Problem 108 - Intermediate Algebra

Question

Simplify each expression. $$ \frac{3}{5} t-\frac{2}{3} t $$

Step-by-Step Solution

Answer

The simplified expression is \( \frac{-1}{15}t \).

Step 1: Identify Like Terms

The expression given is \( \frac{3}{5}t - \frac{2}{3}t \). Notice that both terms are like terms since both involve the variable \( t \). Our goal is to combine these terms.

Step 2: Find a Common Denominator

To subtract the fractions, they must have the same denominator. The denominators here are 5 and 3. The least common denominator of these numbers is 15.

Step 3: Convert Fractions to Common Denominator

Convert \( \frac{3}{5} \) and \( \frac{2}{3} \) so that they both have denominators of 15.For \( \frac{3}{5} \): Multiply both the numerator and the denominator by 3 to get \( \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \).For \( \frac{2}{3} \): Multiply both the numerator and the denominator by 5 to get \( \frac{2 \times 5}{3 \times 5} = \frac{10}{15} \).

Step 4: Perform the Subtraction

Now that the fractions have a common denominator, subtract them:\[ \frac{9}{15}t - \frac{10}{15}t = \frac{9 - 10}{15}t = \frac{-1}{15}t \].

Step 5: Simplified Expression

The expression \( \frac{3}{5}t - \frac{2}{3}t \) simplifies to \( \frac{-1}{15}t \). No further simplification is possible as the fraction is already in its simplest form.

Key Concepts

Common Denominator

When subtracting fractions, one key concept is the common denominator. It acts as a uniform base, allowing fractions to be easily compared or combined. Let's break it down:

• Each fraction has a denominator, which is the bottom number, telling us into how many equal parts the whole is divided.
• To perform addition or subtraction with fractions, it is critical that both fractions share the same denominator, known as the common denominator.
• This step ensures that all fractions are speaking the same 'language'. For example, with fractions \( \frac{3}{5} \) and \( \frac{2}{3} \), we find a common denominator to be 15.
• Finding the least common denominator often involves finding the least common multiple (LCM) of the two denominators.

Achieving a common denominator is a fundamental step in performing fraction subtraction, easing the process of combining the fractional parts.

Like Terms in Algebra

In algebra, recognizing and working with like terms is crucial for simplifying expressions. Let's delve into this concept.

• 'Like terms' are terms within an expression that share the same variable and power, making them similar enough to be combined.
• For instance, in the expression \( \frac{3}{5}t - \frac{2}{3}t \), both terms are like terms because they share the same variable \( t \).
• When combining like terms, only the coefficients (the numerical parts) are affected, while the variable part remains constant.

Working with like terms simplifies problem-solving by reducing the complexity of the expression — making calculations clearer and more streamlined.

Fraction Subtraction

Subtracting fractions is a methodical process that often involves several key steps to ensure accuracy. Here’s an easy way to think about it:

• The primary rule is ensuring that all fractions have a common denominator before attempting to subtract their numerators.
• Once a common denominator is established, like in \( \frac{9}{15}t \) and \( \frac{10}{15}t \), you can focus on just the numerators: \( 9 \) and \( 10 \).
• Subtract one numerator from the other and maintain the common denominator, resulting in a new fraction: \( \frac{-1}{15} \).
• This new fraction is then paired with the original variable \( t \), completing the subtraction process and presenting the simplified result.

By following these steps, you ensure that your subtraction is carried out accurately, maintaining mathematical integrity and arriving at a correct solution.