Find the sum:
`-2 (1)/(3) + 4 (3)/(5)`

Answer

To find the sum of the rational numbers \(-2 \frac{1}{3}\) and \(4 \frac{3}{5}\), we will follow these steps:

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

For \(-2 \frac{1}{3}\):
- Multiply the whole number (2) by the denominator (3): \(2 \times 3 = 6\).
- Add the numerator (1) to this product: \(6 + 1 = 7\).
- Since the original number is negative, we write it as \(-\frac{7}{3}\).

For \(4 \frac{3}{5}\):
- Multiply the whole number (4) by the denominator (5): \(4 \times 5 = 20\).
- Add the numerator (3) to this product: \(20 + 3 = 23\).
- Thus, we write it as \(\frac{23}{5}\).

So we have:
\[ -2 \frac{1}{3} = -\frac{7}{3} \quad \text{and} \quad 4 \frac{3}{5} = \frac{23}{5} \]

### Step 2: Find a Common Denominator
Next, we need to find a common denominator to add the two fractions. The denominators are 3 and 5. The least common multiple (LCM) of 3 and 5 is 15.

### Step 3: Convert Fractions to Have the Same Denominator
Now we convert both fractions to have the denominator of 15.

For \(-\frac{7}{3}\):
- Multiply the numerator and denominator by 5:
\[ -\frac{7 \times 5}{3 \times 5} = -\frac{35}{15} \]

For \(\frac{23}{5}\):
- Multiply the numerator and denominator by 3:
\[ \frac{23 \times 3}{5 \times 3} = \frac{69}{15} \]

### Step 4: Add the Fractions
Now that both fractions have the same denominator, we can add them:
\[ -\frac{35}{15} + \frac{69}{15} = \frac{-35 + 69}{15} = \frac{34}{15} \]

### Step 5: Determine the Sign of the Result
Since \(69\) is greater than \(35\), the result is positive. Therefore, the final answer is:
\[ \frac{34}{15} \]

### Final Answer
The sum of \(-2 \frac{1}{3} + 4 \frac{3}{5}\) is \(\frac{34}{15}\).

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