Problem 73 - Intermediate Algebra

Question

Simplify by combining like terms. See Example 5 . $$\frac{3}{5} t+\frac{1}{3} t$$

Step-by-Step Solution

Answer

Combine the fractions to obtain \( \frac{14}{15}t \).

Step 1: Identify Like Terms

Observe that both terms \( \frac{3}{5}t \) and \( \frac{1}{3}t \) involve the variable \( t \). These terms are like terms because they share the same variable raised to the same power.

Step 2: Find a Common Denominator

The fractions \( \frac{3}{5} \) and \( \frac{1}{3} \) have different denominators, so we need to find a common denominator. The least common denominator (LCD) of 5 and 3 is 15.

Step 3: Rewrite Fractions with Common Denominator

Rewrite each fraction so that both have the common denominator of 15: \( \frac{3}{5}t = \frac{9}{15}t \) and \( \frac{1}{3}t = \frac{5}{15}t \).

Step 4: Combine the Like Terms

Now that both terms have the same denominator, we can combine them: \( \frac{9}{15}t + \frac{5}{15}t = \frac{9+5}{15}t = \frac{14}{15}t \).

Step 5: Simplify the Expression

The fractions have been combined, and \( \frac{14}{15}t \) is already in its simplest form because 14 and 15 have no common factors other than 1.

Key Concepts

Fractions

Fractions are a way to represent numbers that are not whole. They consist of two parts: a numerator and a denominator. The numerator is the top number, and it indicates how many parts we have. The denominator is the bottom number, showing the total number of equal parts the whole is divided into. For instance, in the fraction \(\frac{3}{5}\), 3 is the numerator and 5 is the denominator.
It is crucial when solving problems involving fractions to understand how they work together with variables. Fractions in algebra can have variables attached, like in \(\frac{3}{5}t\). This means \(t\) is multiplied by the fraction \(\frac{3}{5}\).
When dealing with multiple fractions, especially with different denominators, you will often need to find a way to bring them to common terms to perform operations like addition or subtraction.

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions. It is the shared multiple of the denominators of the fractions you are working with.
For example, when you have fractions such as \(\frac{3}{5}\) and \(\frac{1}{3}\), their denominators are 5 and 3. To add these fractions, they must have the same denominator. The least common denominator is the smallest multiple that both denominators divide into without a remainder.

• Identify the denominators: 5 and 3.
• List the multiples of each:
  - 5, 10, 15, 20...
  - 3, 6, 9, 12, 15...
• The smallest number that appears in both lists is 15, so this is the common denominator.

With the common denominator, the original fractions can be rewritten with equivalent values that allow them to be easily combined.

Simplifying Expressions

Simplifying expressions involves combining like terms to make the expression easier to work with. Like terms are terms that have the same variable to the same power, so they can be summed or subtracted from each other.
In an expression like \(\frac{3}{5}t + \frac{1}{3}t\), both terms are like terms because they share the variable \(t\). Once the fractions have been rewritten with a common denominator, they can be combined by adding their numerators:

• Rewritten terms: \(\frac{9}{15}t\) and \(\frac{5}{15}t\).
• Combine the numerators: \(9 + 5 = 14\).
• The combined expression is \(\frac{14}{15}t\).

It is important to always check if the resulting fractions can be simplified further by dividing both the numerator and numerator by their greatest common factor. In this case, \(\frac{14}{15}t\) is already simplified since 14 and 15 have no common factors other than 1.