Find the sum. Write your answer in simplest form.
$8 \frac{5}{12}+7 \frac{5}{8}=$
To find the sum of \(8 \frac{5}{12}\) and \(7 \frac{5}{8}\), we can decompose each mixed number into its integer and fractional parts. A mixed number \(a \frac{b}{c}\) is equal to \(a + \frac{b}{c}\). Thus, we can write the expression as \((8 + \frac{5}{12}) + (7 + \frac{5}{8})\). This allows us to handle the integer and fractional components independently.
We add the whole numbers: \(8 + 7 = 15\). Next, we add the fractions \(\frac{5}{12} + \frac{5}{8}\). To add these, we need a common denominator. The least common multiple (LCM) of 12 and 8 is 24. We rewrite the fractions:
\(\frac{5}{12} = \frac{5 \times 2}{12 \times 2} = \frac{10}{24}\)
\(\frac{5}{8} = \frac{5 \times 3}{8 \times 3} = \frac{15}{24}\)
Now, adding the fractions: \(\frac{10}{24} + \frac{15}{24} = \frac{25}{24}\).
The sum of the fractions is \(\frac{25}{24}\). Since this is an improper fraction, we convert it to a mixed number: \(\frac{25}{24} = 1 \frac{1}{24}\). We then add this back to the sum of the whole numbers: \(15 + 1 \frac{1}{24} = 16 \frac{1}{24}\). Thus, the final sum is \(16 \frac{1}{24}\).
Imagine you are baking cookies and need two different quantities of flour measured in cups. If the recipe calls for \(8 \frac{5}{12}\) cups for one batch and \(7 \frac{5}{8}\) cups for a second, adding these quantities allows you to know exactly how much flour you need in total to complete both batches. Understanding how to handle mixed numbers is essential for cooking, DIY projects, or any situation involving measurement where you need to combine fractional parts accurately.