How to find the elapsed time for a medication when two people have to met a certain dose?

The problem is as follows:

In a children's hospital in Tokyo a doctor has prescribed to two kids a medication to alleviate their sore throat infection. According to this Hiroto and Midori must begin a treatment taking an amoxicillin oral suspension. Hiroto must take $$10$$ mililiters each $$8$$ hours and Midori must take $$7$$ mililiters each $$6$$ hours. They begin their first dose together and they will finish the treatment when the two of them have taken exactly two and a half bottles of this medicine. Assume that each bottle of amoxicillin has a content of 120 militers. Find how long the treatment of Midori lasted.

The alternatives given in my book are as follows:

$$\begin{array}{ll} 1.&\textrm{84 hours}\\ 2.&\textrm{108 hours}\\ 3.&\textrm{96 hours}\\ 4.&\textrm{114 hours}\\ \end{array}$$

I'm confused exactly how to approach this problem, and this arises for the fact that when you take a medication which lasts for a certain time, you must account for each time interval. But the number of doses is those intervals increased by one. And this must be met in order to avoid falling into the telephone post error.

Then this means when the two kids took their dose as follows:

Hiroto:

$$10+\frac{10}{8}t$$

Midori:

$$7+\frac{7}{6}t$$

This is the part where it comes the right interpretation from the words, the two of them have taken the two and a half bottles of that medication.

I'm assuming that those two quanitities must be added in order to get the requested time. For both kids the elapsed time is the same.

Two and a half bottles of that medicine is:

$$120\times 2 + 120\times \frac{1}{2}=240+60=300$$

Then:

$$7+\frac{7}{6}t+10+\frac{10}{8}t=300$$

But this reduces to:

$$\frac{29}{12}t=283$$

Thus the time is:

$$t=\frac{283\cdot 12}{29}$$

But needless to say that this answer does not appear on any of the alternatives. What could be happening here?. Is it just me or what?. Can someone help me here to solve this problem?.

I think I did the right assumptions, but I could be wrong. It would help a lot that the answer could contain a wordy and highlighted explanation of the right interpretation so I can catch up where am I not getting this right.

It does look like there is a problem with the question.

Your functions are continuous, but the dosing is discrete. In other words, your functions are not correct to give the amount received as a function of $$t$$. The correct functions are step functions, not linear functions.

Still, you get close to the right answer, as your functions are similar to the actual dosage step functions. Your answer is a bit over $$117$$ hours, so lets see what the kids have gotten after $$114$$ hours.

By $$t=114$$ hours, Hiroto has taken $$10$$ml at times $$t=0,8,16,\dots112$$, which is $$15$$ doses or $$150$$ml. By the same time, Midori has taken $$7$$ml doses at times $$t=0,6,12,18,\dots,114$$, which is $$20$$ doses, or $$140$$ml. At this point, $$290$$ml of medication has been given.

Six hours later (at $$t=120$$), it’s time for both kids’ next dose, which will take the amount of medication delivered over $$2.5$$ bottles, so it doesn’t seem like it’s possible for there to be a specific point where the children have gotten a total of $$2.5$$ bottles. If when they are scheduled to take doses at the same time, Hiroto goes first, and then the $$2.5$$ bottles run out before Midori’s $$t=120$$ dose, perhaps we could say that Midori's treatment ended after $$114$$ hours. (Hiroto’s treatment ended after $$120$$ hours, but the question did not ask about this.)

Added, in response to a comment: Using Midori as the example, you give this function for the amount of medication received after $$t$$ hours: $$m(t)=10+\frac{10}{8}t$$. As I noted, this is incorrect, because she isn’t given medication continuously. Instead, after $$t$$ hours, she has received $$10$$mg plus $$10$$ additional milligrams for each complete $$8$$-hour period. The correct function is $$m(t)=10+8\displaystyle\left\lfloor{t\over10}\right\rfloor$$, where $$\lfloor\cdot\rfloor$$ is the “floor” or “greatest integer” function.

However, there’s no easy way to solve an equation like $$7+7\displaystyle\left\lfloor{t\over6}\right\rfloor+10+8\displaystyle\left\lfloor{t\over10}\right\rfloor=300$$ in a similar way to solving $$7+\frac{7}{6}t+10+\frac{10}{8}t=300$$

• I'm very sorry I omitted the piece of information from the problem. Yes this was given as a clue. Therefore this is why I made my equations such way. However I'm still confused. If at $114$ hours both kids consumed together $290$ mililiters. Wouldn't it mean that six hours later Midori also consumes her 7 militers as it has elapsed a time which is multiple of $6$?. I mean both coincide with this, and therefore it will mean that Hiroto consumed those $10$ militers and Midori those 6 thus it will make $290+16=306$ so this part is confusing. Commented Feb 27, 2021 at 2:45
• Can you edit your answer to reflect my update and attend my question, please? Commented Feb 27, 2021 at 2:47
• Thanks I'm accepting your answer. I've learned something new from your answer. I wonder, which maybe out of the scope of the question but could a step function be made to solve this problem using such approach?. I'm asking this because you mentioned that my function is linear not discrete, thus this has awaken my curiosity. Commented Feb 28, 2021 at 0:06
• By the way, there is a related problem with this maybe can you help me with that?. I'll appreciate if you could because your answer to this question was very clear to me now. Commented Feb 28, 2021 at 0:12
• I added a bit to my answer in response to your comment. Commented Feb 28, 2021 at 15:16