Problem 20 - Elementary Algebra

Question

Reduce each fraction to lowest terms. $$\frac{-23 a c}{41 c}$$

Step-by-Step Solution

Answer

The fraction \(\frac{-23ac}{41c}\) simplifies to \(\frac{-23a}{41}\).

Step 1: Understand the Problem

We need to simplify the fraction \(\frac{-23ac}{41c}\). This means we have to cancel any common factors in the numerator and the denominator.

Step 2: Identify the Common Factors

Look at the numerator \(-23ac\) and the denominator \(41c\). Observe that both have a common factor of \(c\). We can simplify the fraction by canceling this common factor.

Step 3: Cancel the Common Factor

Divide both the numerator and the denominator by the common factor \(c\). The fraction becomes \(\frac{-23a}{41}\) after cancellation.

Step 4: Verify Simplification

Check if the resultant fraction \(\frac{-23a}{41}\) can be simplified further. Notice that \(-23a\) and \(41\) have no common factors apart from 1, so this is the simplest form.

Key Concepts

Lowest Terms

When we say a fraction is in its "lowest terms," we mean it has been simplified completely so that the numerator and denominator have no common factors other than 1. This process of simplification makes fractions easier to understand and work with. To achieve this, we divide both the top and bottom numbers of the fraction by any common factors they might have. Once there are no more common factors left, the fraction is in its lowest terms meaning it cannot be reduced any further.
This not only simplifies the arithmetic but also helps in comparing and calculating with other fractions seamlessly.

Common Factors

Common factors play an essential role in simplifying fractions. A common factor is any number that will divide exactly both the numerator and the denominator. In fraction simplification, identifying these common factors allows us to "cancel" them out, reducing the fraction to a simpler form.

Here's how you can find common factors:

• List all the factors of the numerator.
• List all the factors of the denominator.
• Identify any numbers that appear in both lists. These are the common factors.

After finding these factors, divide both the numerator and denominator by their greatest common factor (GCF). This simplifies the fraction significantly.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form by dividing both the numerator and the denominator by their common factors, especially the greatest common factor. This makes fractions easier to interpret and work with in calculations. The necessary steps are fairly straightforward but require careful attention to detail.

• First, examine the numerator and the denominator for any shared factors.
• Determine the greatest common factor (GCF) of these numbers.
• Divide both the numerator and the denominator by the GCF.
• The result should be a fraction in its simplest form, where no further simplification is possible.

For example, with the fraction \(-\frac{23ac}{41c}\), we identified "c" as a common factor. By canceling "c", we simplified the fraction to \(-\frac{23a}{41}\), which cannot be reduced further due to lack of additional common factors. Thus, understanding how to simplify fractions is a fundamental math skill.