Question

rational equations: mastery test
3
select the correct answer.
mr. koger is making copies of the school literary magazine to distribute to students and faculty. he has access to two copy machines: an older machine and a new machine. if he used only the older machine, it would take 80 minutes longer than if he used only the new machine. if he used both machines, he would finish the job in 30 minutes.
when m is the number of minutes it would take to finish the job using only the new machine, the situation is modeled by this rational equation:
\\(\frac{1}{m} + \frac{1}{m + 80} = \frac{1}{30}\\)
how many minutes would it take to finish the job if mr. koger used only the new machine?
a. 20 minutes
b. 40 minutes
c. 50 minutes
d. 60 minutes

Explanation:

Step1: Multiply through by \( m(m + 80)(30) \)

To eliminate the denominators, we multiply each term by the least common multiple of \( m \), \( m + 80 \), and \( 30 \), which is \( m(m + 80)(30) \).

$$ 30(m + 80) + 30m = m(m + 80) $$

Step2: Expand and simplify

Expand the left - hand side: \( 30m+2400 + 30m=m^{2}+80m \)
Combine like terms on the left: \( 60m + 2400=m^{2}+80m \)
Rearrange to form a quadratic equation: \( m^{2}+80m-60m - 2400 = 0 \)
Simplify the quadratic equation: \( m^{2}+20m - 2400=0 \)

Step3: Factor the quadratic equation

We need to find two numbers that multiply to \( - 2400 \) and add up to \( 20 \). The numbers are \( 60 \) and \( - 40 \).
So, \( m^{2}+20m - 2400=(m + 60)(m - 40)=0 \)

Step4: Solve for \( m \)

Set each factor equal to zero:

• If \( m+60 = 0 \), then \( m=-60 \). But time cannot be negative, so we discard this solution.
• If \( m - 40=0 \), then \( m = 40 \).

Answer:

B. 40 minutes