Working Together: Combined Rate Problems

What This Problem Teaches

• Converting individual work times to work rates (jobs per unit time)
• Understanding that work rates add when people work simultaneously
• Applying the combined work formula: time = 1 ÷ (sum of individual rates)
• Recognizing the relationship between rates and reciprocals
• Working with fractions in practical contexts and converting to decimals

Solution: Method 1 — The Rate Addition Approach

The key insight is that work rates add directly when people work together. A work rate tells us what fraction of the job gets completed per hour.

Step 1 — Find each person's work rate

If Mandy completes 1 chapter in 5 hours, her rate is:

If Phillip completes 1 chapter in 11 hours, his rate is:

Step 2 — Add their work rates

When working together, their combined rate is the sum of their individual rates:

To add these fractions, we need a common denominator. The LCD of 5 and 11 is 55:

Step 3 — Find the time for one complete chapter

If they work at 16/55 chapters per hour, then to complete 1 chapter takes:

Step 4 — Convert to decimal and round

Converting the fraction to decimal:

Rounded to two decimal places: 3.44 hours

Solution: Method 2 — The Combined Work Formula

There's a shortcut formula for two-person work problems that comes from the algebra we just did.

Step 1 — Apply the harmonic mean formula

For two people with individual times t₁ and t₂, their combined time is:

Step 2 — Substitute our values

With Mandy's time t₁ = 5 hours and Phillip's time t₂ = 11 hours:

Step 3 — Calculate the result

This formula works because it's algebraically equivalent to adding the reciprocal rates, but it's often faster to apply directly.

Verification

Let's verify this makes sense by checking that in 3.44 hours, they complete exactly one chapter.

Check Mandy's contribution

In 3.44 hours at 1/5 chapters per hour, Mandy completes:

Check Phillip's contribution

In 3.44 hours at 1/11 chapters per hour, Phillip completes:

Verify the total

The slight overage (0.001) is due to rounding 3.4375 to 3.44. Using the exact value 55/16 gives exactly 1 chapter.

Does This Seem Reasonable?

Let's do a sanity check on our answer of 3.44 hours.

The combined time should be less than either individual time: ✓ Our answer (3.44 hours) is less than both Mandy's time (5 hours) and Phillip's time (11 hours). This makes sense — two people working together should finish faster than either person alone.

The combined time should be more than half the faster person's time: ✓ Mandy is faster at 5 hours, and half of that would be 2.5 hours. Our answer (3.44 hours) is greater than 2.5 hours, which is correct — Phillip is slower than Mandy, so their combined time can't be as good as having two Mandys.

Comparison to the wrong answer: Students often mistakenly average the times: (5 + 11) ÷ 2 = 8 hours. But this would mean two people working together take longer than either person alone — which is absurd!

Common Pitfalls

The General Pattern

Work rate problems follow a consistent structure that applies whether you have two people, three people, or even machines working together.

For exactly two workers with times t₁ and t₂, this simplifies to the harmonic mean formula:

How to Spot This Problem Type

Work rate problems have distinctive language patterns. Watch for these phrases:

• "How long would it take them working together?" — This is the classic question
• "It takes [person A] X hours and [person B] Y hours" — Individual times given
• "Working at the same rate..." — Different phrasing for the same concept
• "If they work together..." — Signal that rates should be combined

The key structural clue is that you're given individual completion times and asked for the combined completion time. This immediately tells you to convert to rates, add them, then convert back to time.

You'll see this same mathematical structure in physics (parallel resistors), computer science (parallel processing), and economics (production capacity).

Real Applications

• Software development: Estimating how long a coding project takes when multiple programmers work on different modules simultaneously
• Manufacturing: Determining production capacity when running multiple assembly lines or machines in parallel
• Medicine: Calculating drug clearance rates when multiple organs (kidneys, liver) eliminate a medication from the bloodstream
• Network engineering: Computing total bandwidth when multiple internet connections are load-balanced