Evaluate each expression for $p=-4$, $q=8$, and $r=-10$. $\frac{\frac{q}{2}-\frac{r}{3}}{\frac{3p}{4}+\frac{q}{8}}$

Verified step by step guidance

1. First, substitute the given values into the expression: replace \( p \) with \( -4 \), \( q \) with \( 8 \), and \( r \) with \( -10 \). The expression becomes \( \frac{\frac{q}{2} - \frac{r}{3}}{\frac{3p}{4} + \frac{q}{8}} \).
2. Calculate each part of the numerator separately: compute \( \frac{q}{2} \) and \( \frac{r}{3} \) using the substituted values.
3. Calculate each part of the denominator separately: compute \( \frac{3p}{4} \) and \( \frac{q}{8} \) using the substituted values.
4. Simplify the numerator by subtracting the two fractions obtained in step 2. Simplify the denominator by adding the two fractions obtained in step 3.
5. Finally, divide the simplified numerator by the simplified denominator to evaluate the entire expression.

Key Concepts

Order of Operations

The order of operations dictates the sequence in which parts of a mathematical expression are evaluated. It follows the PEMDAS rule: Parentheses, Exponents, Multiplication and Division (left to right), Addition and Subtraction (left to right). Correctly applying this ensures accurate evaluation of complex expressions.

Substitution of Variables

Substitution involves replacing variables in an expression with given numerical values. This step is essential to evaluate expressions numerically. For example, substituting p = -4, q = 8, and r = -10 allows the expression to be simplified to a numerical value.

Simplifying Complex Fractions

A complex fraction has fractions in its numerator, denominator, or both. Simplifying involves combining terms carefully, often by finding common denominators or rewriting the expression to a single fraction. This process is crucial to correctly evaluate expressions like the one given.