Adding Fractions with unlike denominators exercise - Free Printable

To solve the given fraction addition problems, we will follow the steps outlined in the instructions:

Steps to Add Fractions with Unlike Denominators:

1. Write out the multiples of each denominator.
2. Find the Lowest/Least Common Multiple (LCM).
3. Write the equivalent fractions with the LCM as the denominator for both fractions.
4. Solve the equation.

Let's solve each problem step by step.

---

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

#### Step 1: Write out the multiples of each denominator.
- Multiples of 2: \( 2, 4, 6, 8, \ldots \)
- Multiples of 3: \( 3, 6, 9, 12, \ldots \)

#### Step 2: Find the LCM.
The smallest common multiple is \( 6 \).

#### Step 3: Write equivalent fractions with the LCM as the denominator.
- For \( \frac{1}{2} \): Multiply numerator and denominator by 3 to get \( \frac{1 \times 3}{2 \times 3} = \frac{3}{6} \).
- For \( \frac{2}{3} \): Multiply numerator and denominator by 2 to get \( \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \).

#### Step 4: Solve the equation.
\[ \frac{1}{2} + \frac{2}{3} = \frac{3}{6} + \frac{4}{6} = \frac{3 + 4}{6} = \frac{7}{6} \]

Answer: \( \frac{7}{6} \)

---

2. \( \frac{3}{4} + \frac{1}{8} \)

#### Step 1: Write out the multiples of each denominator.
- Multiples of 4: \( 4, 8, 12, 16, \ldots \)
- Multiples of 8: \( 8, 16, 24, \ldots \)

#### Step 2: Find the LCM.
The smallest common multiple is \( 8 \).

#### Step 3: Write equivalent fractions with the LCM as the denominator.
- For \( \frac{3}{4} \): Multiply numerator and denominator by 2 to get \( \frac{3 \times 2}{4 \times 2} = \frac{6}{8} \).
- For \( \frac{1}{8} \): Already has the denominator 8, so it remains \( \frac{1}{8} \).

#### Step 4: Solve the equation.
\[ \frac{3}{4} + \frac{1}{8} = \frac{6}{8} + \frac{1}{8} = \frac{6 + 1}{8} = \frac{7}{8} \]

Answer: \( \frac{7}{8} \)

---

3. \( \frac{4}{9} + \frac{1}{3} \)

#### Step 1: Write out the multiples of each denominator.
- Multiples of 9: \( 9, 18, 27, \ldots \)
- Multiples of 3: \( 3, 6, 9, 12, \ldots \)

#### Step 2: Find the LCM.
The smallest common multiple is \( 9 \).

#### Step 3: Write equivalent fractions with the LCM as the denominator.
- For \( \frac{4}{9} \): Already has the denominator 9, so it remains \( \frac{4}{9} \).
- For \( \frac{1}{3} \): Multiply numerator and denominator by 3 to get \( \frac{1 \times 3}{3 \times 3} = \frac{3}{9} \).

#### Step 4: Solve the equation.
\[ \frac{4}{9} + \frac{1}{3} = \frac{4}{9} + \frac{3}{9} = \frac{4 + 3}{9} = \frac{7}{9} \]

Answer: \( \frac{7}{9} \)

---

4. \( \frac{3}{5} + \frac{1}{2} \)

#### Step 1: Write out the multiples of each denominator.
- Multiples of 5: \( 5, 10, 15, \ldots \)
- Multiples of 2: \( 2, 4, 6, 8, 10, \ldots \)

#### Step 2: Find the LCM.
The smallest common multiple is \( 10 \).

#### Step 3: Write equivalent fractions with the LCM as the denominator.
- For \( \frac{3}{5} \): Multiply numerator and denominator by 2 to get \( \frac{3 \times 2}{5 \times 2} = \frac{6}{10} \).
- For \( \frac{1}{2} \): Multiply numerator and denominator by 5 to get \( \frac{1 \times 5}{2 \times 5} = \frac{5}{10} \).

#### Step 4: Solve the equation.
\[ \frac{3}{5} + \frac{1}{2} = \frac{6}{10} + \frac{5}{10} = \frac{6 + 5}{10} = \frac{11}{10} \]

Answer: \( \frac{11}{10} \)

---

5. \( \frac{1}{6} + \frac{2}{3} \)

#### Step 1: Write out the multiples of each denominator.
- Multiples of 6: \( 6, 12, 18, \ldots \)
- Multiples of 3: \( 3, 6, 9, 12, \ldots \)

#### Step 2: Find the LCM.
The smallest common multiple is \( 6 \).

#### Step 3: Write equivalent fractions with the LCM as the denominator.
- For \( \frac{1}{6} \): Already has the denominator 6, so it remains \( \frac{1}{6} \).
- For \( \frac{2}{3} \): Multiply numerator and denominator by 2 to get \( \frac{2 \times 2}{3 \times 2} = \frac{4}{6} \).

#### Step 4: Solve the equation.
\[ \frac{1}{6} + \frac{2}{3} = \frac{1}{6} + \frac{4}{6} = \frac{1 + 4}{6} = \frac{5}{6} \]

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

---

6. \( \frac{1}{4} + \frac{1}{6} \)

#### Step 1: Write out the multiples of each denominator.
- Multiples of 4: \( 4, 8, 12, 16, \ldots \)
- Multiples of 6: \( 6, 12, 18, \ldots \)

#### Step 2: Find the LCM.
The smallest common multiple is \( 12 \).

#### Step 3: Write equivalent fractions with the LCM as the denominator.
- For \( \frac{1}{4} \): Multiply numerator and denominator by 3 to get \( \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \).
- For \( \frac{1}{6} \): Multiply numerator and denominator by 2 to get \( \frac{1 \times 2}{6 \times 2} = \frac{2}{12} \).

#### Step 4: Solve the equation.
\[ \frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{3 + 2}{12} = \frac{5}{12} \]

Answer: \( \frac{5}{12} \)

---

7. \( \frac{1}{4} + \frac{1}{2} \)

#### Step 1: Write out the multiples of each denominator.
- Multiples of 4: \( 4, 8, 12, \ldots \)
- Multiples of 2: \( 2, 4, 6, 8, \ldots \)

#### Step 2: Find the LCM.
The smallest common multiple is \( 4 \).

#### Step 3: Write equivalent fractions with the LCM as the denominator.
- For \( \frac{1}{4} \): Already has the denominator 4, so it remains \( \frac{1}{4} \).
- For \( \frac{1}{2} \): Multiply numerator and denominator by 2 to get \( \frac{1 \times 2}{2 \times 2} = \frac{2}{4} \).

#### Step 4: Solve the equation.
\[ \frac{1}{4} + \frac{1}{2} = \frac{1}{4} + \frac{2}{4} = \frac{1 + 2}{4} = \frac{3}{4} \]

Answer: \( \frac{3}{4} \)

---

8. \( \frac{3}{12} + \frac{1}{4} \)

#### Step 1: Write out the multiples of each denominator.
- Multiples of 12: \( 12, 24, 36, \ldots \)
- Multiples of 4: \( 4, 8, 12, 16, \ldots \)

#### Step 2: Find the LCM.
The smallest common multiple is \( 12 \).

#### Step 3: Write equivalent fractions with the LCM as the denominator.
- For \( \frac{3}{12} \): Already has the denominator 12, so it remains \( \frac{3}{12} \).
- For \( \frac{1}{4} \): Multiply numerator and denominator by 3 to get \( \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \).

#### Step 4: Solve the equation.
\[ \frac{3}{12} + \frac{1}{4} = \frac{3}{12} + \frac{3}{12} = \frac{3 + 3}{12} = \frac{6}{12} \]

Simplify \( \frac{6}{12} \) to \( \frac{1}{2} \).

Answer: \( \frac{1}{2} \)

---

Final Answers:

\[
\boxed{
\begin{aligned}
1. & \ \frac{7}{6} \\
2. & \ \frac{7}{8} \\
3. & \ \frac{7}{9} \\
4. & \ \frac{11}{10} \\
5. & \ \frac{5}{6} \\
6. & \ \frac{5}{12} \\
7. & \ \frac{3}{4} \\
8. & \ \frac{1}{2}
\end{aligned}
}
\]