Find the sum. Write your answer in simplest form.
$3 \frac{3}{5}+7 \frac{1}{2}=$
To find the sum of \(3 \frac{3}{5}\) and \(7 \frac{1}{2}\), we first break down each mixed number into its integer and fractional parts. A mixed number \(a \frac{b}{c}\) is equivalent to \(a + \frac{b}{c}\). Thus, we can write the given expression as:
\[\left(3 + \frac{3}{5}\right) + \left(7 + \frac{1}{2}\right)\]
Next, we group the integer parts and the fractional parts separately to make the addition more straightforward:
\[(3 + 7) + \left(\frac{3}{5} + \frac{1}{2}\right)\]
Summing the integers gives:
\[3 + 7 = 10\]
Now, we focus on adding the fractions \(\frac{3}{5}\) and \(\frac{1}{2}\). To add these, we need a common denominator. The least common multiple of 5 and 2 is 10. We rewrite each fraction with a denominator of 10:
\[\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}\]
\[\frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10}\]
Adding them together:
\[\frac{6}{10} + \frac{5}{10} = \frac{11}{10}\]
The result \(\frac{11}{10}\) is an improper fraction. We convert this into a mixed number by dividing 11 by 10:
\[\frac{11}{10} = 1 \frac{1}{10}\]
Now, we add this result to the sum of the integers obtained in Step 2:
\[10 + 1 \frac{1}{10} = 11 \frac{1}{10}\]
Therefore, the sum in its simplest form is \(11 \frac{1}{10}\).
Understanding how to add mixed numbers is a foundational skill useful in everyday life, such as in cooking or home improvement projects. For instance, if a recipe calls for \(3 \frac{3}{5}\) cups of flour and another part of the recipe requires \(7 \frac{1}{2}\) cups, knowing how to sum these correctly ensures you measure the total ingredients accurately. Similarly, in woodworking, when you need to add lengths of wood measured in inches and fractions of an inch, this method of splitting and combining integers and fractions makes the task simpler and prevents errors.