Problem 141 - Introductory Algebra for College Students

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I find it easier to multiply \(\frac{1}{3}\) and \(\frac{3}{4}\) than to add them.

Step-by-Step Solution

Answer

It does make sense since multiplying fractions is simpler than adding them as it does not require calculation of the least common multiple.

Step 1: Analyze the Complexity of the Addition Operation

Adding \(\frac{1}{3}\) and \(\frac{3}{4}\) requires the calculation of the common denominator, in this case, 12. These fractions then become \(\frac{4}{12}\) and \(\frac{9}{12}\), and their sum is \(\frac{13}{12}\). Thus, the addition operation involves finding the least common multiple of the denominators and then adding.

Step 2: Analyze the Complexity of the Multiplication Operation

Multiplying \(\frac{1}{3}\) and \(\frac{3}{4}\) is simpler. You multiply the numerators together to get the new numerator and the denominators together to get the new denominator. The result is \(\frac{3}{12}\) or \(\frac{1}{4}\).

Key Concepts

Adding Fractions

When you add fractions, the first step is to ensure that the fractions have a common denominator. A common error among students is to add the numerators and denominators directly without finding a common denominator, leading to incorrect results. To add fractions correctly, follow these steps:

• Determine the Least Common Denominator (LCD): Find the least common multiple of both denominators to get the LCD.
• Convert Fractions: Rewrite each fraction with the LCD as the new denominator by multiplying both the numerator and the denominator of each fraction by the same number that will make the denominator equal to the LCD.
• Add Numerators: Once the fractions have the same denominator, you can add the numerators while keeping the denominator the same.
• Simplify the Result: If necessary, simplify the resulting fraction by dividing both the numerator and the denominator by their greatest common factor.

Incorporating these steps will help students avoid common pitfalls and understand the process of adding fractions systematically.

Multiplying Fractions

Multiplying fractions is generally considered more straightforward than adding them because you don't need a common denominator to proceed. Here's how you can multiply two fractions:

• Multiply the Numerators: Take the two numerators (the top numbers of the fractions) and multiply them together to get the new numerator.
• Multiply the Denominators: Similarly, multiply the denominators (the bottom numbers) together to determine the new denominator.
• Simplify the Result: If the new fraction can be reduced, divide both numerator and denominator by their greatest common factor to simplify the fraction.

Ensuring the concept of simplifying is clear is important, as it's a common area where students might stumble.

Common Denominators

Finding common denominators is essential in adding, subtracting, and comparing fractions. A common denominator refers to a common multiple of two or more denominators. To find a common denominator, use the following approach:

• List Multiples: Start by listing several multiples of each denominator.
• Find Commonality: Look for the lowest multiple that appears in each list—this is your least common multiple (LCM).
• Convert Fractions: Once you have the LCM, convert all fractions involved to equivalent fractions with the LCM as their new denominator.

Understanding how to find common denominators will considerably ease the process of adding and subtracting fractions for students.

Least Common Multiple

The least common multiple, or LCM, is the smallest multiple shared by two or more numbers. It's particularly useful in fraction operations, where having a common denominator is required to perform calculations. To calculate the LCM:

• Prime Factorization: Break down each number into its prime factors.
• Compare Factors: For each prime factor, use the highest power of that factor found in any of the numbers.
• Multiply Prime Factors: Multiply together those chosen prime factors to find the LCM.

By understanding and finding the LCM, students will be better prepared to tackle a variety of algebraic fraction problems effectively.