Problem 8 - Algebra for College Students

Question

For Problems 1-12, perform the indicated operations involving rational numbers. Be sure to express your answers in reduced form. \(\frac{5}{9}-\frac{11}{12}\)

Step-by-Step Solution

Answer

\(\frac{-13}{36}\) is the simplified form.

Step 1: Find the Least Common Denominator (LCD)

To subtract fractions, they must have the same denominator. Find the least common denominator (LCD) of 9 and 12. The prime factors of 9 are 3 x 3, and for 12, they are 2 x 2 x 3. The LCD is the smallest number that both denominators divide into, which in this case is 36.

Step 2: Convert Fractions to Have the Same Denominator

Convert both fractions to have the LCD of 36:\[\frac{5}{9} = \frac{5 \times 4}{9 \times 4} = \frac{20}{36}\]\[\frac{11}{12} = \frac{11 \times 3}{12 \times 3} = \frac{33}{36}\]

Step 3: Subtract the Fractions

Now subtract the converted fractions:\[\frac{20}{36} - \frac{33}{36} = \frac{20 - 33}{36} = \frac{-13}{36}\]

Step 4: Simplify the Fraction

Check if the result can be simplified. Since 13 is a prime number and does not divide into 36, \(\frac{-13}{36}\) is already in its simplest form.

Key Concepts

Understanding the Least Common Denominator

When subtracting fractions, having a different denominator for each fraction can make the task challenging. This is where the least common denominator (LCD) comes into play. The LCD is the smallest number that is a multiple of each of the denominators involved.
To find it, you need to identify the prime factors of each denominator. For example, for the denominators 9 and 12:

• The prime factors of 9 are 3 and 3 (i.e., 9 = 3 x 3).
• The prime factors of 12 are 2, 2, and 3 (i.e., 12 = 2 x 2 x 3).

The LCD is determined by taking the highest power of each prime that appears in any factorization. Therefore, for 9 and 12, the LCD is 36, because it includes:

• Both 3 x 3 from 9 (since 3 appears twice)
• Each 2 from 12
• One 3 from 12's factors

Mastering how to find the least common denominator is crucial, as it simplifies the process of working with fractions and reduces potential errors.

The Art of Fraction Subtraction

Once you've determined the least common denominator, you're ready to perform the subtraction. To subtract fractions, align them to have the same denominator using the LCD. This involves adjusting each fraction so that they are expressed over this common denominator.
In our example with \[\frac{5}{9} - \frac{11}{12}\]Once the LCD of 36 is found:

• Adjust \(\frac{5}{9}\) to \(\frac{20}{36}\)
• Adjust \(\frac{11}{12}\) to \(\frac{33}{36}\)

This adjustment means multiplying the numerator and denominator of each fraction by the same number, ensuring the equality of the value of the original fractions. Once aligned, the fractions can be easily subtracted:\[\frac{20}{36} - \frac{33}{36} = \frac{20 - 33}{36} = \frac{-13}{36}\]After subtraction, you have a new fraction that might require simplification.

The Process of Simplifying Fractions

Simplifying a fraction involves reducing it to its simplest form, where the greatest common divisor (GCD) of the numerator and the denominator is 1. This simplifies calculations and makes fractions easier to understand.
After subtracting the fractions:\[\frac{-13}{36}\]Check whether the result can be simplified. For simplification, examine the numerator and the denominator:

• 13 is a prime number, meaning its only divisors are 1 and 13.
• The denominator, 36, does not share any common factor with 13 besides 1.

Thus, the fraction \(-\frac{13}{36}\) remains as it is, since it does not simplify further. This outcome underscores how essential it is to detect and understand prime numbers in the simplifying process, ensuring mathematical expressions are as concise as possible.