Adding and Subtracting Mixed Numbers with Unlike Denominators

A mixed number has two parts: a whole number and a fraction (like $3\frac{1}{4}$). When you need to add or subtract mixed numbers that have different (unlike) denominators, you cannot just combine them right away. You must first make the fraction parts share the same denominator.

Adding Mixed Numbers

To add mixed numbers with unlike denominators, follow these steps:

1. Find a Common Denominator: Change the fraction parts so they have the same denominator.
2. Add the Fractions: Add the numerators and keep the denominator the same.
3. Add the Whole Numbers: Add the whole number parts together.
4. Simplify (if needed): If your fraction part becomes an improper fraction (numerator is larger than the denominator), convert it and add it to the whole number.

Example: $3\frac{1}{4} + 2\frac{2}{3}$

• Step 1: The denominators are $4$ and $3$. The least common multiple is $12$. Change the fractions: $3\frac{1}{4} = 3\frac{3}{12}$ $2\frac{2}{3} = 2\frac{8}{12}$
• Step 2 & 3: Add the whole numbers and the fractions: $3 + 2 = 5$ $\frac{3}{12} + \frac{8}{12} = \frac{11}{12}$
• Final Answer: $5\frac{11}{12}$

Example with Regrouping: $4\frac{2}{5} + 1\frac{5}{6}$

• Step 1: The common denominator for $5$ and $6$ is $30$. $4\frac{2}{5} = 4\frac{12}{30}$ $1\frac{5}{6} = 1\frac{25}{30}$
• Step 2 & 3: Add them together: $4 + 1 = 5$ $\frac{12}{30} + \frac{25}{30} = \frac{37}{30}$ This gives us $5\frac{37}{30}$.
• Step 4: Because $\frac{37}{30}$ is improper, convert it to a mixed number ($1\frac{7}{30}$) and combine it with the $5$: $5 + 1\frac{7}{30} = 6\frac{7}{30}$

Subtracting Mixed Numbers (with Borrowing)

Subtracting follows a similar process, but sometimes the fraction you are trying to subtract is larger than the fraction you have. When this happens, you have to borrow from the whole number.

Example: $5\frac{1}{3} - 2\frac{3}{4}$

• Step 1: The common denominator for $3$ and $4$ is $12$. $5\frac{1}{3} = 5\frac{4}{12}$ $2\frac{3}{4} = 2\frac{9}{12}$
• Step 2: Try to subtract. You cannot subtract $\frac{9}{12}$ from $\frac{4}{12}$. You need to borrow $1$ whole from the $5$. Take $1$ away from $5$ (making it $4$). Since $1 = \frac{12}{12}$, add that to your fraction: $\frac{4}{12} + \frac{12}{12} = \frac{16}{12}$. Now, $5\frac{4}{12}$ becomes $4\frac{16}{12}$.
• Step 3: Now subtract the new mixed numbers: $4\frac{16}{12} - 2\frac{9}{12}$ Subtract whole numbers: $4 - 2 = 2$ Subtract fractions: $\frac{16}{12} - \frac{9}{12} = \frac{7}{12}$
• Final Answer: $2\frac{7}{12}$