I'm going in circles with this riddle. It doesn't appear too complicated but I got tangled with the right interpretation of the time mentioned.
The problem is as follows:
At a TV station a program director decides to set a new schedule for the morning news show. He decides that the show is to start after 5 am but before 8 am. If we know that the elapsed time between 5 am until 25 minutes before the show begins is equal to two thirds of the time which will be to 8 am, but in 25 minutes. What time does the show starts?.
The existing alternatives in my book are as follows:
$\begin{array}{ll} 1.& \textrm{7:17 AM}\\ 2.& \textrm{5:37 AM}\\ 3.& \textrm{6:27 AM}\\ 4.& \textrm{5:47 AM}\\ 5.& \textrm{6:17 AM}\\ \end{array}$
What I tried to do was the following. It's a bit tricky but I thought that the unknown time to be $x$ and built the equation from there given these interpretations:
Time elapsed between $25$ minutes before the start of the show and $\textrm{5 AM}$:
$\left(x-\frac{1}{4}\right)-5$
Time which is two thirds which will be to $8$ am but in $25$ minutes:
$8-\left(\frac{2}{3}+\frac{25}{60}\right)$
For this part and as well for the previous I'm working using hours and not minutes so by the end I can get a straight answer.
Since it mentions that both are equal then it is just plugging in together:
$\left(x-\frac{1}{4}\right)-5=8-\left(\frac{2}{3}+\frac{25}{60}\right)$
$x=13+\frac{1}{4}-\frac{2}{3}-\frac{5}{12}$
$x=13+\frac{3-8-5}{12}$
$x=13-\frac{16}{12}=13-\frac{4}{3}=12-1-\frac{1}{3}=11-\frac{1}{3}$
Then I interpreted the last line as:
$11 - \frac{1}{3}\times 60 = 11 - 20m$
Hence the time would be:
$\textrm{10 AM 40 mins}$
But this is clearly outside the boundary which is established on the problem.
Then I thought that the right interpretation would be:
Time which is two thirds which will be to $8$ am but in $25$ minutes:
$\frac{2}{3}\left(8-\frac{25}{60}\right)$
Hence:
$\left(x-\frac{1}{4}\right)-5=\frac{2}{3}\left(8-\frac{5}{12}\right)$
Multiplying by $36$ all:
$36x-9-180=192-10$
$36x=189+182$
$36x=371$
Simplifying the last result yields: $\textrm{1h 18m 20s}$. This result is not even close. Can somebody help me which is the part where I got lost or what I did not interpreted correctly?. I reviewed the equations over and over and still I cannot figure out what I did wrong. Can somebody help me with this?
Apparently this is a matter of getting the right interpretation, having that ah-ha moment and also a bit of playing around like Where's Wally?
In the passage the key as mentioned in the comments is what to do with but in $25$ minutes. After the edit, it is more likely that it meant the time after the show started.
To better illustrate this situation below are the variables:
$\textrm{Time when the show started = x}$
$\textrm{But in 25 minutes =} x + \frac{25}{60}$
Time elapsed between 5 AM and 25 minutes before the show started:
$\left(x-\frac{25}{60}\right)-5$
Two thirds of the time which will be to 8 AM, but in 25 minutes.
$\frac{2}{3}\left(8-\left(x+\frac{25}{60}\right)\right)$
Finally all that is left to do is to equate both expressions:
$\left(x-\frac{25}{60}\right)-5=\frac{2}{3}\left(8-\left(x+\frac{25}{60}\right)\right)$
Simplifying:
$\left(x-\frac{5}{12}\right)-5=\frac{2}{3}\left(8-\left(x+\frac{5}{12}\right)\right)$
$\frac{12x-5-60}{12}=\frac{2}{3}\left(\frac{96-12x-5}{12}\right)$
$12x-65=\frac{2}{3}\left(91-12x\right)$
$36x-195=182-24x$
$60x=377$
$x= 6 \frac{17}{60}$
Hence the time when the show began was $\textrm{6:17 AM}$
This checks with the alternatives. Needless to say that probably this was the intended meaning.