Solve Work-Rate Problems: Individual vs. Combined Time

Skills This Problem Builds

• Converting between work rates and work times using reciprocals
• Setting up equations where rates add when workers collaborate
• Solving rational equations with variables in denominators
• Interpreting relationships like "twice as long" in mathematical terms
• Verifying answers through substitution and logical reasoning

Solution: Method 1 — The Rate Addition Approach

The key insight is that work rates add when multiple workers collaborate. If we know how much of a job each worker completes per unit time, their combined rate is simply the sum of individual rates.

Step 1 — Define the variable

Let t = the time (in minutes) for the newer computer to send the email alone.

Since the older computer takes twice as long, it needs 2t minutes to send the email alone.

Step 2 — Express individual work rates

Work rate = jobs completed per unit time = 1 ÷ (time for one complete job)

• Newer computer's rate: 1/t jobs per minute
• Older computer's rate: 1/(2t) jobs per minute

Step 3 — Set up the combined work equation

When working together, their rates add: 1/t + 1/(2t) jobs per minute.

In 10 minutes, they complete exactly 1 email, so:

Step 4 — Solve for t

First, simplify the rate sum by finding a common denominator:

Substitute back into our equation:

Step 5 — Find the older computer's time

The older computer takes 2t = 2(15) = 30 minutes to send the email alone.

Solution: Method 2 — The Fraction-of-Job Approach

Instead of thinking about rates, we can think about what fraction of the total job each computer completes during the 10-minute collaboration.

Step 1 — Set up fraction variables

Let x = the fraction of the email the newer computer sends in one minute.

Since the older computer is half as fast, it sends x/2 of the email in one minute.

Step 2 — Write the 10-minute equation

In 10 minutes working together, they complete the entire email (100% = 1):

Step 3 — Convert to individual times

If the newer computer sends 1/15 of the email per minute, it needs 15 minutes for the complete job.

If the older computer sends 1/30 of the email per minute, it needs 30 minutes for the complete job.

Verification

Let's check that our answer is correct by substituting back into the original conditions:

Also verify the "twice as long" relationship: 30 ÷ 15 = 2 ✓

Both conditions are satisfied, confirming our answer is correct.

Where Students Go Wrong

The General Formula

For any two-worker problem where one worker takes time a and the other takes time b, their combined time T follows the harmonic mean formula:

In our problem: a = 15, b = 30, so T = (15 × 30)/(15 + 30) = 450/45 = 10 ✓

This formula works for any number of workers: the reciprocal of the combined time equals the sum of the reciprocals of individual times. This same mathematics appears in electrical circuits (parallel resistors) and fluid flow (parallel pipes).

Why This Matters

Project management: Estimating completion times when assigning tasks to team members with different productivity levels.

Manufacturing: Calculating throughput when multiple machines with different speeds work on the same production line.

Network engineering: Determining data transfer rates when multiple connections with different bandwidths work in parallel.

Healthcare: Modeling how multiple medications with different clearance rates are processed by the body simultaneously.