Problem 97 - Beginning and Intermediate Algebra

Question

Simplify. $$ \frac{6-2(-3)}{4-3(-2)} $$

Step-by-Step Solution

Answer

\(\frac{6}{5}\)

Step 1: Simplify Inside the Numerator Parentheses

Focus on simplifying the expression inside the parentheses in the numerator first. We have the expression \(-2(-3)\). Remember that multiplying two negative numbers results in a positive number: \(-2 \times -3 = 6\). So, rewrite the numerator as \(6 + 6\).

Step 2: Perform Operations in the Numerator

Now that the expression inside the parentheses is simplified, add the terms in the numerator: \(6 + 6 = 12\). So the simplified numerator is \(12\).

Step 3: Simplify Inside the Denominator Parentheses

Now focus on the denominator. Simplify the expression inside the parentheses \(-3(-2)\). Again, multiplying two negatives gives a positive number: \(-3 \times -2 = 6\). So rewrite the denominator as \(4 + 6\).

Step 4: Perform Operations in the Denominator

Add the terms in the denominator: \(4 + 6 = 10\). So, the simplified denominator is \(10\).

Step 5: Simplify the Fraction

Now substitute the simplified numerator and denominator back into the fraction: \(\frac{12}{10}\). Divide both the numerator and the denominator by their greatest common factor, which is 2: \(\frac{12 \div 2}{10 \div 2} = \frac{6}{5}\).

Key Concepts

Numerator and Denominator

In a fraction, there are two main parts: the numerator and the denominator. The numerator is the number above the fraction line, and the denominator is the number below it. Think of the numerator as the part of the fraction that shows how many parts we have. In contrast, the denominator indicates how many equal parts the whole is divided into. For example, in the fraction \( \frac{12}{10} \), the numerator is 12, signifying twelve parts we are considering, while the denominator is 10, showing these parts are divided from a whole that is 10 parts.

When simplifying fractions, understanding the role of the numerator and the denominator is crucial because it helps us to see where we can reduce or simplify numbers efficiently. By making both parts simpler, we create an easier representation of the same value. Therefore, focusing on simplifying each component makes the entire fraction less complicated to understand and work with.

Multiplying Negative Numbers

Multiplying negative numbers is an important arithmetic concept that often surprises students. When two negative numbers are multiplied, they produce a positive result. This might be confusing at first, but it's a consistent rule in mathematics.

Here's why: a negative number indicates a direction opposite to a positive number. When multiplying two negatives, you essentially "flip" the direction twice, bringing it back to a positive.

• For instance, \(-2 \times -3 = 6\). While it might seem like a contradiction, it's helpful to remember: two negatives make a positive.
• This rule is incredibly useful, as seen in the process of simplifying the expression \(6 - 2(-3)\), which becomes \(6 + 6\) after recognizing that \(-2 \times -3\) equals \(6\).

This understanding assists in working confidently with equations and fractions involving negative numbers.

Greatest Common Factor

The greatest common factor (GCF) is the largest number that can evenly divide both the numerator and the denominator of a fraction. It simplifies the fraction without changing its value, making it easier to work with.

Let's walk through an example. Consider the fraction \( \frac{12}{10} \). Here, the GCF is 2 since 2 is the highest number that can evenly divide both 12 and 10. By dividing both the numerator and denominator by their GCF:

• 12 divided by 2 gives 6
• 10 divided by 2 gives 5

Thus, the simplified fraction is \( \frac{6}{5} \). This method of finding and using the greatest common factor is a time-saving trick that helps present fractions in their simplest form, making them prettier and simpler to understand.