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When you're working with fractions, especially when adding them, finding a common denominator is super important. A common denominator is a number that is a multiple of all the denominators involved.
For two fractions with denominators 3 and 2, like in our exercise, you want to find the least common multiple (LCM) of these numbers. The LCM of 3 and 2 is 6.
Why? Because 6 is the smallest number that both 3 and 2 can divide without leaving a remainder. This common denominator allows you to rewrite each fraction so they can easily be added together.

• Multiply the top and bottom of \(\frac{1}{3}\) by 2 to get \(\frac{2}{6}\).
• Multiply the top and bottom of \(\frac{1}{2}\) by 3 to get \(\frac{3}{6}\).

Finding a common denominator is crucial because it transforms different-sized pieces into matching-sized pieces. This way, you can combine the fractions smoothly.

Adding fractions becomes straightforward once you have a common denominator. After converting \(\frac{1}{3}\) and \(\frac{1}{2}\) to \(\frac{2}{6}\) and \(\frac{3}{6}\) respectively, you're ready to add them!

The denominators are now the same, so you can focus on adding the numerators alone.

• Think of it like counting pieces: \(2\) parts out of \(6\), plus \(3\) parts out of \(6\) means \(5\) parts out of \(6\).

Thus, \(\frac{2}{6} + \frac{3}{6} = \frac{5}{6}\).
Adding fractions with a common denominator is like simple addition where you only add the top parts (numerators) while keeping the bottom (the common-sized denominator) the same.

After adding fractions, in our exercise, we multiplied the sum, \(\frac{5}{6}\), by 6. Multiplying a fraction by a whole number is like multiplying the whole number by the fraction's numerator while dividing by the denominator.

• So, \(6 \times \frac{5}{6}\) means \(\frac{6 \times 5}{6}\).
• Here, 6 in the numerator and denominator cancel each other out effectively. It's like taking a pizza cut into six slices, where multiplying by six is just taking back all slices. So, you're left with the slices numbered.\(5\).

Simplifying in this context involves recognizing when a number is both in a numerator and a denominator, allowing stepping back to simpler arithmetic directly. Remember that multiplication with fractions might involve simplification, and noticing such opportunities makes solving much easier!