. For each fraction, create an equivalent fraction.

3. We segment each whole on the number line into as many parts as the value of the common denominator.

4. We place the 1^st fraction on the number line according to its numerator.

5. We subtract the 2^nd fraction from the 1^st.

Find the difference between: ||\frac{3}{8}-\frac{1}{4}||

1. We find the common denominator of the two fractions.
   
   Multiples of |8=\{\underbrace{\color{red}{8}}_{\color{blue}{1^{st} \ \text{multiple}}}, 16, 24, 32, ... \}|
   
   Multiples of |4=\{4, \underbrace{\color{red}{8}}_{\color{green}{2^{nd} \ \text{multiple}}​}, 12, 16, ...\}|
   
   Thus, the common denominator is |\color{red}{8}.|

2. For each fraction, we create an equivalent fraction. ||\frac{3}{8}^{\color{blue}{\times 1}}_{\color{blue}{\times 1}} =\frac{3}{\color{red}{8}} \ \ \text{and} \ \ \frac{1}{4}^{\color{green}{\times 2}}_{\color{green}{\times 2}} =\frac{2}{\color{red}{8}}||

3. We segment each whole on the number line into as many parts as the value of the common denominator.

4. Place the 1^st fraction according to its numerator.

5. Subtract the 2^nd fraction from the 1^st.

Therefore, |\dfrac{3}{8} - \dfrac{1}{4} = \dfrac{3}{8} - \dfrac{2}{8} = \dfrac{1}{8}.|

Find the difference between the following: ||5 \dfrac{1}{3} - 2 \dfrac{2}{5}||

0. We convert mixed numbers into fractions. ||\begin{align} &5 \dfrac{1}{3} && \text{and} && \quad \ \ 2 \dfrac{2}{5} \\ =\ &\dfrac{5 \times 3 + 1}{3} && \text{and} && =\dfrac{2 \times 5 + 2}{5} \\ = \ &\dfrac{16}{\color{blue}{3}} && \text{and} && =\dfrac{12}{\color{green}{5}} \end{align}||

1. We look for a common denominator.
   
   Multiply the denominators to find the common denominator |\color{blue}{3} \times \color{green}{5} = \color{red}{15}.|

2. For each fraction, we create an equivalent fraction. ||\dfrac{16}{\color{blue}{3}}^{\color{green}{\times 5}}_{\color{green}{\times 5}} =\dfrac{80}{\color{red}{15}}\ \ \text{and} \ \ \dfrac{12}{\color{green}{5}}^{\color{blue}{\times 3}}_{\color{blue}{\times 3}} = \dfrac{36}{\color{red}{15}}||

3. We subtract only the numerators. ||\begin{align} \dfrac{16}{\color{blue}{3}} - \dfrac{12}{\color{green}{5}} &= \dfrac{80}{\color{red}{15}} - \dfrac{36}{\color{red}{15}} \\[3pt] &= \dfrac{80-36}{\color{red}{15}} \\[3pt] &= \dfrac{44}{\color{red}{15}} \\[3pt] &=2\dfrac{14}{15}\end{align}||

It is always a good idea to write an answer as a fraction in its simplest form at the end of a calculation when possible.