Finding Individual Work Rate from Combined Rate

What This Problem Teaches

• Rate addition principle: Individual work rates add together to give the combined rate
• Converting between time and rate: If a job takes t hours, the rate is 1/t jobs per hour
• Setting up rational equations: Work problems create equations with variables in denominators
• Solving for unknown rates: Using algebra to find the missing piece of a rate relationship
• Real-world rate reasoning: Understanding how cooperation speeds up task completion

Visualizing the Work Rates

In work problems, think of each person contributing a fraction of the job per hour. These rates add together when people work simultaneously.

Solution: Method 1 — The Rate Addition Approach

Step 1 — Define the variable

Let t = the number of hours it takes Ashley to shovel the driveway alone.

Step 2 — Express individual work rates

Work rate = fraction of job completed per hour:

• Shaun's rate: 1/6 driveway per hour (since he takes 6 hours alone)
• Ashley's rate: 1/t driveway per hour (since she takes t hours alone)

Step 3 — Set up the rate equation

When working together, their rates add up. Since they complete the job in 2 hours together, their combined rate is 1/2 driveway per hour:

Step 4 — Solve for t

Subtract Shaun's rate from both sides:

Find a common denominator (6):

Therefore:

Solution: Method 2 — Work Accomplished Analysis

Step 1 — Calculate work done in the joint time

In the 2 hours they work together, we know the entire driveway gets shoveled. Let's see how much each person contributes.

Step 2 — Find Shaun's contribution

Shaun works at 1/6 driveway per hour. In 2 hours:

Step 3 — Find Ashley's contribution

Since the whole job is done, Ashley must do the remaining portion:

Step 4 — Calculate Ashley's rate and time

Ashley completes 2/3 of the job in 2 hours, so her rate is:

To complete 1 whole driveway at this rate:

Verification

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

This matches their combined rate of 1/2 driveway per hour (finishing in 2 hours). ✓

Alternative check: In 2 hours working together:

• Shaun completes: 2 × (1/6) = 1/3 of the driveway
• Ashley completes: 2 × (1/3) = 2/3 of the driveway
• Total completed: 1/3 + 2/3 = 1 complete driveway ✓

Common Pitfalls

Does This Answer Make Sense?

Ashley taking 3 hours alone is completely reasonable. Here's why:

• She's faster than Shaun: 3 hours vs. 6 hours means Ashley works twice as fast
• The improvement makes sense: Adding a faster worker should significantly reduce the combined time from 6 hours to 2 hours
• Rate comparison: Ashley's rate (1/3 per hour) is indeed twice Shaun's rate (1/6 per hour)
• Boundary check: If Ashley were infinitely fast, they'd finish in close to 0 hours. If she were infinitely slow, they'd still take about 6 hours. Our answer of 2 hours falls reasonably between these extremes.

The General Formula

For any work problem where you know one person's time and the combined time, here's the pattern:

In our problem: a = 6, c = 2, so:

Important limitation: This formula only works when the faster worker's time is less than the combined time. If a ≤ c, the problem setup is impossible.

Where This Shows Up in Real Life

• Computer processing: When multiple CPU cores work on the same task, their processing rates combine similarly
• Manufacturing: Assembly lines with workers of different speeds—knowing combined output to optimize staffing
• Network bandwidth: Combining multiple internet connections doubles your effective download rate
• Medical dosing: When multiple drugs clear from the body at different rates, calculating combined clearance