Verified Answer :
- Separate the whole number and fractional components: sum the whole parts \((2 + 3 = 5)\) and the fractional parts \(\left(\frac{3}{4} + \frac{5}{6}\right)\).
- Find a common denominator to add the fractions, resulting in \(\frac{19}{12}\) or \(1 \frac{7}{12}\).
- Combine these results to get \(5 + 1 \frac{7}{12} = 6 \frac{7}{12}\).
- The final answer is \(\boxed{6 \frac{7}{12}}\).
Explanation
- Problem Analysis
We are asked to calculate the sum of two mixed numbers: \(2 \frac{3}{4}\) and \(3 \frac{5}{6}\). A mixed number consists of a whole number part and a proper fractional part. To add two mixed numbers, we can either sum the whole numbers and fractions separately or convert both numbers into improper fractions first. We will use both methods to ensure our result is consistent and accurate.
- Method 1: Summing Whole and Fractional Parts Separately
In this method, we sum the whole number parts and the fractional parts independently.
The whole parts are \(2\) and \(3\). Adding them together:
\[2 + 3 = 5\]
The fractional parts are \(\frac{3}{4}\) and \(\frac{5}{6}\). To add these, we find a common denominator, which is \(12\):
\[\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}\]
\[\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}\]
Summing the fractions:
\[\frac{9}{12} + \frac{10}{12} = \frac{19}{12}\]
Since \(\frac{19}{12}\) is an improper fraction, we convert it to a mixed number:
\[\frac{19}{12} = 1 \frac{7}{12}\]
Finally, adding the sum of the whole numbers to this result:
\[5 + 1 \frac{7}{12} = 6 \frac{7}{12}\]
- Method 2: Converting to Improper Fractions
To verify our result, we convert each mixed number to an improper fraction and then add them.
For \(2 \frac{3}{4}\):
\[2 \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{8+3}{4} = \frac{11}{4}\]
For \(3 \frac{5}{6}\):
\[3 \frac{5}{6} = \frac{3 \times 6 + 5}{6} = \frac{18+5}{6} = \frac{23}{6}\]
Now, sum the improper fractions \(\frac{11}{4}\) and \(\frac{23}{6}\) by finding a common denominator of 12:
\[\frac{11}{4} = \frac{11 \times 3}{12} = \frac{33}{12}\]
\[\frac{23}{6} = \frac{23 \times 2}{12} = \frac{46}{12}\]
Summing them:
\[\frac{33}{12} + \frac{46}{12} = \frac{79}{12}\]
Converting the improper fraction \(\frac{79}{12}\) back into a mixed number:
\[79 \div 12 = 6 \text{ remainder } 7, \text{ so } \frac{79}{12} = 6 \frac{7}{12}\]
Both methods yield the same result, confirming our calculation.
- Final Conclusion
The sum of the two mixed numbers \(2 \frac{3}{4}\) and \(3 \frac{5}{6}\) is calculated to be \(6 \frac{7}{12}\). Therefore, the final answer is \(\boxed{6 \frac{7}{12}}\).
Examples
Imagine you are baking cookies and need to combine two recipes. The first recipe requires \(2 \frac{3}{4}\) cups of flour, and the second recipe requires \(3 \frac{5}{6}\) cups of flour. To ensure you have enough flour, you need to find the total sum by adding the two measurements: \(2 \frac{3}{4} + 3 \frac{5}{6} = 6 \frac{7}{12}\) cups. This simple calculation helps you prepare correctly and avoid running out of ingredients while baking your favorite treats.