Problem 12 - 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. \(\frac{5}{6}-\frac{7}{9}-\frac{3}{10}\)

Step-by-Step Solution

Answer

\(-\frac{11}{45}\)

Step 1: Find the Least Common Denominator (LCD)

To perform operations with fractions, we need a common denominator. The denominators in this problem are 6, 9, and 10. The least common multiple (LCM) of these numbers is 90, so the LCD is 90.

Step 2: Convert Each Fraction

Convert each fraction to an equivalent fraction with the LCD as the denominator. For \(\frac{5}{6}\), multiply the numerator and denominator by 15 to get \(\frac{75}{90}\).For \(\frac{7}{9}\), multiply the numerator and denominator by 10 to get \(\frac{70}{90}\).For \(\frac{3}{10}\), multiply the numerator and denominator by 9 to get \(\frac{27}{90}\).

Step 3: Perform the Subtractions

Now that all fractions have a common denominator, perform the subtractions:\[\frac{75}{90} - \frac{70}{90} - \frac{27}{90}\]First, subtract \(\frac{70}{90}\) from \(\frac{75}{90}\), which gives \(\frac{5}{90}\).Then, subtract \(\frac{27}{90}\) from \(\frac{5}{90}\) to get \(-\frac{22}{90}\).

Step 4: Simplify the Result

Simplify the fraction \(-\frac{22}{90}\) by finding the greatest common divisor (GCD) of 22 and 90, which is 2. Divide both the numerator and the denominator by 2:\(-\frac{22 \div 2}{90 \div 2} = -\frac{11}{45}\).

Key Concepts

Least Common Denominator

When working with fractions, especially those with different denominators, finding a common ground is crucial in performing mathematical operations. The least common denominator (LCD) allows each fraction in a problem to be expressed with the same denominator, making addition or subtraction possible.
This is achieved by identifying the least common multiple (LCM) of the individual denominators. In the given problem, the denominators were 6, 9, and 10.

• To find the LCM, list the prime factors or use a multiplication method starting with the largest number, checking its multiples.
• The LCM of 6, 9, and 10 is 90, so the LCD is 90.

Finding the LCD is fundamental as it standardizes the fractions, ensuring the operations are straightforward. This step is achieved by converting each fraction to an equivalent one with the LCD as the denominator.

Fraction Simplification

Simplifying fractions is akin to cleaning up to find results in the simplest form, i.e., with the smallest possible numerator and denominator. Simplification is essential after performing operations to make the results more interpretable and manageable.
The task involves finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by this number. This reduces the fraction to its lowest terms.
In our earlier steps:

• We completed the subtraction, obtaining \(-\frac{22}{90}\).
• The GCD of 22 and 90 is 2, hence divide both: \-\frac{22 \div 2}{90 \div 2} = -\frac{11}{45}\.

This process is critical as it reveals the most reduced form of a fraction, ensuring precision in representations and calculations.

Greatest Common Divisor

The greatest common divisor (GCD) is a key mathematical concept used extensively in fraction simplification. The GCD of two numbers is the largest positive integer that divides both without leaving a remainder. It is crucial in reducing fractions, ensuring they are presented in their simplest form.
Here's how to find the GCD:

• One common method is the Euclidean algorithm, which involves a sequence of division steps.
• Another straightforward approach is listing the factors of each number and picking the largest common factor.

In our mathematical exercise related to simplifying \(-\frac{22}{90}\), the GCD was determined to be 2. By dividing both the numerator and denominator by this GCD, the fraction was reduced effectively to \(-\frac{11}{45}\). This shows the power of understanding and using the GCD in the simplification process.