Evaluate:
$-2 \frac{1}{3}-(-5)=$
We are asked to evaluate the expression \(-2 \frac{1}{3} - (-5)\). First, recall that a mixed number like \(2 \frac{1}{3}\) represents \(2 + \frac{1}{3}\). Therefore, \(-2 \frac{1}{3}\) is equivalent to \(-\left(2 + \frac{1}{3}\right)\), which can be written as an improper fraction:
\[-2 \frac{1}{3} = -\left(\frac{2 \times 3 + 1}{3}\right) = -\frac{7}{3}.\]
The expression becomes \(-\frac{7}{3} - (-5)\). Recall that subtracting a negative number is equivalent to adding its positive counterpart, so \(a - (-b) = a + b\). Our expression simplifies to:
\[-\frac{7}{3} + 5.\]
To add the fraction \(-\frac{7}{3}\) and the integer \(5\), we express \(5\) as a fraction with a denominator of \(3\):
\[5 = \frac{5 \times 3}{3} = \frac{15}{3}.\]
Now, add the two fractions:
\[-\frac{7}{3} + \frac{15}{3} = \frac{-7 + 15}{3} = \frac{8}{3}.\]
Finally, convert the improper fraction \(\frac{8}{3}\) back into a mixed number. Dividing \(8\) by \(3\) gives \(2\) with a remainder of \(1\):
\[\frac{8}{3} = 2 \frac{1}{3} \text{... wait, let me re-check: } 2 \times 3 + 1 = 7. \text{ No, } 2 \times 3 + 2 = 8. \text{ So } \frac{8}{3} = 2 \frac{2}{3}.\]
Thus, the correct value is \(2 \frac{2}{3}\). Comparing this to the given options, we find it corresponds to the first option.
\[\boxed{2 \frac{2}{3}}\]
Imagine you are tracking your bank balance. If you start with a debt of \(\$2 \frac{1}{3}\) (or about \(\$2.33\)) and you receive a deposit of \(\$5\), your new balance can be calculated using the same logic: \(-2 \frac{1}{3} + 5\). By converting to fractions, you find you have \(\$2 \frac{2}{3}\) (or about \(\$2.67\)) remaining in your account. Understanding how to work with mixed numbers and signs is essential for managing personal finances and solving everyday arithmetic problems.