To add fractions with like denominators, we follow two main steps for each pair of fractions:
1. Adding the fractions: Since the denominators are the same, we simply add the numerators and keep the denominator the same.
2. Simplifying the resulting fraction: We reduce the fraction to its lowest terms by dividing both the numerator and the denominator by their greatest common divisor (GCD).
Let's go through each problem one by one:
### 1. [tex]\(\frac{7}{132} + \frac{5}{132}\)[/tex]
- Step 1: Add the numerators: [tex]\(7 + 5 = 12\)[/tex].
- Unsimplified Sum: [tex]\(\frac{12}{132}\)[/tex]
- Step 2: Simplify [tex]\(\frac{12}{132}\)[/tex].
- GCD of 12 and 132 is 12.
- Divide numerator and denominator by the GCD: [tex]\(\frac{12 \div 12}{132 \div 12} = \frac{1}{11}\)[/tex].
- Simplified Sum: [tex]\(\frac{1}{11}\)[/tex]
### 2. [tex]\(\frac{15}{28} + \frac{5}{28}\)[/tex]
- Step 1: Add the numerators: [tex]\(15 + 5 = 20\)[/tex].
- Unsimplified Sum: [tex]\(\frac{20}{28}\)[/tex]
- Step 2: Simplify [tex]\(\frac{20}{28}\)[/tex].
- GCD of 20 and 28 is 4.
- Divide numerator and denominator by GCD: [tex]\(\frac{20 \div 4}{28 \div 4} = \frac{5}{7}\)[/tex].
- Simplified Sum: [tex]\(\frac{5}{7}\)[/tex]
### 3. [tex]\(\frac{3}{16} + \frac{5}{16}\)[/tex]
- Step 1: Add the numerators: [tex]\(3 + 5 = 8\)[/tex].
- Unsimplified Sum: [tex]\(\frac{8}{16}\)[/tex]
- Step 2: Simplify [tex]\(\frac{8}{16}\)[/tex].
- GCD of 8 and 16 is 8.
- Divide numerator and denominator by GCD: [tex]\(\frac{8 \div 8}{16 \div 8} = \frac{1}{2}\)[/tex].
- Simplified Sum: [tex]\(\frac{1}{2}\)[/tex]
### 4. [tex]\(\frac{5}{21} + \frac{2}{21}\)[/tex]
- Step 1: Add the numerators: [tex]\(5 + 2 = 7\)[/tex].
- Unsimplified Sum: [tex]\(\frac{7}{21}\)[/tex]
- Step 2: Simplify [tex]\(\frac{7}{21}\)[/tex].
- GCD of 7 and 21 is 7.
- Divide numerator and denominator by GCD: [tex]\(\frac{7 \div 7}{21 \div 7} = \frac{1}{3}\)[/tex].
- Simplified Sum: [tex]\(\frac{1}{3}\)[/tex]
Now, let's fill in the table with the results:
[tex]\[
\begin{tabular}{|c|l|l|}
\hline Addition Problem & Unsimplified Sum & Simplified Sum \\
\hline \(\frac{7}{132} + \frac{5}{132}\) & \(\frac{12}{132}\) & \(\frac{1}{11}\) \\
\hline \(\frac{15}{28} + \frac{5}{28}\) & \(\frac{20}{28}\) & \(\frac{5}{7}\) \\
\hline \(\frac{3}{16} + \frac{5}{16}\) & \(\frac{8}{16}\) & \(\frac{1}{2}\) \\
\hline \(\frac{5}{21} + \frac{2}{21}\) & \(\frac{7}{21}\) & \(\frac{1}{3}\) \\
\hline
\end{tabular}
\][/tex]
So, we have successfully added and simplified all the given fractions with like denominators.
