Problem 74 - Elementary and Intermediate Algebra

Question

Perform the operations and, if possible, simplify. $$ \frac{7}{15}+\frac{1}{5}-\frac{4}{9} $$

Step-by-Step Solution

Answer

The simplified result is \(\frac{2}{9}\).

Step 1: Finding a common denominator

To add or subtract fractions, we need a common denominator. The denominators we have are 15, 5, and 9. Start by finding the least common multiple (LCM) of these numbers. Break it down: 15 = 3 × 5, 5 = 5, and 9 = 3 × 3. The LCM will be the highest power of each prime: 3² × 5 = 45.

Step 2: Converting to common denominators

Convert each fraction to have the denominator 45. For \(\frac{7}{15}\), multiply both the numerator and denominator by 3 to get \(\frac{21}{45}\), For \(\frac{1}{5}\), multiply both by 9 to get \(\frac{9}{45}\), For \(\frac{4}{9}\), multiply both by 5 to get \(\frac{20}{45}\).

Step 3: Performing the addition and subtraction

Now that all fractions have the same denominator, add and subtract their numerators: \[ \frac{21}{45} + \frac{9}{45} - \frac{20}{45} = \frac{21 + 9 - 20}{45} = \frac{10}{45} \].

Step 4: Simplifying the fraction

Simplify \(\frac{10}{45}\) by finding the greatest common divisor (GCD) of 10 and 45, which is 5. Divide both the numerator and the denominator by 5 to get \(\frac{2}{9}\).

Key Concepts

Least Common Multiple (LCM)

When we perform operations with fractions, like addition or subtraction, we need a common baseline or denominator to work with. This is where the Least Common Multiple, or LCM, comes in handy. The LCM of a set of numbers is the smallest number that all those numbers can divide without leaving a remainder. Let's see how this applies:

• Consider the denominators we have: 15, 5, and 9.
• Factor these numbers into their prime factors:
  • 15 is 3 × 5
  • 5 is simply 5
  • 9 is 3 × 3
• The LCM must include each prime number raised to the highest power that appears in the factorizations. Here, that's 3² (from 9) and 5 (from both 5 and 15).

This gives us the LCM, which is 45. By using the LCM of the denominators, we convert each fraction to equivalent fractions, where all the denominators are the same, allowing easy addition or subtraction.

Simplifying Fractions

Once you have performed all the arithmetic with fractions, it's a good idea to simplify them, if possible. Simplifying a fraction means to make it as "simple" as possible. This involves making sure that the numerator and the denominator have no common factors other than 1. Here's how you simplify:

• After combining the fractions, we ended up with \(\frac{10}{45}\).
• To simplify, we seeking the greatest common divisor (GCD) of the numerator and the denominator.

Simplifying is an important step because it reduces the fraction to its most basic form, making it easier to understand and compare.
Once simplified, \(\frac{10}{45}\) becomes \(\frac{2}{9}\), which is the simplest form.

Greatest Common Divisor (GCD)

Finding the GCD is crucial when simplifying fractions. The GCD of two numbers is the largest positive integer that divides both numbers without a remainder. Here's the process for finding the GCD:

• Begin with the numbers 10 (from the numerator) and 45 (from the denominator).
• List the factors of each number:
  • Factors of 10 are 1, 2, 5, and 10.
  • Factors of 45 are 1, 3, 5, 9, 15, and 45.
• The greatest common factor they share is 5.

Divide both top and bottom of the fraction by the GCD, 5, for \(\frac{10}{45}\) to reach the simplest form, \(\frac{2}{9}\).
This step of simplifying fractions with the GCD ensures that the fraction represents the smallest and simplest version of the numbers involved.