Work Rate Problem: Calculate Job Completion Time with Staggered Start

Work Rate 9th-10th Grade
PROBLEM
One copier can do a job in 42 minutes. A newer machine can do the job in 30 minutes. If the older machine works for 5 minutes before being joined by the new machine, how long after the new machine joins will the job be finished?
One copier can do a job in 42 minutes. A newer machine can do the job in 30 minutes. If the older machine works for 5...

What This Problem Teaches

  • Converting completion times to work rates using reciprocals
  • Setting up equations for sequential work phases
  • Adding work rates when machines operate simultaneously
  • Solving fractional equations with multiple unknowns
  • Understanding how staggered starts affect total completion time

Solution: Method 1 — The Work Rate Approach

Step 1 — Find each machine's work rate

Work rate equals 1 job divided by the time to complete that job.

Old copier rate = 1/42 jobs per minute
New copier rate = 1/30 jobs per minute

Step 2 — Calculate work completed during the solo phase

The old copier works alone for 5 minutes at its rate of 1/42 jobs per minute.

Work completed alone = (1/42) × 5 = 5/42 of the job

Step 3 — Find the remaining work

After the old copier works alone, some portion of the job remains.

Remaining work = 1 - 5/42 = 42/42 - 5/42 = 37/42 of the job

Step 4 — Calculate the combined work rate

When both machines work together, their rates add.

Combined rate = 1/42 + 1/30
= 5/210 + 7/210
= 12/210 = 2/35 jobs per minute

Step 5 — Solve for the remaining time

Use the equation: Remaining work = Combined rate × Time together

37/42 = (2/35) × t
t = (37/42) ÷ (2/35)
t = (37/42) × (35/2)
t = (37 × 35)/(42 × 2) = 1295/84

Step 6 — Simplify the answer

Convert the fraction to a mixed number or decimal.

t = 1295/84 = 15 25/84 minutes
t ≈ 15.42 minutes

Solution: Method 2 — The Remaining Job Equation

Step 1 — Set up the work completion equation

Let t = time after new machine joins. The total work equals 1 job.

(Old copier work alone) + (Combined work together) = 1
(1/42) × 5 + (1/42 + 1/30) × t = 1

Step 2 — Simplify the solo work term

5/42 + (1/42 + 1/30) × t = 1

Step 3 — Find the common denominator for the combined rate

1/42 + 1/30 = 5/210 + 7/210 = 12/210 = 2/35

Step 4 — Substitute and solve

5/42 + (2/35)t = 1
(2/35)t = 1 - 5/42
(2/35)t = 42/42 - 5/42 = 37/42

Step 5 — Isolate t

t = (37/42) × (35/2) = 1295/84 ≈ 15.42 minutes
The new machine must work with the old machine for approximately 15.42 minutes (or exactly 15 25/84 minutes) to complete the job.

Verification

Let's check our answer by calculating the total work completed.

Old copier alone for 5 minutes:
Work = (1/42) × 5 = 5/42 of the job
Both machines together for 1295/84 minutes:
Work = (2/35) × (1295/84) = (2 × 1295)/(35 × 84) = 2590/2940
Simplifying: 2590/2940 = 37/42 of the job
Total work:
5/42 + 37/42 = 42/42 = 1 complete job ✓

Common Pitfalls

✗ Averaging the completion times
Students often think: "42 + 30 = 72, so together they take 72/2 = 36 minutes." This is wrong because you add rates, not times. The correct combined time starting together would be 35/2 = 17.5 minutes.
✗ Forgetting about the solo work phase
Some students calculate only the combined work time and forget that the old copier already completed 5/42 of the job alone. This leads to solving (2/35)t = 1 instead of (2/35)t = 37/42.
✗ Incorrect fraction arithmetic
When finding 1/42 + 1/30, students might add numerators and denominators: "1/42 + 1/30 = 2/72." Always find a common denominator: LCM(42,30) = 210, giving 5/210 + 7/210 = 12/210 = 2/35.

The Underlying Pattern

All staggered work problems follow this general structure:

(Rate₁ × Time_alone) + (Rate₁ + Rate₂) × Time_together = 1 job

Where Rate = 1/completion_time. This formula works for any number of workers joining at different times. The key insight is that work rates are additive when machines or workers operate simultaneously.

Important: This approach only works when the rates remain constant. If machines slow down due to wear, overheating, or shared resources, you need more complex models.

Where This Shows Up in Real Life

  • Manufacturing: Production lines where additional machines come online to meet deadlines
  • Computing: Parallel processing where multiple CPUs work together, or when additional servers join a computational task
  • Construction: Projects where additional crews join partway through to accelerate completion

What If?

1
Different Solo Time
The old copier (42 min) works alone for 8 minutes before the new copier (30 min) joins. How long after the new machine joins will the job be finished?
Step 1 — Calculate work completed alone

Old copier works at rate 1/42 jobs/min for 8 minutes: (1/42) × 8 = 8/42 = 4/21 of the job

Step 2 — Find remaining work

Remaining = 1 - 4/21 = 17/21 of the job

Step 3 — Calculate combined rate

Combined rate = 1/42 + 1/30 = 5/210 + 7/210 = 2/35 jobs/min

Step 4 — Solve for remaining time

(2/35)t = 17/21
t = (17/21) × (35/2) = 595/42 ≈ 14.17 minutes

Verification

Total work: 4/21 + (2/35) × (595/42) = 4/21 + 17/21 = 1 ✓

2
Reverse the Unknown
The older copier (42 min) starts alone. After some time, the newer copier (30 min) joins, and together they finish the job 25 minutes after the older one started. How long did the older copier work alone?
Step 1 — Set up the equation

Let s = solo time for old copier, 25-s = time working together

Step 2 — Write work equation

(1/42)s + (2/35)(25-s) = 1

Step 3 — Expand and simplify

s/42 + 50/35 - (2s)/35 = 1
s/42 + 10/7 - 2s/35 = 1

Step 4 — Solve for s

Common denominator 210: 5s + 300 - 12s = 210
-7s = -90, so s = 90/7 ≈ 12.86 minutes

Answer

The old copier worked alone for 90/7 ≈ 12.86 minutes

3
Three Machines
Machine A (60 min) works alone for 10 minutes. Machine B (40 min) joins for 5 minutes. Then Machine C (30 min) joins both. How long do all three work together to finish?
Step 1 — Calculate Phase 1 work

Machine A alone: (1/60) × 10 = 1/6 of the job

Step 2 — Calculate Phase 2 work

A & B together: (1/60 + 1/40) × 5 = (2/120 + 3/120) × 5 = (5/120) × 5 = 25/120 = 5/24

Step 3 — Find remaining work

Completed so far: 1/6 + 5/24 = 4/24 + 5/24 = 9/24 = 3/8
Remaining: 1 - 3/8 = 5/8

Step 4 — Calculate three-machine rate

Combined rate = 1/60 + 1/40 + 1/30 = 2/120 + 3/120 + 4/120 = 9/120 = 3/40

Step 5 — Solve final phase

(3/40)t = 5/8
t = (5/8) × (40/3) = 200/24 = 25/3 ≈ 8.33 minutes

4
Machine Breakdown
Both copiers start together and work for 6 minutes. Then the new machine (30 min) breaks down and the old machine (42 min) must finish alone. How much longer does the old machine work by itself?
Step 1 — Calculate combined work phase

Combined rate = 1/42 + 1/30 = 2/35 jobs/min
Work completed together: (2/35) × 6 = 12/35

Step 2 — Find remaining work

Remaining work = 1 - 12/35 = 23/35 of the job

Step 3 — Calculate old machine solo time

Old machine rate = 1/42 jobs/min
Time needed: (23/35) ÷ (1/42) = (23/35) × 42 = 966/35

Step 4 — Convert to mixed number

966 ÷ 35 = 27 21/35 = 27 3/5 = 27.6 minutes

Answer

The old machine works alone for 27.6 minutes

Frequently Asked Questions

How do you solve work rate problems when machines start at different times? +
Calculate each machine's work rate (jobs per minute), find how much work the first machine completes alone, then set up an equation for the remaining work when both machines work together. In this problem, the old copier completes 5/42 of the job alone, leaving 37/42 for both machines working at their combined rate of 2/35 jobs per minute.
What's the formula for combined work rates? +
Add the individual rates: if machine A completes 1/a jobs per minute and machine B completes 1/b jobs per minute, together they complete (1/a + 1/b) jobs per minute. Here, the combined rate is 1/42 + 1/30 = 2/35 jobs per minute.
Why can't you just average the completion times in work rate problems? +
Averaging times gives the wrong answer because rates add, not times. Two machines that take 42 and 30 minutes individually don't work together in (42+30)/2 = 36 minutes. Instead, their combined rate is 1/42 + 1/30 = 2/35 jobs per minute, completing the job in 35/2 = 17.5 minutes when starting together.
NJ
Neven Jurkovic, PhD

Professor of Computer Science, Palo Alto College, Alamo Colleges District, San Antonio, TX

Developer of Algebrator

Contact

This solution was prepared with AI assistance and reviewed by Dr. Jurkovic for mathematical accuracy and pedagogical clarity.

2026-08-10

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