Problem 26 - Elementary and Intermediate Algebra

Question

Subtract and simplify the result, if possible. \(\frac{16 c}{11}-\frac{4 c}{11}\)

Step-by-Step Solution

Answer

\(\frac{12c}{11}\)

Step 1: Identify the Fractions

We are given two fractions with the same denominator: \(\frac{16c}{11}\) and \(\frac{4c}{11}\). When fractions have a common denominator, they can be subtracted directly.

Step 2: Subtract the Numerators

Subtract the numerators of the fractions while keeping the denominator the same. This looks like: \(16c - 4c = 12c\). Therefore, the new fraction is \(\frac{12c}{11}\).

Step 3: Simplify the Result

Check if the fraction \(\frac{12c}{11}\) can be simplified. Since 12 and 11 have no common factors other than 1, the fraction is already in its simplest form.

Key Concepts

Common Denominator

When working with fractions, it's crucial to understand the concept of a common denominator. A denominator is the bottom part of a fraction and represents the total number of equal parts. When two fractions have the same denominator, they are said to have a common denominator. This commonality makes performing operations like addition or subtraction easier.

In our exercise, we have the fractions \(\frac{16c}{11}\) and \(\frac{4c}{11}\). Both of these fractions have the same denominator, 11.

• The advantage of a common denominator is that it allows us to directly subtract the numerators, which are the top parts of the fractions, while keeping the denominator unchanged.
• If the denominators were different, we would first need to find a common denominator before proceeding with subtraction.

Recognizing when denominators are already common is an important skill, saving you time and helping you perform calculations accurately.

Simplifying Fractions

Simplifying fractions is another essential step in fraction problems. It involves reducing the fraction to its simplest form. A fraction is in its simplest form when the numerator and the denominator have no common factors besides 1. This means you can't evenly divide both the numerator and the denominator by any common number except 1.

In our exercise, after subtracting, we obtained the fraction \(\frac{12c}{11}\). Checking for simplification involves trying to find a number other than 1 that divides both 12c (numerator) and 11 (denominator).

• Upon examination, we find that 11 is a prime number and doesn't divide 12 evenly.
• This tells us the fraction is already in its simplest form.

Simplifying fractions helps ensure answers are more readable and easier to work with in future calculations.

Numerators and Denominators

The roles of numerators and denominators are fundamental in understanding fractions. The numerator is the top number of a fraction and shows how many parts of the whole are being considered. The denominator, as mentioned earlier, is the bottom number and indicates the total number of equal parts.

In our given fractions \(\frac{16c}{11}\) and \(\frac{4c}{11}\), the numerators are 16c and 4c respectively. During subtraction, we keep the denominator (11) constant and perform the operation on the numerators:

• Subtracting the second numerator from the first gives us 12c.
• This step highlights that numerators impact the actual size of the fraction's value in relation to the denominator.

Understanding numerators and denominators helps decipher the size and value of fractions, aiding in quick and accurate arithmetic operations.