- To add the fractions \(\frac{1}{a}\) and \(\frac{1}{b}\), identify the denominators \(a\) and \(b\).
- Find the least common denominator by calculating the product of the denominators: \(a \times b = ab\).
- Rewrite each fraction with the common denominator: \(\frac{b}{ab} + \frac{a}{ab}\).
- The common denominator is \(\boxed{ab}\).
Explanation
- Analysis of the Problem
The objective is to find the common denominator when adding the fractions \(\frac{1}{a}\) and \(\frac{1}{b}\) found in the expression \(\frac{\frac{1}{a} - \frac{1}{b}}{\frac{1}{a} + \frac{1}{b}}\). To add two fractions with different denominators, we need to find a common denominator, which is typically the least common multiple of the two denominators. In this case, the denominators are \(a\) and \(b\).
- Calculating the Common Denominator
To add the fractions \(\frac{1}{a}\) and \(\frac{1}{b}\), we multiply the first fraction by \(\frac{b}{b}\) and the second fraction by \(\frac{a}{a}\) to ensure they share the same denominator:
\[\frac{1}{a} + \frac{1}{b} = \frac{1 \cdot b}{a \cdot b} + \frac{1 \cdot a}{b \cdot a} = \frac{b}{ab} + \frac{a}{ab}\]
By combining the numerators over the common denominator, we get:
\[\frac{b + a}{ab}\]
The denominator that allows these two fractions to be summed is the product of the individual denominators, which is \(ab\).
- Conclusion
Comparing the result \(ab\) with the given choices:
- \(a^2 b^2\)
- \(a - b\)
- \(a + b\)
- \(ab\)
The common denominator is \(ab\). Thus, the correct option is the fourth one. The final answer is \(\boxed{ab}\).
Examples
In many real-world scenarios, such as calculating the combined rate of two machines working together, we encounter sums of reciprocals. For example, if Machine A can complete a task in \(a\) hours and Machine B in \(b\) hours, their combined hourly rate is \(\frac{1}{a} + \frac{1}{b}\). To add these rates, we find a common denominator, \(ab\), resulting in a combined rate of \(\frac{a+b}{ab}\) tasks per hour. This simplification is fundamental in project management, manufacturing, and engineering to optimize workflows and predict completion times.