Student's solution for a math problem calculating the individual printing times for two printers based on their combined work rate.

Problem Statement:

Two printers, \( A \) and \( B \), are available to print the school newspaper. It takes Printer \( A \) twice as long to print the newspaper as it takes Printer \( B \). If the two printers can print a copy of the school newspaper in 3 hours when working together, how long would it take each printer alone to print a copy of the school newspaper?

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Solution:

#### Step 1: Define Variables
Let:
- \( x \): Time (in hours) it takes Printer \( B \) to print one copy of the newspaper.
- Since Printer \( A \) takes twice as long as Printer \( B \), the time for Printer \( A \) is \( 2x \).

#### Step 2: Work Rates
The work rate of a printer is the fraction of the job completed per hour. Therefore:
- Printer \( A \)'s work rate: \( \frac{1}{2x} \) (since it takes \( 2x \) hours to complete one job).
- Printer \( B \)'s work rate: \( \frac{1}{x} \) (since it takes \( x \) hours to complete one job).

When both printers work together, their combined work rate is the sum of their individual work rates:
\[
\text{Combined work rate} = \frac{1}{2x} + \frac{1}{x}
\]

#### Step 3: Combined Work Rate Equation
We are given that the two printers together can complete the job in 3 hours. This means their combined work rate is \( \frac{1}{3} \) (since they complete 1 job in 3 hours). Therefore:
\[
\frac{1}{2x} + \frac{1}{x} = \frac{1}{3}
\]

#### Step 4: Solve the Equation
To solve for \( x \), first find a common denominator for the fractions on the left-hand side. The common denominator is \( 2x \):
\[
\frac{1}{2x} + \frac{1}{x} = \frac{1}{2x} + \frac{2}{2x} = \frac{1 + 2}{2x} = \frac{3}{2x}
\]
So the equation becomes:
\[
\frac{3}{2x} = \frac{1}{3}
\]

Next, cross-multiply to solve for \( x \):
\[
3 \cdot 3 = 1 \cdot 2x
\]
\[
9 = 2x
\]
\[
x = \frac{9}{2} = 4.5
\]

#### Step 5: Interpret the Results
- Printer \( B \) takes \( x = 4.5 \) hours to print one copy of the newspaper.
- Printer \( A \) takes \( 2x = 2 \times 4.5 = 9 \) hours to print one copy of the newspaper.

#### Step 6: Verify the Solution
If Printer \( A \) takes 9 hours and Printer \( B \) takes 4.5 hours, their work rates are:
- Printer \( A \): \( \frac{1}{9} \)
- Printer \( B \): \( \frac{1}{4.5} = \frac{2}{9} \)

Their combined work rate is:
\[
\frac{1}{9} + \frac{2}{9} = \frac{3}{9} = \frac{1}{3}
\]
This matches the given information that they complete the job in 3 hours when working together. Thus, the solution is verified.

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Final Answer:

\[
\boxed{9 \text{ hours (Printer A)}, 4.5 \text{ hours (Printer B)}}
\]