- Begin by evaluating the expression inside the parentheses: \(-\frac{2}{5} + \frac{1}{5} = -\frac{1}{5}\).
- Multiply this result by the term outside: \(-\frac{1}{4} \times (-\frac{1}{5}) = \frac{1}{20}\).
- Subtract the resulting product from \(-1\): \(-1 + \frac{1}{20} = -\frac{20}{20} + \frac{1}{20} = -\frac{19}{20}\).
- The final result is \(\boxed{-\frac{19}{20}}\).
Explanation
- Understanding the Problem and Order of Operations
The problem asks us to evaluate the arithmetic expression \(-1 - \frac{1}{4} \times \left(-\frac{2}{5} + \frac{1}{5}\right)\). This requires us to strictly follow the order of operations, often referred to as PEMDAS or BODMAS. This rule dictates that we must first evaluate the expressions within parentheses, then perform multiplications or divisions, and finally carry out additions and subtractions.
- Evaluating the Expression inside Parentheses
First, we look inside the parentheses: \(\left(-\frac{2}{5} + \frac{1}{5}\right)\). Since these fractions share a common denominator, we can directly combine their numerators:
\[-\frac{2}{5} + \frac{1}{5} = \frac{-2 + 1}{5} = -\frac{1}{5}\]
- Performing the Multiplication
Next, we proceed with the multiplication part of the expression: \(-\frac{1}{4} \times \left(-\frac{1}{5}\right)\). Multiplying two negative numbers results in a positive value, and multiplying the numerators and denominators separately gives:
\[-\frac{1}{4} \times \left(-\frac{1}{5}\right) = \frac{-1 \times -1}{4 \times 5} = \frac{1}{20}\]
- Final Calculation
Finally, we substitute this result back into the original expression:
\[-1 - \left(-\frac{1}{4} \times \left(-\frac{1}{5}\right)\right) = -1 + \frac{1}{20}\]
To combine these, we convert \(-1\) into a fraction with a denominator of 20:
\[-1 = -\frac{20}{20}\]
Adding them together:
\[-\frac{20}{20} + \frac{1}{20} = \frac{-20 + 1}{20} = -\frac{19}{20}\]
The final result is \(\boxed{-\frac{19}{20}}\).
Examples
Order of operations is fundamental in everyday life, such as when calculating discounts or total costs. For example, if you have a budget of \(\$100\) and decide to buy two books for \(\$20\) each, but you have a coupon for \(\$5\) off, you calculate your total spent as \(100 - (2 \times 20 - 5)\). Following the steps correctly, you evaluate the parentheses first: \(2 \times 20 = 40\), then \(40 - 5 = 35\), and finally subtract that from your initial budget: \(100 - 35 = 65\). Knowing the order ensures you never miscalculate your remaining balance.