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When adding or subtracting fractions, it’s essential that the fractions have the same denominator. The denominator is the bottom part of the fraction, which tells us into how many parts the whole is divided. If the denominators are different, we need to find a common denominator. This is a shared multiple of the original denominators.

In our exercise, the fractions are \(\frac{\frac{\pi}}{6}\) and \(\frac{\frac{\pi}}{4}\). The denominators here are 6 and 4. Without a common denominator, it is impossible to directly add these fractions. Finding a common denominator ensures that we are working with the same sized parts of the whole.

To find a common denominator, we use the least common multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of both denominators. For example, the denominators 6 and 4 can be broken down into their prime factors:

• 6 = 2 * 3
• 4 = 2 * 2

The LCM is found by taking the highest powers of all prime factors present, so the LCM of 6 and 4 is 12

In our example, using 12 as the common denominator means we convert both fractions to have this denominator.

Once we have the common denominator, we rewrite each fraction with this denominator. This process is called converting fractions. For \(\frac{\frac{\pi}}{6}\), we multiply both numerator and denominator by 2 to get \(\frac{2\frac{\pi}}{12}\). For \(\frac{\frac{\pi}}{4}\), we multiply both by 3 to get \(\frac{3\frac{\pi}}{12}\).

Now that both fractions have the same denominator, we can add them simply by adding their numerators: \(\frac{2\frac{\pi}}{12} + \frac{3\frac{\pi}}{12} = \frac{(2\frac{\pi} + 3\frac{\pi})}{12} = \frac{5}{6}\). This result, \(\frac{5\frac{\pi}}{12}\), is the sum of the original fractions. By using common denominators and fraction addition methods, solving such problems becomes straightforward.