How does one practice mindfulness during thinking - e.g. thinking about how to solve a problem? This question is in context of practicing mindfulness in one's day-to-day activities (e.g. walking, talking, washing dishes etc.), where one is supposed to be "aware of" / "observe" / "be mindful of" these activities.

But how does one be "mindful" during thinking, since the very act of "observing" thinking would interrupt the thought process? And, say, if the thought process is about solving a specific problem, the very act of "being aware of /observing" the thoughts would result in one being unable to effectively solve that problem - when compared to not observing/ not being mindful about those thoughts. Or, be unable to solve it all.

The same question for when reading something very interesting, or studying, where one is fully engrossed? Wouldn't being mindful about these activities interrupt the reading/study?

New to mindfulness here. Apologies if the question sounds foolish, or irrelevant due to my misunderstanding of mindfulness.

Thanks in advance.
A.K. 
It's not a foolish question at all,  I think it's something everyone spends sometime trying to understand when they first start learning meditation and mindfulness. 

Mindfulness is being present.  If you're sitting you know you're sitting, if you're standing you know you're standing. If you are washing dishes you are only washing dishes, you are not thinking about the what happened at work, about the bills, or about any problems you may have had with your spouse or friend. Eventually with enough practice you can stay present while sleeping though I make no claims on being able to do this. 

This can extend to studying, shopping, for groceries, driving, etc.  I meditate most days while driving into work, as I'm completely mindful of what I'm doing and undistracted,  I'm probably the safest driver on the road. As an "intellectual worker", I stay mindful and undistracted with the work I do.  I think many people do this and do not understand they are being mindful. Mindfulness is not "non-thinking" though it can be as being present doesn't require thinking, it's being completely present with and aware of what you are doing. Decisions can be part of the process but should be void of I or my, rather the appropriateness of the decision to completing the task in the physical world is the factor of action. 

A suggestion is that you don't draw a distinction between being mindful in meditation and when you get up from the mat, try to find things to extend your mindfulness to as the day goes on, pick simple things at first like taking a shower, going to the bathroom, driving, and start working on extending it through the entire day.  You will find things like basic planning and paying bills will take care of themself to a large extent.  You won't forget your social security number or to pay the bills..no worries. It is also a process that takes years -> decades, etc to fully develop but it is well worth the effort. 

Addendum: I realized I hadn't addressed part of your question. About being mindful with "engrossing" or something or really interesting. This is probably the unpopular part of the answer but truth is just what it is.  For monastics, entertainment is not allowed.  The reason for this isn't that it's evil, but because by design it "hooks you",  creating desire (i.e. Dukka - suffering).  There is no problem with experience happiness or suffering - but mindfulness is to help you not cling to good things "engrossing" and desire (Dukka - suffering) push away bad things.  If you are fairly well down the path you may be able to do this quite easily without suffering, but most of us aren't.  So if you want to know how "engrossing" things cause suffering, imagine being denied to watch the next season of "Game of Thrones" if you are a fan.  This is a gross example, there are much more subtle forms of suffering also, things that re-enforce I and my. So though some could be mindful watching "Game of Thrones" without desire/attachment/suffering, it's really tough for most of us.  

2nd Addendum:
I believe there are two different issues that are being mixed, one is thinking or concentration, the other is actively encouraging suffering. When you are mindful,  you are not bored as you are present and observing, and as noted taking action (decisions and thinking) such as braking and not hit the stopped car in front of you when driving. Working on a really hard problem is not much different than washing dishes and deciding to wipe more or not, except that it's more likely to be iterative. If you are really mindful, it's not necessarily more focused as you can be really focused on simple tasks as well as difficult problems. Working difficult problems does not encourage I or my  (dukka - suffering), unless of course you scold or complement yourself on how stupid or smart you are.  

The case of entertainment again is something designed to make you unmindful, to rouse desire and emotions (dukka), if they didn't then shows like the "Out of Focus Pages of the Telephone Book" would be very popular.  The "designed to make you unmindful" part again doesn't make it evil - it's just with Buddhism the long game is rejiggering your mind, taking advantage of neuroplasticity to break life long habits that lead to suffering.  If you meditate alot, you will find watching something like GoT will make it really difficult to meditate for sometime afterwards. You find that avoiding things like this make you a more contented person.  I know that if you watched GoT in a completely mindful way, it would be like having a teflon mind, nothing would stick, but you would also not get the "high" (or "low") from it, you would not feel jubilation when the bad guy gets killed or depression when the good guy gets killed - you would just be observing and noting.  Renunciation is a process, you don't do it in a day, it happens naturally while developing your practice. It turns out not to be a sacrifice or hardship when you really understand on a deeper level more than an intellectual level within yourself that you're really gaining peace by doing so. In the mean time go and enjoy GoT and note how if it disturbs your mind and do not take my word for it. If you find this is true as you develop a practice (meditation,etc), it will make you naturally draw back from watching other programs of this sort in the future. 

3rd Addendum: From your last comment, I'll try my best to answer but I admit I am working somewhat out of my pay grade so take it as as a potentially flawed answer.  The purpose of meditation is as a vehicle to enlightenment which is the end of suffering by extinguishing the ego.  It is the ego that causes suffering,  not consciousness and not emotions.  With or without the ego emotions are still with us, it's just without the ego there is no suffering because the emotions are experienced without clinging and aversion. You can be happy, but as it passes since everything is temporary, you don't try to hold on to it or make it last longer, and when you're unhappy, you're not pushing this away. Without an ego there is no one to suffer. So an enlightened person can laugh and cry and is not an emotionless robot.  The difference is that there is no clinging or aversion. 

An enlightened person may not detached from life at all, or may detach completely.  Though the next comment may cause some ire of others, I believe it to be true.  Many persons who very likely were enlightened have done things that people thought were unenlightened.  Examples are Suziki Roshi crying when his wife died, he mourned for a few days and some criticized him for it asking how could he be enlightened and do such a thing. He later commented that he mindfully observed his mourning. I've gone to a lecture with the Dalai Lama and he is very engaged, he often tells funny stories and jokes. Some of the best dharma talks I've heard could almost have been standup comedy routines. Metta meditation intentionally works to create compassion and raise positive emotions as does Tonglen meditation.  Probably more controversial would be Trungpa Rimpoche who drank, smoked, and was a womanizer, he never tried to hide any of this. You'd have to study his life and his work and try to figure out if you think he was authentic or not. I do personally. It's thought that this was at least in part to confound the minds of his followers as to what an enlightened being was and break their wrong preconceived notions.  When you read his writings and look at his accomplishments they were authentic and significant.  It would be tough to have much more of a less (apparently) detached life than his. 

Monks are students, just like the lay Buddhist, it's just they've pledged their life exclusively to the path.  If they actually become a fully realized being, there are no longer bound by restrictions as those restrictions were only there to help them become a fully realized being. 

I hope I answered you're question rather than creating more confusion. 

I hope this helps..