Question

Neil and Yuki run a data entry service. Neil starts at 9:00am and can type 45 words per minute. Yuki arrives at 10:30am and can type 60 words per minute. Write and solve an inequality to find at what time Yuki will have typed more words than Neil. Let x represent the time in minutes.

Answer 1

270m

Explanation:

Lets define two functions that describe Neil and Yuri productivity, depending on the hours.

As Neil starts at 9 and types 45 words per minute (wpm), he will have typed the following words at the respective hours:

at 9:30 -> 45 * 30 (as he have worked 30 minutes typing 45 wpm)

at 10:00 -> 45 * 60

at 10:30 -> 45 * 90 = 4,050

at 11:00 -> 45 * 120 and so on.

The same we can do for Yuki:

at 11:00 -> 60* 30

at 12:00 -> 60*60

and so on.

Let X be any time of the day measured in minutes after 10:30, which is the minimun time we can consider (our answer wont be a time beofre 10:30 as Yuki will have 0 words typed). This is, if X=20, the hour is 10:30 plus 20 minutes, it is, 10:50. We need to measure this way as its difficult to work with the sexgesimal system.

So, now we can define the wpm production of both Neil and Yuki as a function of the time worked. Lets call Neil production N(X) and it will be equal to the words typed at 10:30 plus the extra words by minute:

N(X) = 4050 + 45X

The same for Yuki. Y(X) is the words typed at 10:30 plus his production per minute:

Y(X) = 0 + 60X = 60X

Now we need Yuki to tyoe more words than Neil, it is, Y(X) > N(X):

60X > 4050 + 45X

Lets subtract 45X in both sides to leave Xs all at one side:

60X - 45X > 4050 + 45X - 45X

15X > 4050

Dividing both sides by 15:

X > 4050/15

X > 270

So, after 270 minutes Yuki will hace typed more than Neil.