Solve: Combining (3/8)a + (14/9)b + 1(1/9)b + (6/8)a with Like Terms

To solve this problem, we'll follow these steps:

• Step 1: Group and simplify terms with the same variable.
• Step 2: Convert any mixed numbers to improper fractions.
• Step 3: Find a common denominator to combine fractions.
• Step 4: Simplify the expression.

Let's work through the steps:
Step 1: Start by grouping like terms. The expression is:
$\frac{3}{8}a + \frac{6}{8}a + \frac{14}{9}b + 1\frac{1}{9}b$.
Step 2: Convert the mixed number to an improper fraction. For $1\frac{1}{9}b$: $1\frac{1}{9}b = \frac{10}{9}b$.
Rewrite the expression: $\frac{3}{8}a + \frac{6}{8}a + \frac{14}{9}b + \frac{10}{9}b$.
Step 3: Combine the $a$-terms and $b$-terms separately:
The $a$-terms: $\frac{3}{8}a + \frac{6}{8}a = \left(\frac{3}{8} + \frac{6}{8}\right)a = \frac{9}{8}a$.
For the $b$-terms: $\frac{14}{9}b + \frac{10}{9}b = \left(\frac{14}{9} + \frac{10}{9}\right)b = \frac{24}{9}b$.
Simplify $\frac{24}{9}$: $\frac{24}{9} = \frac{8}{3}$ after dividing by the greatest common divisor 3.
Step 4: Combine simplified terms: $\frac{9}{8}a + \frac{8}{3}b$.
Convert $\frac{9}{8}a$ to a mixed number: $\frac{9}{8}a = 1\frac{1}{8}a$.
Convert $\frac{8}{3}b$ to a mixed number: $\frac{8}{3}b = 2\frac{2}{3}b$.
Thus, the simplified expression is: $1\frac{1}{8}a + 2\frac{2}{3}b$.

Therefore, the solution to the problem is $1\frac{1}{8}a + 2\frac{2}{3}b$.