Problem 144 - 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 saved money by buying a computer for \(\frac{3}{2}\) of its original price.

Step-by-Step Solution

Answer

The statement 'I saved money by buying a computer for \(\frac{3}{2}\) of its original price' does not make sense, as this means paying more, not less.

Step 1: Understanding the fraction

The fraction \(\frac{3}{2}\) is equivalent to 1.5 or 150% when expressed as a decimal or percentage respectively.

Step 2: Applying the Fraction to the Price

If you buy a computer for \(\frac{3}{2}\) or 1.5 times its original price, it means you're actually paying 50% more than the original price.

Step 3: Evaluating the Statement

Based on the interpretation of the fraction \(\frac{3}{2}\), the statement 'I saved money by buying a computer for \(\frac{3}{2}\) of its original price' does not make sense, because paying 1.5 times the original price means you paid more, not less.

Key Concepts

Fractions in Algebra

Fractions are commonly used in algebra to represent parts of a whole. In algebraic expressions, fractions might denote coefficients or ratios. For instance, the fraction \( \frac{3}{2} \) appears frequently to express a value greater than one whole. Understanding this fraction involves recognizing it as 1.5, meaning one whole and one half.

Important Points:

• Fractions like \( \frac{3}{2} \) indicate parts greater than a whole—3 parts divided by 2.
• This fraction can be expressed as a mixed number \( 1 \frac{1}{2} \) or a decimal (1.5).
• Converting a fraction to a decimal can make algebraic operations easier to visualize.

It’s crucial to practice converting between fractions and other forms, as this skill aids in understanding complex algebraic expressions.

Understanding Percentages

Percentages are another mathematical way to express proportions, comparing a number to 100. When dealing with percentages, understanding conversion between fractions and percentages is key. Let's look at \( \frac{3}{2} \) again—the same value can be expressed as 150%.

Steps to Convert Fractions to Percentages:

• Multiply the fraction by 100 to find its percentage form.
• So, \( \frac{3}{2} \times 100 = 150\% \).

Percentages can depict increases or decreases in value:

• Paying 150% of something's price indicates an increase of 50% over its original value.

Recognizing how percentages relate to fractions aids in evaluating statements involving savings or costs, and helps clarify mathematical problems.

Mathematical Logic

Logic in math involves critical thinking and ensuring statements make sense. Evaluating statements like "I saved money by buying a computer for \( \frac{3}{2} \) of its original price" requires basic logical reasoning.The original sentence falsely claims savings but actually shows a cost increase. Here's why:

• \( \frac{3}{2} \) indicates 1.5 times the original cost, meaning you've paid more.
• Logical Conclusion: Paying 150% doesn’t equate to saving, it's spending extra.

Logical reasoning helps detect inaccuracies in statements. Critical thinking skills allow students to distinguish between logical truths and falsehoods in mathematical assertions.