Problem 72 - Algebra A Combined Function

Question

Perform each indicated operation. Write all results in lowest terms. $$ \frac{5}{9}-\frac{5}{12} $$

Step-by-Step Solution

Answer

The result is \(\frac{5}{36}\).

Step 1: Find the Least Common Denominator (LCD)

To perform the operation, we need to have the same denominator for both fractions. The denominators are 9 and 12. The Least Common Denominator is the smallest number that both 9 and 12 divide evenly into. For 9 and 12, the LCD is 36.

Step 2: Convert Fractions to Equivalent Fractions

Adjust both fractions so they have the same denominator. To convert \(\frac{5}{9}\), multiply the numerator and denominator by 4: \[\frac{5}{9} = \frac{5 \times 4}{9 \times 4} = \frac{20}{36}\]For \(\frac{5}{12}\), multiply the numerator and denominator by 3: \[\frac{5}{12} = \frac{5 \times 3}{12 \times 3} = \frac{15}{36}\]

Step 3: Subtract the Fractions

Now that both fractions have the same denominators, subtract the second fraction from the first: \[\frac{20}{36} - \frac{15}{36} = \frac{20 - 15}{36} = \frac{5}{36}\]

Step 4: Simplify the Result

Ensure the resulting fraction is in its simplest form. The fraction \(\frac{5}{36}\) is already in the lowest terms because 5 and 36 have no common factors besides 1.

Key Concepts

Understanding Least Common Denominator

When subtracting fractions, having the same denominator for all fractions involved is crucial. This common denominator is known as the Least Common Denominator (LCD). It is the smallest number that serves as a shared denominator for two or more fractions. This number allows the fractions to be directly added or subtracted easily. For example, with denominators like 9 and 12, we need to find their LCD.
First, list the multiples of each denominator. For 9, these are 9, 18, 27, 36, and so on. For 12, they are 12, 24, 36, and so on. By comparing these lists, you can see that the first common multiple is 36; hence, it is our LCD.
Once you have the LCD, it becomes easier to convert the fractions. This ensures that both denominators match, enabling you to perform operations like addition or subtraction without complications.

Transforming into Equivalent Fractions

After finding the LCD, the next step is to convert each fraction into an equivalent fraction with this common denominator. Equivalent fractions are essentially different representations of the same value but with different numerators and denominators.
To create an equivalent fraction, you multiply the numerator and the denominator by the same number. For instance:

• To adjust \( \frac{5}{9} \) to a denominator of 36, you multiply both the numerator and the denominator by 4, because 9 times 4 equals 36. Thus, \( \frac{5}{9} \) becomes \( \frac{20}{36} \).
• Similarly, to adjust \( \frac{5}{12} \) to have the same common denominator, multiply both the numerator and denominator by 3, resulting in \( \frac{15}{36} \).

Converting to equivalent fractions with the LCD is crucial as it sets the stage for straightforward subtraction or addition of the fractions.

Simplifying Fractions

After subtracting the fractions with a common denominator, you may need to simplify the result. Simplifying a fraction means reducing it to its simplest form by ensuring that the numerator and denominator have no common factors other than 1.
To check for simplification:

• Find the greatest common factor (GCF) of the numerator and denominator.
• Divide both the numerator and denominator by this GCF.

In our subtraction example, after computing, we get \( \frac{5}{36} \). The numbers 5 and 36 do not share any prime factors other than 1, indicating that the fraction is already in its simplest form.
When fractions are simplified, it’s easier to understand their value and compare them to other fractions. It's always good practice to ensure your final answer is in the lowest terms.