Problem 60 - Algebra 2

Question

REASONING In the expression \(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}, a, b,\) and \(c\) are nonzero real numbers. Determine whether each statement is sometimes, always, or never true. Explain your answer. a. \(a b c\) is a common denominator. b. \(a b c\) is the LCD. c. \(a b\) is the LCD. d. \(b\) is the LCD. e. The sum is \(\frac{b c+a c+a b}{a b c}\)

Step-by-Step Solution

Answer

Statements a, b, and e are always true; statements c and d are never true.

Step 1: Analyze Statement a

Statement: \(abc\) is a common denominator.A common denominator is any shared multiple of all individual denominators. Here, the individual denominators are \(a\), \(b\), and \(c\). Since multiplying these, \(abc\), provides a multiple of each, it can indeed serve as a common denominator for the fractions \(\frac{1}{a}\), \(\frac{1}{b}\), and \(\frac{1}{c}\). Thus, this statement is always true.

Step 2: Examine Statement b

Statement: \(abc\) is the LCD (Least Common Denominator).The least common denominator is the smallest expression that can serve as a common denominator for the fractions. Since \(abc\), being the product of relatively prime numbers (assuming no overlap in factors), is the smallest such multiple of \(a\), \(b\), and \(c\), it acts as the least common denominator. This statement is always true.

Step 3: Review Statement c

Statement: \(ab\) is the LCD.The least common denominator must be divisible by each individual denominator, which in this case are \(a\), \(b\), and \(c\). Since \(ab\) lacks the factor \(c\), it cannot be divisible by \(c\). Therefore, \(ab\) cannot be the least common denominator. This statement is never true.

Step 4: Analyze Statement d

Statement: \(b\) is the LCD.To serve as the least common denominator, a value must be divisible by each fraction's denominator (\(a\), \(b\), and \(c\)). \(b\) lacks factors \(a\) and \(c\) (unless it's a special case where \(a\) or \(c\) are factors), preventing \(b\) from generally equaling the LCD. This statement is never true.

Step 5: Assess Statement e

Statement: The sum is \(\frac{bc + ac + ab}{abc}\).To add fractions \(\frac{1}{a}\), \(\frac{1}{b}\), and \(\frac{1}{c}\), first convert them to a common denominator (\(abc\)).\[\frac{1}{a} = \frac{bc}{abc},\] \[\frac{1}{b} = \frac{ac}{abc},\] and \[\frac{1}{c} = \frac{ab}{abc}.\]Adding these fractions:\[\frac{bc}{abc} + \frac{ac}{abc} + \frac{ab}{abc} = \frac{bc + ac + ab}{abc}.\]Thus, the expression is correctly represented, meaning this statement is always true.

Key Concepts

Fraction Addition

Adding fractions might seem daunting at first, but it becomes simple when you follow a structured process. To add fractions such as \( \frac{1}{a} + \frac{1}{b} + \frac{1}{c} \), it's crucial first to ensure all fractions share a common denominator. This makes them compatible for addition.
To achieve this, convert each fraction into equivalent forms with the shared denominator and then add the numerators. The initial step is crucial, as without a common ground, these fractions can't align correctly to reveal a unified sum.

• Find a common denominator for all fractions.
• Convert each fraction.
• Add the numerators together, keeping the common denominator.

This streamlined method allows fractions to be added seamlessly, ensuring no piece of the expression is lost in translation.

Common Denominator

A common denominator is a fundamental aspect of fraction addition. It refers to a multiple that's shared among all the denominators involved in the sum, providing a foundation upon which each fraction can balance.
For instance, with fractions like \( \frac{1}{a} \), \( \frac{1}{b} \), and \( \frac{1}{c} \), the product \( abc \) of these denominators is a common denominator.
Even if \( a \), \( b \), and \( c \) differ greatly, multiplying them ensures every denominator is found within the product. Thus, it paves the way for adding fractions by bringing everything to a common platform. Simplicity follows when each fraction is expressed over this shared denominator. This is why looking for a common denominator isn't just a rule; it's the key to unlocking the solution.

LCD (Least Common Denominator)

Unlike a regular common denominator, the Least Common Denominator (LCD) is the smallest possible denominator all fractions can share. It's the smallest number or expression that holds each original denominator as a factor.
Using the fractions \( \frac{1}{a} \), \( \frac{1}{b} \), and \( \frac{1}{c} \), the product \( abc \) often turns out to be the LCD, especially when \( a \), \( b \), and \( c \) don't share factors.

• Consider it as the smallest common ground for addition.
• It reduces the complexity of expressions.
• Ensures no unnecessary terms bolster the solution.

The LCD makes calculations not only possible but efficient by preventing cumbersome arithmetic with larger numbers.

Rational Expressions

Rational expressions are fractions where numerators and denominators are algebraic expressions. Similar to simple fractions, operations like addition depend heavily on common denominators.
For example, adding \( \frac{1}{a} \), \( \frac{1}{b} \), and \( \frac{1}{c} \) isn't just about slapping numerators and denominators together. It involves careful manipulation to reflect their algebraic nature, ensuring factors align properly before performing operations.

• Express each part over a shared common denominator.
• Factorize where possible to simplify the expression.
• Manage similar terms to streamline the equation.

Understanding rational expressions adjusts fraction handling by focusing on simplifying the complexity of algebraic relationships within the expressions.