Find the sum. Write your answer in simplest form.
$10+5 \frac{1}{2}$
The goal is to calculate the sum of an integer, 10, and a mixed number, \(5 \frac{1}{2}\). A mixed number is a shorthand way of writing the sum of its integer part and its fractional part. Specifically, \(5 \frac{1}{2} = 5 + \frac{1}{2}\). Thus, our expression can be rewritten as \(10 + (5 + \frac{1}{2})\).
By the associative property of addition, we can group the integer parts together to simplify the calculation:
\[10 + 5 \frac{1}{2} = 10 + 5 + \frac{1}{2}.\]
Performing the addition of the integers, we get:
\[(10 + 5) + \frac{1}{2} = 15 + \frac{1}{2}.\]
This can be expressed as the mixed number \(15 \frac{1}{2}\).
To express the result as a single improper fraction, we can convert \(15 \frac{1}{2}\) by multiplying the integer part by the denominator and adding the numerator:
\[15 \frac{1}{2} = \frac{15 \times 2 + 1}{2} = \frac{30 + 1}{2} = \frac{31}{2}.\]
Both \(15 \frac{1}{2}\) and \(\frac{31}{2}\) represent the same value in simplest form. We will provide the result in mixed number form. Thus, the final answer is \(\boxed{15 \frac{1}{2}}\).
Mixed numbers are commonly used in daily life, such as in cooking or construction. For instance, if a recipe calls for 10 cups of flour and you already have 5 and a half cups, calculating the total amount needed involves adding these quantities. Mastering this operation allows you to quickly determine ingredient amounts or material requirements, ensuring accuracy in your projects.