Yesterday it took 5 cleaners 4 and ½ hours to clean all the rooms in a hotel. There are only 3 cleaners to clean all the rooms in the hotel today. How much time will it take them?

Firstly we need to identify what is the underlying principle of this question and what are the assumptions we have to make. We are given some information about the cleaning process in a hotel. The variables in this question are number of cleaners and the time in which the hotel is cleaned. To be able to deduce something from the given information, we have to assume that all the cleaners work at the same pace every time and the amount of work they have to do is also the same each day. The question doesn’t specify this, so it is expected from us to make these assumptions by ourselves. There is some thought process required to understand the concept of this question. As we assumed above, there is some fixed amount of work, which corresponds to the time it would take 1 cleaner to clean all the rooms in the hotel. We can then imagine that the work is split among more cleaners. So let's consider these cases: a) Only one cleaner cleans the hotel. (To make it more illustrative, assume it's at one go, although this is unrealistic.) It will take him/her some amount of time, which we denote by T. b) More cleaners clean the hotel and they work one at a time, consecutively. We recognize that it is not that different from the case when only one cleaner does all the work, so it should take them the same amount of time in total to clean the hotel. But each cleaner does only some part of the work, and hence each cleaner works for less time. If we distribute the work evenly among the cleaners, then each cleaner works for the same amount of time, and the total time is sum of all these time intervals. The number of time intervals is the same as number of workers, and so we can say that the total time is product of number of workers (denote n) and the time each one of them spends cleaning (denote t_n): T = n * t_n (1). We can rearrange to get the time each one of them spends cleaning, so if we divide both sides of the equation by n and swap the sides, we get: t_n = T/n (2). Here t_n is dependent on n, and we say it is inversely proportional to n, because it is proportional (in the sense that it is the same as the second variable multiplied by some constant) to the (multiplicative) inverse of n. This means the more workers we have, the less time each one of them spends working. c) More cleaners clean the hotel and they all work at the same time. This is the most relevant case for our question. If all the cleaners start working at the same time and all finish at the same time, then they all work for the same amount of time. In some other questions this might not be the case, so it is important to pay attention to it. This case is very closely related to case b). If we have the same number of workers in both these cases, we can even imagine each one of them cleans exactly the same rooms in exactly the same way, and hence it should also take him/her the same amount of time. The only difference is in timing, i.e. when he/she does his/her work. In case a), there was only one cleaner working at a time, whereas in this case, they are all working at the same time. This means that (as mentioned above), they all start and finish at the same time. So if we are interested in the time interval in which the whole hotel is cleaned, it will be the same time interval as the interval during which one cleaner works. So the same equations (1) and (2) apply, and again the time it takes to clean the hotel is inversely proportional to the number of cleaners. This is important to note, because it is common type of relation, and there are various questions involving this. Another common type is direct proportionality, which means the bigger one variable is, the bigger the dependent variable is, and their ratio remains the same. For example the more hours a cleaner works, the more rooms he cleans. Solution and answer: Now we apply what we have deduced to this specific question. We are given the number of cleaners who worked yesterday (denote n₁), and the time it took them to clean the hotel (denote t₁, use hours as units). We are also given the number of cleaners who are working today (denote n₂) and we are asked to find the time it will take them to clean the hotel (denote t₂, use hours as units). So we have: n₁ = 5, t₁ = 4.5 h, n₂ = 3, t₂ = ? We can use equation (1) to find the total time required for one cleaner to clean the hotel, T, (note that we are using slightly different notation now, when t₁ does not mean n = 1), so: T = n₁ * t₁ = 5 * 4.5 h = 22.5 h. Now we can use equation (2) to find t₂: t₂ = T/n₂ = 22.5 h / 3 = 7.5 h. Therefore we conclude that today 3 cleaners will clean all the rooms in the hotel in 7.5 hours.