Lesson 2: Adding Mixed Numbers

Property

A mixed number $A \frac{b}{c}$ is visually represented by $A$ wholes, where each whole is equivalent to $\frac{c}{c}$, plus the fractional part $\frac{b}{c}$.

To convert a mixed number to an improper fraction:

1. Multiply the whole number by the denominator.
2. Add the numerator to the product found in Step 1.
3. Write the final sum over the original denominator.

Property

To add multiple fractions with like denominators, find pairs whose numerators sum to the denominator.
Group these pairs to make a whole ($1 = \frac{n}{n}$), then add the remaining fractions to form a mixed number.

Examples

Property

Decompose one addend to make the next whole number with the other addend. This is also known as the "make a whole" or "make one" strategy.

Examples

• $4\frac{7}{8} + 2\frac{3}{8} = (4\frac{7}{8} + \frac{1}{8}) + 2\frac{2}{8} = 5 + 2\frac{2}{8} = 7\frac{2}{8}$
• $3\frac{2}{3} + 3\frac{2}{3} = (3\frac{2}{3} + \frac{1}{3}) + 3\frac{1}{3} = 4 + 3\frac{1}{3} = 7\frac{1}{3}$
• $5\frac{5}{6} + 4\frac{4}{6} = (5\frac{5}{6} + \frac{1}{6}) + 4\frac{3}{6} = 6 + 4\frac{3}{6} = 10\frac{3}{6}$

Explanation

This strategy simplifies addition by focusing on creating whole numbers, which are easier to work with. To use this method, identify how much the first mixed number needs to become the next whole number. Then, decompose the second mixed number to provide that amount, and add the remaining part to the new whole.

Property

When adding mixed numbers, first add the whole numbers and then add the fractions.
If the sum of the fractions is an improper fraction (greater than or equal to 1), convert it to a mixed number and add it to the sum of the whole numbers.

If $\frac{b+e}{d} \geq 1$, regroup.

Examples

• $2\frac{3}{4} + 1\frac{3}{4} = (2+1) + (\frac{3}{4} + \frac{3}{4}) = 3 + \frac{6}{4} = 3 + 1\frac{2}{4} = 4\frac{2}{4}$
• $5\frac{2}{3} + 2\frac{2}{3} = (5+2) + (\frac{2}{3} + \frac{2}{3}) = 7 + \frac{4}{3} = 7 + 1\frac{1}{3} = 8\frac{1}{3}$

Explanation

This skill builds on adding like units. You first combine the whole numbers and then the fractions separately. When the sum of the fractions is an improper fraction, you must regroup. To do this, you convert the improper fraction into a mixed number and then add this new whole number to your original sum of whole numbers.