Problem 45 - Elementary Algebra

Question

Add or subtract as indicated and express your answers in simplest form. (Objective 3) $$\frac{7 n}{8}-\frac{3 n}{9}$$

Step-by-Step Solution

Answer

\( \frac{13n}{24} \)

Step 1: Find a Common Denominator

To add or subtract fractions, we need a common denominator. The denominators given are 8 and 9. The least common multiple (LCM) of 8 and 9 is 72. Therefore, we will convert both fractions to have a denominator of 72.

Step 2: Convert Fractions to Common Denominator

Convert each fraction to an equivalent form with the denominator of 72. For \( \frac{7n}{8} \), multiply the numerator and denominator by 9 to get \( \frac{63n}{72} \). For \( \frac{3n}{9} \), multiply the numerator and denominator by 8 to get \( \frac{24n}{72} \).

Step 3: Subtract the Fractions

With common denominators, subtract the second fraction from the first. Subtract \( \frac{24n}{72} \) from \( \frac{63n}{72} \) to get: \( \frac{63n - 24n}{72} = \frac{39n}{72} \).

Step 4: Simplify the Result

Simplify \( \frac{39n}{72} \) by finding the greatest common divisor (GCD) of 39 and 72, which is 3. Divide both the numerator and the denominator by 3: \( \frac{39n}{72} = \frac{13n}{24} \).

Key Concepts

Least Common Multiple (LCM)

The least common multiple (LCM) is a fundamental concept that helps us work with fractions. The idea is to find the smallest number that is a common multiple of two or more numbers. This allows us to perform operations like addition or subtraction of fractions with different denominators.

For example, in the expression \( \frac{7n}{8} - \frac{3n}{9} \), the denominators are 8 and 9. To subtract these fractions, we first need a common denominator, which is where the LCM comes in. We find the smallest number that is a multiple of both 8 and 9.

• List multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72...
• List multiples of 9: 9, 18, 27, 36, 45, 54, 63, 72...

In this case, 72 is the least common multiple because it appears in both lists. Now, with 72 as a common denominator, we can proceed to adjust the fractions accordingly.

Subtraction of Fractions

Subtracting fractions requires that both fractions share the same denominator. With an LCM of 72 found for the denominators 8 and 9, the next step is to express both fractions with this new common denominator.
For \( \frac{7n}{8} \), multiply the numerator and denominator by 9 to maintain equivalence, resulting in \( \frac{63n}{72} \). Similarly, convert \( \frac{3n}{9} \) to an equivalent fraction by multiplying its numerator and denominator by 8, resulting in \( \frac{24n}{72} \).

Once both fractions have a common denominator, subtraction becomes a straightforward process of subtracting the numerators:

• \( \frac{63n}{72} - \frac{24n}{72} = \frac{63n - 24n}{72} \)

This yields the new fraction \( \frac{39n}{72} \). With both numerators and denominators aligned, subtraction of fractions is simplified to mere arithmetic on the numerators.

Simplifying Fractions

Simplifying is the final step in handling fractions. After subtracting the fractions \( \frac{63n}{72} - \frac{24n}{72} \), we arrive at \( \frac{39n}{72} \). This new fraction needs to be simplified to its simplest form.

Simplification involves dividing both the numerator and the denominator by their greatest common divisor (GCD). For the numbers 39 and 72, the GCD is 3. Simplifying \( \frac{39n}{72} \) entails dividing both parts by 3:

• \( \frac{39n \div 3}{72 \div 3} = \frac{13n}{24} \)

Thus, \( \frac{13n}{24} \) is the simplest form of the fraction. It's always beneficial to check that the resulting numerator and denominator do not share other common factors aside from 1, ensuring the fraction is fully simplified. Simplifying fractions not only makes them easier to understand but also sets the stage for performing further calculations efficiently.