Solving Complex Fractional Expressions: A Guide to the Mathematical Operations
In this article, we will explore the solution to a complex fractional expression, breaking down each step of the process using the order of operations (PEMDAS).
Understanding the Expression: 3/2 1/2 of 15/16 - 1
The given expression is:
(frac{3}{2} frac{1}{2} cdot frac{15}{16} - 1)
Breaking Down the Problem: Order of Operations (PEMDAS)
Order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), guides us on how to simplify and solve the expression. Let's go through each step.
Step 1: Interpret the Expression
The phrase "of" in the expression translates to multiplication. Thus, the expression becomes:
(frac{3}{2} cdot frac{1}{2} cdot frac{15}{16} - 1)
Step 2: Perform the Multiplication
First, we multiply (frac{1}{2} cdot frac{15}{16}):
(frac{1}{2} cdot frac{15}{16} frac{15}{32})
Step 3: Substitute and Simplify
Substitute (frac{15}{32}) back into the original expression:
(frac{3}{2} cdot frac{15}{32} - 1)
Step 4: Convert (frac{3}{2}) and 1 to a Common Denominator
Convert (frac{3}{2}) and (1) to have a common denominator of 32:
(frac{3}{2} frac{3 cdot 16}{2 cdot 16} frac{48}{32})
(1 frac{32}{32})
Step 5: Combine the Fractions and Perform Multiplication
Substitute these values into the expression:
(frac{48}{32} cdot frac{15}{32} - frac{32}{32})
Combine the fractions:
(frac{48 cdot 15 - 32}{32} frac{720 - 32}{32} frac{688}{32} frac{31}{32})
Thus, the final result is:
(frac{31}{32})
Conclusion
Understanding the order of operations and correctly interpreting phrases like "of" is crucial for solving complex fractional expressions. By following the PEMDAS rule, we can confidently simplify and solve such problems.