- The goal is to add two fractions \(\frac{1}{a}\) and \(\frac{1}{b}\) by finding their common denominator.
- By multiplying the numerator and denominator of \(\frac{1}{a}\) by \(b\) and \(\frac{1}{b}\) by \(a\), both fractions share the common denominator \(ab\).
- Adding the results gives \(\frac{b}{ab} + \frac{a}{ab} = \frac{a+b}{ab}\).
- The common denominator identified is \(\boxed{ab}\).
Explanation
- Analyzing the fractions
To simplify the sum of fractions \(\frac{1}{a} + \frac{1}{b}\), we need to find a common denominator. The denominators of these two fractions are \(a\) and \(b\). The smallest common denominator that both \(a\) and \(b\) divide into is their product, \(ab\).
- Rewriting with a common denominator
To express the sum \(\frac{1}{a} + \frac{1}{b}\) as a single rational expression, we rewrite each fraction so that they share the denominator \(ab\).
For the first fraction \(\frac{1}{a}\), we multiply the numerator and the denominator by \(b\):
\[\frac{1}{a} = \frac{1 \times b}{a \times b} = \frac{b}{ab}\]
For the second fraction \(\frac{1}{b}\), we multiply the numerator and the denominator by \(a\):
\[\frac{1}{b} = \frac{1 \times a}{b \times a} = \frac{a}{ab}\]
- Adding the fractions and identifying the result
Now that both fractions have the common denominator \(ab\), we can easily add them:
\[\frac{1}{a} + \frac{1}{b} = \frac{b}{ab} + \frac{a}{ab} = \frac{b + a}{ab}\]
The resulting denominator for the combined expression is \(ab\). Therefore, the common denominator used is \(ab\).
Examples
In daily life, finding a common denominator is like ensuring you are using the same units when adding quantities. For example, if you have one-third of a cup of flour and one-fourth of a cup of sugar, you need to find a common unit (like twelfths) to combine them into a single measurement: \(\frac{1}{3} + \frac{1}{4} = \frac{4}{12} + \frac{3}{12} = \frac{7}{12}\). This is exactly how we simplify algebraic expressions by finding a common denominator to combine terms!