Find the sum. Write your answer in simplest form.
$6 \frac{5}{8}+6 \frac{5}{6}$
To find the sum of \(6 \frac{5}{8}\) and \(6 \frac{5}{6}\), we can break each mixed number into an integer part and a fractional part. A mixed number \(A \frac{B}{C}\) is equivalent to \(A + \frac{B}{C}\). Therefore, the sum can be written as:
\[6 \frac{5}{8} + 6 \frac{5}{6} = \left(6 + \frac{5}{8}\right) + \left(6 + \frac{5}{6}\right).\]
By rearranging the terms, we group the integer parts together and the fractional parts together:
\[(6 + 6) + \left(\frac{5}{8} + \frac{5}{6}\right).\]
The sum of the integers is \(6 + 6 = 12\). Our task is now to compute the sum of the fractions.
To add the fractions \(\frac{5}{8}\) and \(\frac{5}{6}\), we need to find a common denominator. The least common multiple (LCM) of the denominators 8 and 6 is 24.
Now, we convert each fraction to an equivalent fraction with the denominator 24:
For \(\frac{5}{8}\): multiply the numerator and the denominator by 3:
\[\frac{5 \times 3}{8 \times 3} = \frac{15}{24}.\]
For \(\frac{5}{6}\): multiply the numerator and the denominator by 4:
\[\frac{5 \times 4}{6 \times 4} = \frac{20}{24}.\]
Now we add the converted fractions:
\[\frac{15}{24} + \frac{20}{24} = \frac{15 + 20}{24} = \frac{35}{24}.\]
The result of the fractional addition is \(\frac{35}{24}\).
Since \(\frac{35}{24}\) is an improper fraction (where the numerator is greater than the denominator), we can simplify it into a mixed number. Dividing 35 by 24 gives a quotient of 1 and a remainder of 11:
\[\frac{35}{24} = 1 \frac{11}{24}.\]
Now, we add this back to the sum of the integer parts we calculated in Step 1:
\[12 + 1 \frac{11}{24} = 13 \frac{11}{24}.\]
Thus, the sum of \(6 \frac{5}{8}\) and \(6 \frac{5}{6}\) in its simplest form is \(13 \frac{11}{24}\).
In everyday life, knowing how to add mixed numbers is essential for cooking and carpentry. For instance, if you are following a recipe that calls for \(6 \frac{5}{8}\) cups of flour for a cake base and \(6 \frac{5}{6}\) cups of flour for the frosting, you need to add these values to know the total amount of flour required for your baking project. Mastering this technique ensures accuracy in measurements and helps you manage your supplies effectively.