Problem 62 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{9}{8}-\frac{5}{6} $$

Step-by-Step Solution

Answer

\(\frac{7}{24}\)

Step 1: Find the Least Common Denominator

To subtract fractions with different denominators, we first need to find a common denominator. The denominators we have are 8 and 6. The smallest number that both 8 and 6 divide into evenly is their least common multiple (LCM), which is 24.

Step 2: Convert Each Fraction

Convert each fraction to an equivalent fraction with the common denominator of 24. Start with \( \frac{9}{8} \):Multiply the numerator and denominator by 3 to get \( \frac{27}{24} \).Then, for \( \frac{5}{6} \):Multiply the numerator and denominator by 4 to get \( \frac{20}{24} \).

Step 3: Subtract the Fractions

With both fractions having the same denominator, subtract the numerators:\[ \frac{27}{24} - \frac{20}{24} = \frac{7}{24} \]

Step 4: Simplify the Result (if possible)

Check if \( \frac{7}{24} \) can be simplified. Since 7 and 24 have no common factors other than 1, the fraction \( \frac{7}{24} \) is already in its simplest form.

Key Concepts

Understanding the Least Common Denominator

When subtracting fractions like \( \frac{9}{8} \) and \( \frac{5}{6} \), the first step is finding the least common denominator, also known as the LCD. This is a key concept because fractions can only be added or subtracted when they have the same denominator. The LCD is the smallest number that can be evenly divided by both denominators you are working with. In this example:

• The denominators are 8 and 6.
• The smallest number they both divide into is 24, which is the LCM of 8 and 6.

To find the LCD, list multiples of each denominator until you find the smallest common multiple. This ensures you can rewrite each fraction so they can be combined easily. Remembering how to find the least common denominator is a fundamental skill for working with fractions.

Simplifying Fractions Made Easy

Simplifying fractions is all about making them as simple as possible. It involves reducing fractions to their simplest form where the numerator and denominator have no common factors other than 1. In our exercise, after subtracting \( \frac{27}{24} - \frac{20}{24} \), we get \( \frac{7}{24} \). To check if a fraction can be simplified, follow these steps:

• Identify if there's any common factor between the numerator and denominator.
• If there is a common factor, divide both by the greatest common factor (GCF).

In this case, 7 and 24 share no common factors other than 1, meaning \( \frac{7}{24} \) is already in its simplest form. Simplifying ensures that fractions are easy to interpret and understand.

Exploring Equivalent Fractions

Equivalent fractions are fractions that represent the same value or portion, even though they may look different. They are essential in operations like addition and subtraction of fractions. To create equivalent fractions, multiply or divide both the numerator and the denominator by the same non-zero number. For instance, turning \( \frac{9}{8} \) into \( \frac{27}{24} \) involves:

• Multiplying both the numerator and the denominator by 3.

This gives the fraction a new form without changing its value. Understanding equivalent fractions lets you change denominators so fractions can be added or subtracted smoothly. Practicing creating equivalent fractions helps in recognizing various forms of the same fraction.