Question

13
$5\frac{3}{4} + 2\frac{5}{8}$

Explanation:

Step1: Convert mixed numbers to improper fractions

First, convert \(5\frac{3}{4}\) to an improper fraction. The formula for converting a mixed number \(a\frac{b}{c}\) to an improper fraction is \(\frac{a\times c + b}{c}\). So for \(5\frac{3}{4}\), we have \(a = 5\), \(b = 3\), \(c = 4\). Then \(5\frac{3}{4}=\frac{5\times4 + 3}{4}=\frac{20 + 3}{4}=\frac{23}{4}\).

Next, convert \(2\frac{5}{8}\) to an improper fraction. Here, \(a = 2\), \(b = 5\), \(c = 8\). So \(2\frac{5}{8}=\frac{2\times8 + 5}{8}=\frac{16 + 5}{8}=\frac{21}{8}\).

Step2: Find a common denominator

The denominators are 4 and 8. The least common denominator of 4 and 8 is 8. So we need to convert \(\frac{23}{4}\) to a fraction with denominator 8. To do this, we multiply the numerator and denominator of \(\frac{23}{4}\) by 2 (since \(4\times2 = 8\)). So \(\frac{23}{4}=\frac{23\times2}{4\times2}=\frac{46}{8}\).

Step3: Add the fractions

Now we have \(\frac{46}{8}+\frac{21}{8}\). When adding fractions with the same denominator, we add the numerators and keep the denominator the same. So \(\frac{46 + 21}{8}=\frac{67}{8}\).

Step4: Convert back to a mixed number (optional)

Now convert \(\frac{67}{8}\) back to a mixed number. Divide 67 by 8. \(8\times8 = 64\), so \(67\div8 = 8\) with a remainder of \(67 - 64 = 3\). So \(\frac{67}{8}=8\frac{3}{8}\). We can also check by adding the whole numbers and the fractions separately. The whole numbers are 5 and 2, so \(5 + 2 = 7\). The fractions are \(\frac{3}{4}\) and \(\frac{5}{8}\). Convert \(\frac{3}{4}\) to \(\frac{6}{8}\), then \(\frac{6}{8}+\frac{5}{8}=\frac{11}{8}=1\frac{3}{8}\). Then add the whole number part: \(7+1\frac{3}{8}=8\frac{3}{8}\).

Answer:

\(8\frac{3}{8}\) (or \(\frac{67}{8}\))