Combined Work Rate: Two Printers

What You Will Learn

• Converting completion times into work rates (jobs per minute)
• Adding work rates to find combined productivity
• Using the reciprocal relationship between rate and time
• Working with fractions in practical applications
• Understanding why you can't simply average completion times

Solution: Method 1 — The Combined Rate Approach

The key insight is that when machines work together, we add their work rates, not their completion times. Let's convert each machine's performance into a rate.

Step 1 — Convert completion times to work rates

Work rate tells us what fraction of the job gets completed per minute:

Step 2 — Add the individual rates

When working together, their rates combine:

Step 3 — Find a common denominator

To add these fractions, we need a common denominator. The LCM of 10 and 16 is 80:

Step 4 — Convert the rate back to time

To find how long it takes to complete 1 dissertation, we take the reciprocal:

Converting to minutes and seconds: 80/13 = 6 + 2/13 ≈ 6 minutes and 9 seconds

Solution: Method 2 — Work Completed Per Minute (Tabular)

Sometimes it helps to visualize exactly how much work each printer completes each minute. Let's track their progress in a table.

Step 1 — Calculate work completed per minute

Express each printer's contribution as a fraction of the whole job:

Step 2 — Track cumulative progress

Let's see how much of the dissertation is completed after each minute:

Step 3 — Solve for exact completion time

We need to find when the total work equals exactly 1 dissertation:

Verification

Let's verify our answer by checking that the combined work in 80/13 minutes equals exactly one complete dissertation.

Check each printer's contribution

Verify the total

Double-check with decimals

Does This Seem Reasonable?

Let's step back and see if our answer makes intuitive sense.

The OfficeJet alone takes 10 minutes, and the LaserJet alone takes 16 minutes. When working together, they should definitely finish faster than either machine working alone.

What if both printers were equally fast? If both took 10 minutes alone, together they'd take 5 minutes (half the time). Since the LaserJet is slower than the OfficeJet, we'd expect the combined time to be slightly more than 5 minutes. Our answer of 6.15 minutes fits this expectation.

The tempting but wrong approach would be to average the times: (10 + 16)/2 = 13 minutes. This is clearly wrong because two machines working together should be faster than either one alone, not slower than both!

Watch Out For These

The General Formula

For any two-machine work problem, there's a reliable pattern you can memorize.

Applying the formula to our problem

Why this formula works

The formula comes directly from adding rates:

Real Applications

This type of rate-combination calculation appears frequently in real work scenarios:

Network downloads: When downloading a file simultaneously from multiple servers, your total download rate is the sum of the individual connection rates. A 100MB file downloading at 2MB/s from one server and 3MB/s from another will complete in 100/5 = 20 seconds.

Manufacturing and assembly lines: When multiple production lines make the same product, plant managers use this math to calculate total output capacity and plan delivery schedules.

Medical drug clearance: In pharmacology, when the body eliminates a drug through multiple pathways (kidney and liver), the total clearance rate is the sum of the individual clearance rates—the same mathematical structure as our printer problem.

What If?