Adding Fractions with Unlike Denominators for 5th Grade

Adding fractions with unlike denominators requires finding a common denominator so we can add “like” parts—tenths with tenths, sixths with sixths. In Grade 5, students add fractions with different denominators by converting to equivalent fractions with a common denominator (usually the LCM of the denominators), then adding the numerators and keeping the denominator. This skill is used when combining measurements, adding parts of different-sized wholes (when we treat them as equivalent), and in real-world problems like “Maria ran \(\frac{2}{5}\) mile and walked \(\frac{1}{4}\) mile.”

We cannot add \(\frac{1}{2} + \frac{1}{3}\) directly because halves and thirds are different-sized parts. We need a common denominator. The LCM of 2 and 3 is 6. So \(\frac{1}{2} = \frac{3}{6}\) and \(\frac{1}{3} = \frac{2}{6}\). Now \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\). The sum is \(\frac{5}{6}\).

DETAILED EXPLANATION

Steps to add fractions with unlike denominators:

1. Find the least common multiple (LCM) of the denominators—this will be the common denominator.

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2. Convert each fraction to an equivalent fraction with that denominator.

3. Add the numerators; keep the denominator the same.

4. Simplify the result if possible (reduce to lowest terms, convert improper to mixed).

Example: \(\frac{1}{2} + \frac{1}{3}\). LCM(2,3)=6. \(\frac{1}{2}=\frac{3}{6}\), \(\frac{1}{3}=\frac{2}{6}\). \(\frac{3}{6}+\frac{2}{6}=\frac{5}{6}\).

If the sum is improper (numerator ≥ denominator), convert to a mixed number: \(\frac{13}{12} = 1 \frac{1}{12}\).

WORKED EXAMPLES WITH STEP BY STEP SOLUTIONS

Example 1

Add \(\frac{1}{2} + \frac{1}{3}\)

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Solutions:

Step 1: Denominators are 2 and 3. LCM of 2 and 3 is 6.

Step 2: Convert: \(\frac{1}{2} = \frac{1 \times 3}{2 \times 3} = \frac{3}{6}\); \(\frac{1}{3} = \frac{1 \times 2}{3 \times 2} = \frac{2}{6}\).

Step 3: Add numerators: \(\frac{3}{6} + \frac{2}{6} = \frac{5}{6}\).

Step 4: \(\frac{5}{6}\) is already in lowest terms.

Answer: \(\frac{5}{6}\)

Example 2

Maria ran \(\frac{2}{5}\) mile and walked \(\frac{1}{4}\) mile. How far did she go?

Solutions:

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Step 1: Add \(\frac{2}{5} + \frac{1}{4}\). LCM of 5 and 4 is 20.

Step 2: Convert: \(\frac{2}{5} = \frac{8}{20}\); \(\frac{1}{4} = \frac{5}{20}\).

Step 3: Add: \(\frac{8}{20} + \frac{5}{20} = \frac{13}{20}\).

Step 4: Maria went \(\frac{13}{20}\) mile in total.

Answer: \(\frac{13}{20}\) mile

Example 3

Add \(\frac{3}{4} + \frac{2}{6}\)

Solutions:

Step 1: LCM of 4 and 6 is 12. Convert: \(\frac{3}{4} = \frac{9}{12}\); \(\frac{2}{6} = \frac{4}{12}\).

Step 2: Add: \(\frac{9}{12} + \frac{4}{12} = \frac{13}{12}\).

Step 3: \(\frac{13}{12}\) is improper. Convert: \(13 \div 12 = 1\) remainder 1, so \(\frac{13}{12} = 1 \frac{1}{12}\).

Answer: \(1 \frac{1}{12}\)

Example 4

Add \(\frac{2}{3} + \frac{3}{5}\)

Solutions:

Step 1: LCM of 3 and 5 is 15. \(\frac{2}{3} = \frac{10}{15}\); \(\frac{3}{5} = \frac{9}{15}\).

Step 2: \(\frac{10}{15} + \frac{9}{15} = \frac{19}{15} = 1 \frac{4}{15}\).

Answer: \(1 \frac{4}{15}\)