Taken from Project Gutenberg’s Amusements in Mathematics, by Henry Ernest Dudeney, originally published ca. 1917 … #58 – “A Time Puzzle” …
How many minutes is it until six o’clock if fifty minutes ago it was four times as many minutes past three o’clock?
Solution coming soon …
I always like to start a problem like this with a picture. Any kind of picture. In fact, I consider drawing a picture that represents the problem to be a necessary first step toward solving the problem.
As you can see, my very rough approximation is sort of a number line that represents time. Each tick mark represents one hour of time, between 3:00 and 6:00, which are the times used in the problem.
The very next step in solving a word problem is to decide what your variable will represent. Usually, but not always, you can let your variable stand for what you’re looking for. Here we are looking for the number of minutes before 6:00, so as shown in the diagram, I let x represent the number of minutes before 6:00.
I don’t know what that is yet, but letting the variable represent what I’m actually looking for will make the end of my solution process go much easier.
Next, we have to look at what we are told. This is a very short word problem, so every phrase is bound to be important. We need to find how many minutes it is before 6:00 if 50 minutes ago (before the current time), the number of minutes after 3:00 was four times as many as the number of minutes it is now before 6:00.
Look at the top of the diagram. Notice the labels. Starting at 6:00 on the right, I have my variable, which is x minutes before 6:00, so it goes to the left of 6:00.
Next we have the additional 50 minutes “ago”, which also is a movement to the left on the number line (time line) from the current time. See where I marked “50 mins.”?
And finally whatever that time is, we know that it is 4x minutes past (to the right of) 3:00. So we label it as such.
Now we have three consecutive segments of time, starting at 3:00 and ending at 6:00: 4x, 50 minutes, and x.
The total number of minutes between 3:00 and 6:00 is 180, as shown in red, and these three segments of time have to add up to a total of 180 minutes. Therefore, we have the equation
And the time is therefore 26 minutes until 6, or 5:34.