Adding Mixed Numbers with Different Denominators
To solve this problem, we will add the two mixed numbers first by converting them to improper fractions.
Here's the process:
For \(2\frac{3}{4}\):
Convert this to an improper fraction by multiplying the whole number 2 by the denominator 4 and then adding the numerator 3.
\[2 \times 4 + 3 = 8 + 3 = 11\]
So, \(2\frac{3}{4}\) becomes \(\frac{11}{4}\).
For \(6\frac{4}{5}\):
Convert this to an improper fraction by multiplying the whole number 6 by the denominator 5 and then adding the numerator 4.
\[6 \times 5 + 4 = 30 + 4 = 34\]
So, \(6\frac{4}{5}\) becomes \(\frac{34}{5}\).
Now we need a common denominator to add the fractions. The least common denominator (LCD) for 4 and 5 is 20. We will now convert both fractions to have this common denominator.
For \(\frac{11}{4}\), to get a denominator of 20, we multiply both numerator and denominator by 5:
\[\frac{11}{4} \times \frac{5}{5} = \frac{55}{20}\]
For \(\frac{34}{5}\), to get a denominator of 20, we multiply both numerator and denominator by 4:
\[\frac{34}{5} \times \frac{4}{4} = \frac{136}{20}\]
Now we can add these two fractions:
\[\frac{55}{20} + \frac{136}{20} = \frac{55 + 136}{20} = \frac{191}{20}\]
To convert this back to a mixed number, divide the numerator by the denominator:
\[191 \div 20 = 9\text{ R }11\]
So the improper fraction \(\frac{191}{20}\) is equivalent to the mixed number \(9\frac{11}{20}\).
The final answer is \(9\frac{11}{20}\).
