- Simplify the parentheses to find \(-1/4\).
- Calculate the square of \(-4/5\) to get \(16/25\).
- Perform the division \(-\frac{3}{5} \div (-\frac{1}{4})\) resulting in \(12/5\).
- Subtract the squared term from the result: \(\frac{60}{25} - \frac{16}{25} = \frac{44}{25}\).
- Final Answer: \(\boxed{\frac{44}{25}}\).
Explanation
- Data and Problem Analysis
To evaluate the expression \(-\frac{3}{5} \div \left(-\frac{1}{2} + \frac{1}{4}\right) - \left(-\frac{4}{5}\right)^2\), we follow the order of operations, often remembered by the acronym PEMDAS: Parentheses, Exponents, Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). The expression contains fractions, a division, a subtraction, and a power. We will address these components systematically.
- Evaluating the Parentheses
First, we evaluate the expression inside the parentheses: \(\left(-\frac{1}{2} + \frac{1}{4}\right)\). To add these fractions, we find a common denominator, which is 4. Thus, \(-\frac{1}{2} = -\frac{2}{4}\). Adding the two terms gives \(-\frac{2}{4} + \frac{1}{4} = -\frac{1}{4}\).
- Evaluating the Exponent
Next, we evaluate the exponent term \(\left(-\frac{4}{5}\right)^2\). Squaring a fraction involves squaring both the numerator and the denominator. Since the base is negative, squaring it results in a positive value. Thus, \(\left(-\frac{4}{5}\right) \times \left(-\frac{4}{5}\right) = \frac{16}{25}\).
- Performing the Division
Now we perform the division: \(-\frac{3}{5} \div \left(-\frac{1}{4}\right)\). Dividing by a fraction is equivalent to multiplying by its reciprocal. The reciprocal of \(-\frac{1}{4}\) is \(-4\). Therefore, the calculation becomes \(-\frac{3}{5} \times (-4) = \frac{12}{5}\).
- Final Subtraction and Result
Finally, we perform the subtraction: \(\frac{12}{5} - \frac{16}{25}\). To subtract these, we use a common denominator of 25. Converting \(\frac{12}{5}\) to have a denominator of 25, we get \(\frac{12 \times 5}{5 \times 5} = \frac{60}{25}\). Subtracting gives \(\frac{60}{25} - \frac{16}{25} = \frac{44}{25}\). The final result is \(\frac{44}{25}\), which is also \(1 \frac{19}{25}\) or \(1.76\). The simplest fraction form is \(\boxed{\frac{44}{25}}\).
Examples
When managing a budget for a project, you might encounter complex fractional costs or scale factors. For example, if you are calculating the net return on an investment after accounting for fees and tax percentages represented as fractions, you apply these same steps. Evaluating such expressions ensures you correctly handle proportions and ratios, allowing for precise financial planning and analysis. Mastering these operations allows you to confidently solve real-world problems involving scaled quantities or relative adjustments.