Problem 118 - Introductory and Intermediate Algebra for College Students 4th

Question

Perform the indicated operation. Write the answer as an algebraic expression. $$\frac{10}{y}-\frac{6}{y}$$

Step-by-Step Solution

Answer

The algebraic expression for the given operation is \( \frac{4}{y}\).

Step 1: Recognizing the Denominator

It can be observed that in the given problem \(\frac{10}{y} - \(\frac{6}{y}\), the denominator of the two fractions is y. Since the denominators are the same for both fractions, they can be subtracted directly.

Step 2: Subtraction of Numerators

The next step is to subtract the numerators of the fractions. This means subtracting 6 from 10, which yields 4.

Step 3: Writing Final Expression

The final step is to write the result as an algebraic expression, which is \( \frac{4}{y}\).

Key Concepts

Fractions

Fractions are a way to express parts of a whole. They consist of two main components: the numerator and the denominator. The numerator is the top part of the fraction, indicating how many parts we have. The denominator is the bottom part, showing into how many equal parts the whole is divided. For example, in the fraction \( \frac{10}{y} \), 10 is the numerator and \( y \) is the denominator. By understanding these parts, we can perform various operations on fractions, such as addition, subtraction, multiplication, and division. Fractions can represent values less than one, greater than one, or even equal to one, depending on the relationship between the numerator and the denominator.
Fractions are a fundamental part of algebra, and knowing how to manipulate them is crucial for solving many algebraic expressions.

Subtraction of Fractions

Subtracting fractions might seem tricky at first, but it becomes straightforward once you grasp the concept. The key to subtracting fractions is ensuring they have a common denominator. Once the denominators are the same, you can subtract the numerators directly. Take the expression \(\frac{10}{y} - \frac{6}{y}\) as an example.
Here, both fractions already have the common denominator \(y\). This allows us to simply subtract the numerators \(10 - 6\) to get \(4\). There's no need to change the denominator or find a new common one since it's already the same.
Always remember, after subtracting the numerators, the structure of the fraction remains the same with the common denominator, resulting in a new simplified fraction.

Common Denominator

The concept of a common denominator is crucial when working with the addition or subtraction of fractions. A common denominator is a shared multiple of the denominators of two or more fractions. For subtraction, like in \(\frac{10}{y} - \frac{6}{y}\), having a common denominator allows for straightforward operations. Since the denominators are already the same (\(y\)), no further steps are needed to adjust or convert them.
If fractions did not share a common denominator initially, you would need to find one by determining the least common multiple (LCM) of the denominators. Fortunately, in this case, the denominator \(y\) is already common to both fractions.
Efficiently finding and using common denominators simplifies the subtraction process and helps form accurate algebraic expressions.