Solve Mixed Number Addition: 5⅔ + 1⅖ Step by Step

To solve the problem of adding the mixed numbers $5\frac{2}{3}$ and $1\frac{2}{5}$, follow these steps:

Step 1: Convert the mixed numbers to improper fractions.
For $5\frac{2}{3}$:
Multiply the whole number by the denominator, then add the numerator:
$5 \times 3 + 2 = 15 + 2 = 17$.
Thus, $5\frac{2}{3}$ becomes $\frac{17}{3}$.
For $1\frac{2}{5}$:
Multiply the whole number by the denominator, then add the numerator:
$1 \times 5 + 2 = 5 + 2 = 7$.
Thus, $1\frac{2}{5}$ becomes $\frac{7}{5}$.

Step 2: Find a common denominator.
The denominators are 3 and 5. The least common multiple of 3 and 5 is 15. Therefore, we'll convert each fraction to have a denominator of 15:
For $\frac{17}{3}$, multiply both the numerator and the denominator by 5:
$\frac{17 \times 5}{3 \times 5} = \frac{85}{15}$.
For $\frac{7}{5}$, multiply both the numerator and the denominator by 3:
$\frac{7 \times 3}{5 \times 3} = \frac{21}{15}$.

Step 3: Add the fractions.
Add the numerators while keeping the common denominator:
$\frac{85}{15} + \frac{21}{15} = \frac{85 + 21}{15} = \frac{106}{15}$.

Step 4: Convert the improper fraction back to a mixed number.
Divide the numerator by the denominator:
$106 \div 15 = 7$ with a remainder of 1.
Hence, $\frac{106}{15}$ is equivalent to $7\frac{1}{15}$.

Therefore, the solution to the problem is $7\frac{1}{15}$.