Question

Consider $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$ if a, b, and c are real numbers. Determine whether each statement is sometimes, always, or never true. Explain your answer.

a. abc is a common denominator.

b. abc is the LCD.

c. ab is the LCD.

d. b is the LCD.

e. The sum is $\frac{bc+ac+ab}{abc}$.

Short answer

Statements a, b and e are correct always true.

Statements c and d are sometimes true.

Step-by-step solution

Step 1.  Given

Given rational expression is $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$.

Step 2.  To determine

Determine whether each statement is sometimes, always, or never true. Explain your answer.

a. abc is a common denominator.

b. abc is the LCD.

c. ab is the LCD.

d. b is the LCD.

e. The sum is $\frac{bc+ac+ab}{abc}$.

Step 3.  Calculation

The denominators of the rational expression $\frac{1}{a}+\frac{1}{b}+\frac{1}{c}$are a, b and c.

So, the LCM (least common multiple of them is abc.

We try to get abc as denominator for each term and then simplify:

$\begin{array}{l}\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\\ =\frac{bc}{abc}+\frac{ac}{abc}+\frac{ab}{abc}\\ =\frac{bc+ac+ab}{abc}\end{array}$

It means statements a, b, and e are correct.

If b = c then the LCM of a, b, c = b will be ab. So, option c is sometimes true.

If a = b = c then LCM of a = b, b, c = b will be b. So, option d is sometimes true.