Find the sum. Write your answer in simplest form.
$2 \frac{5}{12} + 12 \frac{7}{8} =$
To solve the addition of the two mixed numbers \(2 \frac{5}{12}\) and \(12 \frac{7}{8}\), we first analyze the structure of each mixed number. A mixed number consists of an integer part and a proper fraction part, such that \(n \frac{a}{b} = n + \frac{a}{b}\). Therefore, the expression can be broken down as follows:
\[2 \frac{5}{12} + 12 \frac{7}{8} = \left(2 + \frac{5}{12}\right) + \left(12 + \frac{7}{8}\right)\]
This allows us to handle the integer and fractional parts separately, simplifying the addition process.
Next, we group the integer parts and the fractional parts together:
\[(2 + 12) + \left(\frac{5}{12} + \frac{7}{8}\right) = 14 + \left(\frac{5}{12} + \frac{7}{8}\right)\]
To add the two fractions \(\frac{5}{12}\) and \(\frac{7}{8}\), we need a common denominator. The least common multiple (LCM) of 12 and 8 is 24. We rewrite each fraction to have a denominator of 24:
\[\frac{5}{12} = \frac{5 \cdot 2}{12 \cdot 2} = \frac{10}{24}\]
\[\frac{7}{8} = \frac{7 \cdot 3}{8 \cdot 3} = \frac{21}{24}\]
Now, we add the two fractions:
\[\frac{10}{24} + \frac{21}{24} = \frac{10 + 21}{24} = \frac{31}{24}\]
Since \(\frac{31}{24}\) is an improper fraction, we convert it back into a mixed number:
\[\frac{31}{24} = 1 + \frac{7}{24} = 1 \frac{7}{24}\]
Finally, we add this result back to the sum of the integer parts:
\[14 + 1 \frac{7}{24} = 15 \frac{7}{24}\]
The sum in simplest form is \(15 \frac{7}{24}\).
Adding mixed numbers is a fundamental skill used in various practical situations, such as construction and cooking. For example, if you are building a shelf and need to combine two pieces of wood measuring \(2 \frac{5}{12}\) inches and \(12 \frac{7}{8}\) inches, you would add these lengths to determine the total length required. Mastering this process ensures accuracy when measuring materials for home projects or scaling recipes for a large group.