Find Assistant's Time Working Alone

What You Will Learn

• How to convert completion times into work rates (jobs per hour)
• Why work rates add when people collaborate on the same task
• Setting up and solving equations where the unknown appears in a denominator
• The relationship between individual capabilities and team productivity
• How to verify answers using the original rate equation

Solution: Method 1 — The Rate Addition Approach

The key insight is that when people work together, their work rates add up. Let's think of work rates as "fraction of the job completed per hour."

Step 1 — Define the work rates

Let t = the number of hours for the assistant to complete the job alone.

The worker completes the job in 10 hours, so the worker's rate is 1/10 of the job per hour.

The assistant completes the job in t hours, so the assistant's rate is 1/t of the job per hour.

Step 2 — Set up the combined rate equation

When working together, they complete the job in 6 hours. This means their combined rate is 1/6 of the job per hour.

Since rates add when people work together:

Step 3 — Solve for t

To solve this equation, we'll get a common denominator and cross-multiply:

Therefore, t = 15 hours.

Step 4 — Interpret the result

The assistant, working alone, would take 15 hours to cover the parking lot with asphalt.

Solution: Method 2 — The Work Contribution Method

Instead of thinking about rates, let's think about how much work each person contributes during their 6-hour collaboration.

Step 1 — Calculate the worker's contribution

The worker's rate is 1/10 job per hour. In 6 hours of working together, the worker completes:

Step 2 — Find the assistant's contribution

Since the entire job is completed in 6 hours, and the worker does 3/5 of it, the assistant must do:

Step 3 — Calculate the assistant's rate

The assistant completes 2/5 of the job in 6 hours, so the assistant's rate is:

Step 4 — Find time for complete job

If the assistant works at 1/15 job per hour, then to complete 1 full job:

Verification

Let's check our answer by substituting back into the original rate equation:

• Worker's rate: 1/10 job per hour
• Assistant's rate: 1/15 job per hour
• Combined rate: 1/10 + 1/15

Perfect! A combined rate of 1/6 job per hour means they complete the job in 6 hours together, which matches the problem statement. ✓

The Tell-Tale Signs

You'll recognize work rate problems by these key phrases:

• "Working together" or "with help" — signals that rates need to be added
• "Working alone" — you're looking for individual completion times
• Time statements like "can complete in X hours" — convert these to rates of 1/X per hour
• Phrases like "fill a pool," "paint a house," "complete a project" — any task that can be measured as a fraction of completion
• "How long would it take..." — you're solving for time, which is the reciprocal of rate

Common Pitfalls

Mistake 1: Averaging the times

This logic treats time as if it averages linearly, but that's not how work rates combine. A very fast assistant (2 hours) plus the worker (10 hours) would complete the job much faster than 6 hours.

Mistake 2: Confusing rates and times

You can't add completion times directly. The equation should use rates: 1/10 + 1/t = 1/6. Adding times only works for sequential work, not parallel work.

Mistake 3: Forgetting to take the reciprocal

If 1/t = 1/15, then t = 15, not 1/15. The assistant takes 15 hours, not 4 minutes!

The Pattern Behind This

All work rate problems follow the same fundamental principle:

This formula works because rates (jobs per hour) add when workers collaborate, but completion times don't. If you know any two of these three values, you can solve for the third.

Real Applications

This same mathematical structure appears everywhere:

• Computer processing: When two CPUs work on the same task, their processing rates combine exactly like work rates
• Manufacturing: Assembly lines with multiple workers or machines follow these same principles
• Parallel circuits: The reciprocal formula for electrical resistance in parallel is mathematically identical to work rates
• Pharmacokinetics: When multiple organs process a drug simultaneously, their clearance rates add just like work rates

What If?