To solve $5 \frac{2}{5} + 8 \frac{1}{2}$, we can follow these steps:

Step 1: Convert Mixed Numbers to Improper Fractions

• For $5 \frac{2}{5}$: $5 \frac{2}{5} = \frac{5 \times 5 + 2}{5} = \frac{25 + 2}{5} = \frac{27}{5}$

• For $8 \frac{1}{2}$: $8 \frac{1}{2} = \frac{8 \times 2 + 1}{2} = \frac{16 + 1}{2} = \frac{17}{2}$

Step 2: Find a Common Denominator

The least common denominator (LCD) of 5 and 2 is 10.

• To convert $\frac{27}{5}$ to a denominator of 10: $\frac{27}{5} = \frac{27 \times 2}{5 \times 2} = \frac{54}{10}$

• To convert $\frac{17}{2}$ to a denominator of 10: $\frac{17}{2} = \frac{17 \times 5}{2 \times 5} = \frac{85}{10}$

Step 3: Add the Fractions

Now that both fractions have the same denominator, we can add them: $\frac{54}{10} + \frac{85}{10} = \frac{54 + 85}{10} = \frac{139}{10}$

Step 4: Convert the Improper Fraction Back to a Mixed Number

To convert $\frac{139}{10}$ into a mixed number: $139 \div 10 = 13 \text{ remainder } 9$ So, $\frac{139}{10} = 13 \frac{9}{10}$.

Final Answer:

$5 \frac{2}{5} + 8 \frac{1}{2} = 13 \frac{9}{10}$

Would you like further explanation or details on any steps? Here are some related questions:

1. How do you convert a mixed number into an improper fraction?
2. What are the steps to add fractions with different denominators?
3. How do you simplify a fraction once it's added?
4. What is the least common denominator, and why do we need it?
5. How do you subtract mixed numbers?

Tip: When adding or subtracting fractions, always make sure the denominators are the same before performing the operation!