Work Rate Problem: Staggered Printer Operation

What This Problem Teaches

• Setting up work rate equations when machines operate in different phases
• Converting time-to-complete into fraction-per-minute rates
• Breaking complex scenarios into manageable calculation steps
• Solving fractional equations where the unknown appears in denominators
• Verifying solutions by checking that work phases sum to exactly one complete job

Visualizing the Work Timeline

Solution: Method 1 — The Work Rate Equation

Step 1 — Define the work rates

Let x = time (in minutes) for the smaller printer to complete the job alone.

Step 2 — Calculate work done in Phase 1 (both printers working)

Both printers work together for 10 minutes:

Step 3 — Calculate work done in Phase 2 (small printer only)

After the large printer breaks, the small printer works alone for 50 minutes:

Step 4 — Set up the equation for total work

The sum of work from both phases must equal exactly 1 complete job:

Step 5 — Solve for x

Isolate the term containing x:

Solution: Method 2 — Work Units Approach

Step 1 — Define one complete job as 40 work units

Since the large printer completes the job in 40 minutes, let's say the complete job equals 40 work units. This means:

Step 2 — Calculate units completed in each phase

Phase 1 (both working for 10 minutes):

Phase 2 (small printer alone for 50 minutes):

Step 3 — Set up the equation

Total units must equal 40 (one complete job):

Verification

Let's check that our answer produces exactly one complete job:

With x = 80 minutes for the small printer:

Phase 1 (both working for 10 minutes):

Phase 2 (small printer alone for 50 minutes):

Total work:

The verification confirms our answer is correct.

Common Pitfalls

The Pattern Behind This

This problem follows the general pattern for two-phase work problems:

The key insight is recognizing when a problem has multiple phases with different working conditions. Any time you see "after X minutes, something changes," you're likely dealing with a multi-phase scenario.

For this specific type where one machine breaks down:

Where:

• A = time for machine 1 alone
• B = time for machine 2 alone (unknown)
• t₁ = time both machines work together
• t₂ = time machine 2 works alone after machine 1 breaks

Real Applications

This exact scenario appears in several real-world contexts:

Manufacturing: When a production line has multiple machines and one breaks down mid-shift, requiring calculation of individual machine capacities for maintenance planning.

Network Systems: When servers work in parallel to handle data processing, and one server goes offline partway through a large job batch.

Construction Projects: When multiple crews work together initially, but one crew is reassigned to another project, and you need to determine individual crew productivity rates.

What If?