Problem 56 - Introductory Algebra for College Students

Question

Determine whether statement "makes sense" or "does not make sense" and explain your reasoning. I simplified \(\frac{\frac{1}{2}+\frac{x}{3}}{4}\) by multiplying the numerator by 6.

Step-by-Step Solution

Answer

The statement does not make sense as multiplying just the numerator or denominator of a fraction by a particular number changes the value of the expression. Correct simplification would be making common denominator for fractions in the numerator and then divide it by 4.

Step 1: Understand the given expression

Consider the expression \( \frac{\frac{1}{2}+\frac{x}{3}}{4} \). The numerator is a sum of two fractions, \( \frac{1}{2} \) and \( \frac{x}{3} \). The denominator is \( 4 \), a whole number.

Step 2: Evaluate the proposed simplification

The student claims to have simplified the expression by multiplying the numerator by \( 6 \). If we apply this operation, we get \( \frac{6*(\frac{1}{2}+\frac{x}{3})}{4} = \frac{3+x}{4} \).

Step 3: Check for correctness

What the student did is incorrect as it alters the value of the given expression. Multiplying just the numerator of a fraction is the same as multiplying the whole fraction by that coefficient. Originally, the student was supposed to add the two fractions by finding a common denominator and then divide the total sum by \( 4 \). But instead, just multiplied the numerator by \( 6 \), which changes the value of the expression.

Key Concepts

Fraction Simplification

Fraction simplification is the process of making a fraction as simple as possible, without changing its value. When simplifying fractions, we try to find the smallest whole numbers for the numerator and denominator. Remember, what we do to the numerator, we must also do to the denominator to keep the fraction equivalent.

In our original expression, it's tempting to multiply the numerator by 6 to simplify it. But this doesn't work unless we multiply the denominator by the same amount too, or handle the entire fraction uniformly. This ensures we do not change the value of the fraction. Simplifying should involve combining terms properly, possibly by using a common denominator first.

Understanding how to simplify fractions correctly is essential for solving algebraic expressions accurately.

Algebraic Expressions

Algebraic expressions can seem complex, but they're simply a combination of numbers, variables (like x), and operations (like addition and multiplication). Expressions can vary from simple to quite detailed, containing multiple terms or fractions, like \( \frac{\frac{1}{2}+\frac{x}{3}}{4} \).

When dealing with algebraic expressions, particularly involving fractions, we need to approach them methodically:

• Identify all parts of the expression, such as coefficients and other terms.
• Understand the relationship between the different parts (e.g., numerators vs. denominators).
• Combine like terms when possible.

An understanding of these expressions helps in applying operations correctly and avoiding mistakes during simplification.

Numerator and Denominator

The numerator and denominator are fundamental components of a fraction, which can sometimes be a bit confusing. The numerator is the top part of a fraction, representing how many parts we have. The denominator, on the bottom, tells us the total number of equal parts the whole is divided into.

In the expression \( \frac{\frac{1}{2}+\frac{x}{3}}{4} \), \( \frac{1}{2}+\frac{x}{3} \) is the entire numerator. That means everything in this sum is part of the fractional expression being divided by the denominator, which is 4.

Understanding the roles of numerators and denominators is critical for correctly handling fractions. It ensures that operations, such as simplification, keep the original fraction's value intact. If you only affect the numerator or the denominator without considering the other, you risk changing the expression's value.