Problem 76 - Beginning and Intermediate Algebra

Question

Perform each indicated operation. \(\frac{-12(-5)}{7-(-5)}\)

Step-by-Step Solution

Answer

5

Step 1: Simplify the Numerator

First, simplify the expression in the numerator. The numerator is \ \( -12(-5) \). Multiply these two numbers together to get \ \( 60 \).

Step 2: Simplify the Denominator

Next, simplify the expression in the denominator. The denominator is \ \( 7 - (-5) \). Subtracting a negative number is the same as adding, so this becomes \ \( 7 + 5 = 12 \).

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Now, divide the simplified numerator by the simplified denominator. This means \ \( \frac{60}{12} = 5 \).

Key Concepts

Numerator and Denominator

Understanding the terms numerator and denominator is essential for solving fraction problems. In any fraction, the numerator is the top number and the denominator is the bottom number. The numerator represents how many parts of the whole you have. The denominator shows the total number of equal parts that make up the whole.

For instance, in the fraction \(\frac{3}{4}\), '3' is the numerator and '4' is the denominator. This means you have 3 parts out of 4 equal parts. When performing operations, always start by handling each part separately.

• Numerator: The number of chosen parts.
• Denominator: The total number of parts.

Multiplication

Multiplication plays a crucial role in algebraic operations, especially when simplifying expressions. In this exercise, we first simplified the numerator by multiplying two numbers.

For example, \(-12 \times (-5)\) involves multiplying -12 and -5. Remember:

• Multiplying two negative numbers results in a positive product.
• Multiplying a negative number by a positive number gives a negative product.

Here, \(-12 \times -5 = 60\), since multiplying two negative numbers yields a positive result.

Addition and Subtraction

Addition and subtraction are fundamental operations in algebra. In our exercise, addition and subtraction are used to simplify the denominator.

The given expression has \(7 - (-5)\). When you subtract a negative number, it's the same as adding the positive counterpart. Hence, \(7 - (-5) = 7 + 5 = 12\). Making sense of such operations is key to solving complex algebraic problems effectively.

• Subtracting a Negative Number: Adds the absolute value.
• Adding Numbers: Combine values to get a sum.

Applying these rules ensures accurate and efficient simplification of the fraction.