Evaluate: b. `2 (5)/(9) + 3 (7)/(15)`

Answer

To evaluate the expression \( 2 \frac{5}{9} + 3 \frac{7}{15} \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers into improper fractions.

1. For \( 2 \frac{5}{9} \):
\[ 2 \frac{5}{9} = \frac{2 \times 9 + 5}{9} = \frac{18 + 5}{9} = \frac{23}{9} \]

2. For \( 3 \frac{7}{15} \):
\[ 3 \frac{7}{15} = \frac{3 \times 15 + 7}{15} = \frac{45 + 7}{15} = \frac{52}{15} \]

So, we have:
\[ 2 \frac{5}{9} + 3 \frac{7}{15} = \frac{23}{9} + \frac{52}{15} \]

### Step 2: Find the Least Common Multiple (LCM)
Next, we need to find the LCM of the denominators \( 9 \) and \( 15 \).

- The prime factorization of \( 9 \) is \( 3^2 \).
- The prime factorization of \( 15 \) is \( 3^1 \times 5^1 \).

The LCM is found by taking the highest power of each prime:
\[ \text{LCM}(9, 15) = 3^2 \times 5^1 = 9 \times 5 = 45 \]

### Step 3: Convert Fractions to Have the Same Denominator
Now, we convert both fractions to have the same denominator of \( 45 \).

1. For \( \frac{23}{9} \):
\[ \frac{23}{9} = \frac{23 \times 5}{9 \times 5} = \frac{115}{45} \]

2. For \( \frac{52}{15} \):
\[ \frac{52}{15} = \frac{52 \times 3}{15 \times 3} = \frac{156}{45} \]

Now we have:
\[ \frac{23}{9} + \frac{52}{15} = \frac{115}{45} + \frac{156}{45} \]

### Step 4: Add the Fractions
Now, we can add the two fractions:
\[ \frac{115}{45} + \frac{156}{45} = \frac{115 + 156}{45} = \frac{271}{45} \]

### Step 5: Convert to Mixed Number
Finally, we convert \( \frac{271}{45} \) into a mixed number.

1. Divide \( 271 \) by \( 45 \):
\[ 271 \div 45 = 6 \quad \text{(since \( 45 \times 6 = 270 \))} \]
The remainder is:
\[ 271 - 270 = 1 \]

So, we can write:
\[ \frac{271}{45} = 6 \frac{1}{45} \]

### Final Answer
Thus, the final answer is:
\[ 6 \frac{1}{45} \]

---