Evaluate each expression. $\frac{15 \div 5 \cdot 4 \div 6 - 8}{-6 - (-5) - 8 \div 2}$

Verified step by step guidance

1. Rewrite the expression clearly to understand the order of operations: \(\frac{15}{5} \times \frac{4}{6} - \frac{8}{-6} - (-5) - \frac{8}{2}\).
2. Evaluate each division and multiplication separately: calculate \(\frac{15}{5}\), \(\frac{4}{6}\), \(\frac{8}{-6}\), and \(\frac{8}{2}\).
3. Multiply the results of \(\frac{15}{5}\) and \(\frac{4}{6}\) to handle the multiplication part.
4. Simplify the subtraction and addition of terms, remembering that subtracting a negative number is the same as adding the positive equivalent.
5. Combine all simplified terms step-by-step to write the final simplified expression before evaluating the numerical value.

Key Concepts

Order of Operations

The order of operations is a set of rules that dictate the sequence in which mathematical operations should be performed to ensure consistent results. The common acronym PEMDAS (Parentheses, Exponents, Multiplication and Division from left to right, Addition and Subtraction from left to right) helps remember this order. Applying these rules correctly is essential to evaluate expressions accurately.

Division and Multiplication of Fractions and Whole Numbers

Division and multiplication involving fractions and whole numbers require careful handling. Multiplying fractions involves multiplying numerators and denominators, while division by a fraction is equivalent to multiplying by its reciprocal. Understanding how to simplify and combine these operations is key to evaluating expressions with mixed numbers.

Handling Negative Numbers and Subtraction

Working with negative numbers and subtraction requires attention to signs and the use of parentheses. Subtracting a negative number is equivalent to adding its positive counterpart. Correctly interpreting expressions like '-(-5)' ensures accurate simplification and prevents common errors in evaluation.