Problem 46 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{8 n}{10}-\frac{7 n}{15}$$

Step-by-Step Solution

Answer

\( \frac{n}{3} \)

Step 1: Find a Common Denominator

To add or subtract fractions, they must have the same denominator. The denominators are 10 and 15. Find the least common multiple (LCM) of 10 and 15. The LCM of 10 and 15 is 30.

Step 2: Convert Fractions to Have Common Denominator

Convert \( \frac{8n}{10} \) to a fraction with a denominator of 30. Multiply both numerator and denominator by 3 to get \( \frac{24n}{30} \). Convert \( \frac{7n}{15} \) to a fraction with a denominator of 30 by multiplying both numerator and denominator by 2 to get \( \frac{14n}{30} \).

Step 3: Subtract the Fractions

Now that both fractions have the same denominator, subtract \( \frac{14n}{30} \) from \( \frac{24n}{30} \): \[ \frac{24n}{30} - \frac{14n}{30} = \frac{24n - 14n}{30} = \frac{10n}{30} \].

Step 4: Simplify the Fraction

Simplify \( \frac{10n}{30} \) by dividing both the numerator and the denominator by their greatest common divisor, which is 10. This results in \( \frac{n}{3} \).

Key Concepts

Understanding Common Denominators

A fundamental step when dealing with algebraic fractions is ensuring they have a common denominator. This is important because you can only directly add or subtract fractions when they share the same denominator.
To achieve this, you need to find a number that all denominators can divide into evenly. Once found, you convert each fraction so that they all have this denominator.
In the original problem, the denominators are 10 and 15. It's not possible to subtract directly because they are different.
By finding their common denominator, you can simplify the subtraction of these fractions.

Determining the Least Common Multiple

The least common multiple (LCM) is crucial for finding a common denominator among fractions. It is the smallest number that both denominators can divide into without any remainder. This skill becomes particularly useful when dealing with algebraic fractions, where you might have different numbers or variables.
Here's a quick way to find the LCM. For the numbers in this problem, 10 and 15:

• List the multiples of 10: 10, 20, 30, 40...
• List the multiples of 15: 15, 30, 45...

Notice that 30 is the first number they both share. That's why it's the LCM.
Using the LCM as the common denominator makes it easier to add or subtract fractions.

Practicing Simplifying Fractions

Once you've performed arithmetic operations on fractions, simplifying them is the next essential step. Simplifying means reducing the fraction to its simplest form without changing its value.
To simplify, find the greatest common divisor (GCD) of the numerator and the denominator. Divide both by this number.
For example, after subtracting the algebraic fractions, \( \frac{10n}{30} \), we further simplify by dividing both the numerator and denominator by 10. This results in\( \frac{n}{3} \). This step ensures that your answer is as simple and clear as possible.