Problem 141 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement "makes sense" or "does not make sense" and explain your reasoning. I find it easier to multiply \(\frac{1}{5}\) and \(\frac{3}{4}\) than to add them.

Step-by-Step Solution

Answer

The statement 'It's easier to multiply \(\frac{1}{5}\) and \(\frac{3}{4}\) than to add them' makes sense. This is because adding fractions requires finding a common denominator, which is an extra step compared to multiplication of fractions.

Step 1: Evaluate the Multiplication Operation

First, let's consider the multiplication operation. To multiply two fractions, simply multiply the numerators together for the new numerator and multiply the denominators together for the new denominator. Multiplying \(\frac{1}{5}\) and \(\frac{3}{4}\), you get \(\frac{1 \cdot 3}{5 \cdot 4} = \frac{3}{20}\).

Step 2: Evaluate the Addition Operation

Next, let's consider the addition operation. To add two fractions, you need to find a common denominator and then add the corresponding numerators. Adding \(\frac{1}{5}\) and \(\frac{3}{4}\), you get \(\frac{4 + 15}{20} = \frac{19}{20}\). Finding a common denominator can sometimes be more complex depending on the fractions being added.

Step 3: Compare the Complexity of Both Operations

When comparing the processes of both operations on fractions, it can generally be seen that multiplication is straight-forward and requires less work than addition, particularly when the denominators are not the same. This is due to the extra step of finding a common denominator in addition.

Key Concepts

Multiplying Fractions

Multiplying fractions is often simpler than adding them. Here's how it works: when you multiply fractions, you multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together.
For example, if you are given \( \frac{1}{5} \) and \( \frac{3}{4} \), the multiplication operation looks like this:

• Multiply the numerators: \( 1 \times 3 = 3 \).
• Multiply the denominators: \( 5 \times 4 = 20 \).

Combine these results to get the product: \( \frac{3}{20} \).
Multiplying fractions doesn't require additional steps, which is why some find it simpler. It's a direct path from start to finish.

Adding Fractions

Adding fractions involves more steps than multiplying. The main reason is because fractions must share a common denominator before you can add them.
To add \( \frac{1}{5} \) and \( \frac{3}{4} \):

• Find a common denominator. For \( 5 \) and \( 4 \), the least common denominator is \( 20 \).
• Adjust each fraction:
  • \( \frac{1}{5} \rightarrow \frac{4}{20} \) (multiply top and bottom by 4)
  • \( \frac{3}{4} \rightarrow \frac{15}{20} \) (multiply top and bottom by 5)
• Add the adjusted fractions: \( \frac{4}{20} + \frac{15}{20} = \frac{19}{20} \).

The need to find a common denominator is an extra, often tricky step in the process, making it less straightforward than multiplication.

Common Denominator

A common denominator is essential when adding fractions. It ensures fractions are equivalent, allowing them to be added because they reference the same 'whole'.
Finding a common denominator involves identifying a multiple that each original denominator can divide into without leaving a remainder.
To find the smallest common denominator, calculate the least common multiple (LCM) of the denominators. In the example \( \frac{1}{5} \) and \( \frac{3}{4} \), the LCM of \( 5 \) and \( 4 \) is \( 20 \).
Once you have this common basis, adjust each fraction accordingly before performing the addition process. This step can make adding fractions seem harder than multiplying, especially with different denominators.