Find the sum. Write your answer in simplest form.
$12 \frac{4}{5}+9 \frac{1}{3}=$
We are tasked with computing the sum of two mixed numbers: \(12 \frac{4}{5}\) and \(9 \frac{1}{3}\). A mixed number is the sum of an integer part and a fractional part. For example, \(12 \frac{4}{5} = 12 + \frac{4}{5}\) and \(9 \frac{1}{3} = 9 + \frac{1}{3}\). To add these, we can sum the integer parts and the fractional parts separately, or we can convert both mixed numbers into improper fractions.
First, let's sum the integer parts and the fractional parts separately.
The integer parts are \(12\) and \(9\), so their sum is:
\[12 + 9 = 21\]
Next, the fractional parts are \(\frac{4}{5}\) and \(\frac{1}{3}\). To add these, we need a common denominator. The least common multiple of \(5\) and \(3\) is \(15\).
Converting the fractions:
\[\frac{4}{5} = \frac{4 \times 3}{5 \times 3} = \frac{12}{15}\]
\[\frac{1}{3} = \frac{1 \times 5}{3 \times 5} = \frac{5}{15}\]
Now, sum the fractions:
\[\frac{12}{15} + \frac{5}{15} = \frac{17}{15}\]
Since \(\frac{17}{15}\) is an improper fraction, we convert it to a mixed number:
\[\frac{17}{15} = 1 \frac{2}{15}\]
Finally, add this to the sum of the integer parts:
\[21 + 1 \frac{2}{15} = 22 \frac{2}{15}\]
Alternatively, we can convert both mixed numbers to improper fractions.
For \(12 \frac{4}{5}\):
\[12 \frac{4}{5} = \frac{12 \times 5 + 4}{5} = \frac{60 + 4}{5} = \frac{64}{5}\]
For \(9 \frac{1}{3}\):
\[9 \frac{1}{3} = \frac{9 \times 3 + 1}{3} = \frac{27 + 1}{3} = \frac{28}{3}\]
Sum the improper fractions by using the common denominator of \(15\):
\[\frac{64}{5} + \frac{28}{3} = \frac{64 \times 3}{15} + \frac{28 \times 5}{15} = \frac{192}{15} + \frac{140}{15} = \frac{332}{15}\]
Now, convert \(\frac{332}{15}\) back into a mixed number. Dividing \(332\) by \(15\):
\(332 \div 15 = 22\) with a remainder of \(2\) (since \(15 \times 22 = 330\)).
So, \(\frac{332}{15} = 22 \frac{2}{15}\). Both methods confirm the result.
By summing the integer and fractional parts, or by converting to improper fractions and then back, we find that the sum of \(12 \frac{4}{5}\) and \(9 \frac{1}{3}\) is \(22 \frac{2}{15}\). The final answer is \(\boxed{22 \frac{2}{15}}\).
Mixed numbers are used in various real-life scenarios, such as measuring ingredients for baking or construction materials. For example, if you are following a recipe that calls for \(12 \frac{4}{5}\) cups of flour and you need to add another \(9 \frac{1}{3}\) cups to a larger batch, you would need to calculate the sum \(12 \frac{4}{5} + 9 \frac{1}{3} = 22 \frac{2}{15}\) cups to ensure you have the correct total amount. This basic arithmetic skill is foundational for managing quantities precisely.