- Perform the division first: \(\frac{1}{5} \div \frac{2}{3} = \frac{1}{5} \times \frac{3}{2} = \frac{3}{10}\).
- Express all terms with a common denominator of 10: \(\frac{8}{10} + \frac{3}{10} - \frac{5}{10}\).
- Combine the terms: \(\frac{8+3-5}{10} = \frac{6}{10}\).
- Simplify the final fraction to get \(\boxed{\frac{3}{5}}\).
Explanation
- Analyzing the expression and order of operations
The expression to evaluate is \(\frac{4}{5} + \frac{1}{5} \div \frac{2}{3} - \frac{1}{2}\). Following the order of operations (often remembered by the acronym PEMDAS or BODMAS), we must perform division before addition and subtraction.
- Performing the division
First, we perform the division:
\[ \frac{1}{5} \div \frac{2}{3} \]
Dividing by a fraction is equivalent to multiplying by its reciprocal. Thus:
\[ \frac{1}{5} \div \frac{2}{3} = \frac{1}{5} \times \frac{3}{2} = \frac{1 \times 3}{5 \times 2} = \frac{3}{10} \]
- Finding a common denominator and simplifying
Now, substitute this result back into the original expression:
\[ \frac{4}{5} + \frac{3}{10} - \frac{1}{2} \]
To perform the addition and subtraction, we need a common denominator. The least common multiple of 5, 10, and 2 is 10. We rewrite each fraction with a denominator of 10:
\[ \frac{4}{5} = \frac{4 \times 2}{5 \times 2} = \frac{8}{10} \]
\[ \frac{1}{2} = \frac{1 \times 5}{2 \times 5} = \frac{5}{10} \]
Now the expression is:
\[ \frac{8}{10} + \frac{3}{10} - \frac{5}{10} \]
- Completing the calculation
Perform the addition and subtraction from left to right:
\[ \frac{8}{10} + \frac{3}{10} = \frac{11}{10} \]
\[ \frac{11}{10} - \frac{5}{10} = \frac{6}{10} \]
Finally, simplify the fraction \(\frac{6}{10}\) by dividing the numerator and denominator by their greatest common divisor, which is 2:
\[ \frac{6 \div 2}{10 \div 2} = \frac{3}{5} \]
The final result is \(\boxed{\frac{3}{5}}\).
Examples
Understanding the order of operations is essential in everyday life when calculating costs or measurements. For example, if you are buying items that have different tax rates or discounts applied at different stages of a transaction, you must follow the correct order of operations to ensure the total is accurate. By mastering these rules, you gain the confidence to handle financial transactions, cooking measurements, and technical problems accurately and efficiently.