- Convert all mixed numbers to improper fractions: \(4 \frac{1}{2} = \frac{9}{2}\) and \(3 \frac{1}{2} = \frac{7}{2}\).
- Simplify the parenthesis: \(-6 - (-\frac{7}{2}) = -\frac{5}{2}\).
- Divide the fractions: \(\frac{9}{2} \div (-\frac{5}{2}) = -\frac{9}{5}\).
- Add the final term: \(-\frac{9}{5} - 3 = -\frac{24}{5} = \boxed{-4 \frac{4}{5}}\).
Explanation
- Convert mixed numbers to improper fractions
To evaluate the expression \(4 \frac{1}{2} \div (-6 - (-3 \frac{1}{2})) + (-3)\), we first convert all mixed numbers into improper fractions.
Recall that a mixed number \(a \frac{b}{c}\) is equivalent to \(\frac{a \times c + b}{c}\).
- \(4 \frac{1}{2} = \frac{4 \times 2 + 1}{2} = \frac{9}{2}\)
- \(3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{7}{2}\)
Substituting these into our expression, we get:
\[\frac{9}{2} \div (-6 - (-\frac{7}{2})) + (-3)\]
- Simplify inside the parentheses
Next, we evaluate the expression inside the parentheses: \(-6 - (-\frac{7}{2})\).
Subtracting a negative number is equivalent to adding its positive counterpart:
\[-6 - (-\frac{7}{2}) = -6 + \frac{7}{2}\]
To add these, we find a common denominator, which is 2:
\[-6 = -\frac{12}{2}\]
So,
\[-\frac{12}{2} + \frac{7}{2} = \frac{-12 + 7}{2} = -\frac{5}{2}\]
Now the entire expression becomes:
\[\frac{9}{2} \div (-\frac{5}{2}) - 3\]
- Perform division and final arithmetic
Now we perform the division. Dividing by a fraction is the same as multiplying by its reciprocal:
\[\frac{9}{2} \div (-\frac{5}{2}) = \frac{9}{2} \times (-\frac{2}{5})\]
We multiply the numerators and the denominators:
\[\frac{9 \times (-2)}{2 \times 5} = \frac{-18}{10} = -\frac{9}{5}\]
Finally, we complete the addition (subtraction):
\[-\frac{9}{5} - 3 = -\frac{9}{5} - \frac{15}{5} = -\frac{24}{5}\]
Converting back to a mixed number:
\[-\frac{24}{5} = -4 \frac{4}{5}\]
Examples
This type of arithmetic problem is essential for managing daily financial transactions or project planning. For instance, if you are tracking a budget that involves fractional debt or partial payments—like paying off a loan of \(\$450\) in increments that involve subtracting fractions of a base unit—you need to handle mixed operations accurately. Mastering these steps ensures that you can precisely calculate remaining balances or resources, preventing errors in complex real-world calculations.