Question - Adding Mixed Numbers with Different Denominators
mathdifferent denominatorsmixed numbers additionconverting improper fractionscommon denominator calculation
Solution:
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 $$9\frac{11}{20}$$.
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