Verified Answer :
- Convert the mixed number \(2 \frac{3}{4}\) to the improper fraction \(\frac{11}{4}\).
- Find the least common multiple of 4 and 6, which is 12, and rewrite both fractions: \(\frac{11}{4} = \frac{33}{12}\) and \(\frac{5}{6} = \frac{10}{12}\).
- Add the two fractions to get \(\frac{43}{12}\).
- Convert the improper fraction back to the mixed number \(\boxed{3 \frac{7}{12}}\).
Explanation
- Understanding the Problem
We are asked to calculate the sum of the mixed number \(2 \frac{3}{4}\) and the fraction \(\frac{5}{6}\). A mixed number \(a \frac{b}{c}\) is equivalent to \(a + \frac{b}{c}\). To add these, it is often easiest to first convert the mixed number to an improper fraction, then find a common denominator for the resulting fractions.
- Converting to an Improper Fraction
We convert \(2 \frac{3}{4}\) to an improper fraction using the formula \(a \frac{b}{c} = \frac{ac + b}{c}\). Here, \(a=2\), \(b=3\), and \(c=4\).
\[2 \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{8 + 3}{4} = \frac{11}{4}.\]
Now our expression becomes \(\frac{11}{4} + \frac{5}{6}\).
- Finding a Common Denominator and Adding
To add \(\frac{11}{4}\) and \(\frac{5}{6}\), we need a common denominator. We find the least common multiple (LCM) of the denominators 4 and 6. The multiples of 4 are 4, 8, 12, ... and the multiples of 6 are 6, 12, ... The least common multiple is 12. Now we rewrite both fractions with 12 as the denominator:
\[\frac{11}{4} = \frac{11 \times 3}{4 \times 3} = \frac{33}{12}\]
\[\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}\]
Summing these gives:
\[\frac{33}{12} + \frac{10}{12} = \frac{43}{12}.\]
- Simplifying the Result
Finally, we can convert the improper fraction \(\frac{43}{12}\) back into a mixed number. Dividing 43 by 12 gives a quotient of 3 and a remainder of 7:
\[43 \div 12 = 3 \text{ with a remainder of } 7.\]
Thus, \(\frac{43}{12} = 3 \frac{7}{12}\). The sum is \(\boxed{3 \frac{7}{12}}\).
Examples
In everyday life, adding mixed numbers is very common when dealing with measurements. For example, if you are building a bookshelf and you have a board that is \(2 \frac{3}{4}\) meters long and you need to attach an extension piece that is \(\frac{5}{6}\) of a meter long, you can use these methods to calculate the total length. By converting to a common denominator of 12, you find that the total length of the board will be \(3 \frac{7}{12}\) meters. Understanding this ensures you get your measurements correct for your projects!