It would take a Brooklyn native 10 seconds to tell a stranger they’re from New York, and it would take a Manhattan native 2 seconds to tell a stranger they’re from New York. If the two New Yorkers were able to combine their speaking abilities, how many seconds would it take both natives working together, at their respective rates, to tell a stranger they’re from New York?
This is a group work problem in which we must determine how long it would take multiple entities working together at differing rates to complete a task. To do this, we must determine what fraction of the job each New Yorker will complete in a certain amount of time. For example, given 10 seconds, the Brooklyn native would complete the task at hand. But given 5 seconds, he would only finish 5/10, or 1/2 of the job. We can see we simply divide the time elapsed by how long it takes to finish the job and this tells us the fraction of the job completed by either individual. Thus in t seconds,
represents the fraction of the task the Brooklyn native will complete. Similarly,
represents the fraction of the task the Manhattan native will complete given the same t seconds.
As a result, we can say that if we add together the fractions of the job each New Yorker completes in t seconds, we want them to finish telling the stranger they’re from New York, or finish 1/1 of the job:
To add fractions together, we need a common denominator, in this case 10.
As a result, we see that the two New Yorkers magically combining their speaking abilities together would allow them to inform a stranger of their roots in a mere 5/3 seconds.
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