Convert completion times before combining independent work
Two completion times cannot generally be added or averaged to find the time of simultaneous work. First express each worker or machine in the same output unit per time unit. The model must say that the outputs are independent, count toward the same total and do not duplicate one another. Without those assumptions, a combined-rate calculation may not describe the situation. Worked example: one printer completes 120 pages in 8 minutes, while another completes 120 pages in 12 minutes. Their rates are 15 pages/minute and 10 pages/minute. If they print different pages of one 120-page job simultaneously at those constant rates, with no startup delay, the combined rate is 25 pages/minute. The job takes 120 pages ÷ 25 pages/minute = 4.8 minutes. At that time the first contributes 72 pages and the second 48. The average of the original times is 10 minutes, which does not model simultaneous contributions. Adding the times gives 20 minutes, which describes neither printer sharing this job in the stated way. Adding the rates works because each minute contributes 15 plus 10 new pages. The output unit in the numerator is essential: adding pages/minute to minutes/page would be meaningless. A check can multiply the proposed elapsed time by each rate and add the outputs. It should recover the required total under the stated model. Fractional minutes are permitted in our exercise; if an additional whole-batch or startup condition were supplied, it would require a different model. These are fictional rates for reasoning practice, not claims about actual printer performance.
Printer A can print a 120-page job in 4 minutes; printer B can print the same size job in 6 minutes. For this arithmetic model, both work simultaneously at constant rates on different pages of one 120-page job, with no startup time. Fractional minutes are allowed. How long does the job take?
- A prints 120/4 = 30 pages/minute and B prints 120/6 = 20 pages/minute. Together they print 50 pages/minute because their outputs are independent and distinct. The 120-page job therefore takes 120/50 = 2.4 minutes. At that time A prints 72 pages and B 48, totaling 120.
Compare the other answer choices
This treats the slower printer as if it also matched A and simply halves the faster completion time.
Ten adds the times as if both complete a full job sequentially, contrary to the model.
Five is the average of the two completion times, not the time of simultaneous combined work.
Try it, then explain it.
Complete a short set without looking at the example. After each answer, explain which evidence or rule supports your decision and why the alternatives do not fit.
Sources & authorship
Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.
ASVAB SubtestsSolve Proportions and their Applications