Problem 81 - Beginning and Intermediate Algebra

Question

Perform the following operations. Write answers in lowest terms. $$ \frac{2}{3}-\frac{5}{9}+\frac{5}{6} $$

Step-by-Step Solution

Answer

The answer is \( \frac{17}{18} \).

Step 1: Find a Common Denominator

To perform operations on fractions, they must have a common denominator. The denominators we have are 3, 9, and 6. The least common multiple of these numbers is 18. Convert each fraction to have a denominator of 18.

Step 2: Convert Each Fraction

Convert \( \frac{2}{3} \). Multiply both the numerator and the denominator by 6 to get \( \frac{12}{18} \).Convert \( \frac{5}{9} \). Multiply both the numerator and the denominator by 2 to get \( \frac{10}{18} \).Convert \( \frac{5}{6} \). Multiply both the numerator and the denominator by 3 to get \( \frac{15}{18} \).

Step 3: Perform the Subtraction

Subtract the second fraction from the first: \( \frac{12}{18} - \frac{10}{18} = \frac{2}{18} \).

Step 4: Perform the Addition

Now add the result from Step 3 to the third fraction: \( \frac{2}{18} + \frac{15}{18} = \frac{17}{18} \).

Step 5: Simplify the Final Fraction

The fraction \( \frac{17}{18} \) is already in its simplest form since 17 and 18 have no common factors other than 1.

Key Concepts

Common Denominator

When working with fractions, it's crucial to understand the concept of a common denominator. Fractions can only be added or subtracted when they share the same denominator. Let's consider our example: \(\frac{2}{3} - \frac{5}{9} + \frac{5}{6}\). Here, the denominators are 3, 9, and 6. To perform operations, a common denominator is needed, which encompasses all the given denominators without leaving out any fraction.

To determine this common denominator, often we use the least common multiple (LCM) of these numbers, ensuring that the operations are possible without altering the value of the fractions involved. Once found, each fraction is converted into an equivalent form with this common denominator. This process requires multiplication of both the numerator and the denominator by the same number.For instance, the LCM of 3, 9, and 6 is 18, which serves as our common denominator in this problem. By converting each fraction to have this denominator, it becomes feasible to add or subtract them directly. This step simplifies what could seem a laborious task into a more straightforward operation.

Least Common Multiple

Finding the least common multiple (LCM) is essential when working with fractions like in our exercise. The LCM is the smallest multiple shared by a set of numbers. In our specific case, it aims to align the denominators so that the fractions can be combined. To find the LCM of the denominators 3, 9, and 6, consider their multiples:

• Multiples of 3: 3, 6, 9, 12, 15, 18...
• Multiples of 9: 9, 18, 27, 36...
• Multiples of 6: 6, 12, 18, 24...

The smallest number common to all these sets is 18, making it the LCM. By using this number as the common denominator, the process of adding and subtracting fractions becomes streamlined and manageable. Once identified, each fraction is adjusted to have this common denominator by multiplying both its numerator and denominator appropriately. It maintains the size of each fraction while making addition or subtraction possible. This understanding of LCM is not only useful in fraction operations but also in other areas of mathematics.

Simplifying Fractions

After performing operations on fractions, it’s important to check if the result can be simplified, ensuring the fraction is expressed in its simplest form. Simplifying means to reduce the fraction to its lowest terms, where the numerator and the denominator share no common factors besides 1.Consider the result of our example, \(\frac{17}{18}\). To simplify a fraction:

• Identify any common factors the numerator and the denominator might share.
• Divide both by their greatest common factor (GCF).

In this case, 17 and 18 are successive integers, meaning 17 is a prime number and shares no common factors with 18. Thus, \(\frac{17}{18}\) is already in its simplest form.Simplification is a critical step not just for neatness, but also because it helps to highlight the true size of a fraction, making mathematical results clearer and more understandable. By practicing simplification, one acknowledges the elegance and efficiency inherent in mathematical operations.