Romeo and Juliet

Romeo and Juliet are young lovers, tragically separated by an unspecified distance.

Every morning, at exactly 9:00 am, each of them leaves their respective houses and walks, at their own constant speed, towards each other. They always follow the same route. They meet at exactly 10:00 am and share their first kiss of the day.

Today, Romeo overslept and did not leave his house until 9:30 am. Juliet, unaware of this, left at the usual time (covering some of the distance in this half hour).

Both still walked at their regular individual constant speeds, and this time, their first kiss of the day occurred when they met at 10:21 am.

Romeo felt ashamed of his faux pas, so decided to get up extra early the next day and walk directly to Juliet’s house so that he could surprise her there at 9:00 am. What time should Romeo leave his house?

Let D be the distance between the houses. Let R be the speed that Romeo walks, and J be the speed that Juliet walks.

We know the regular trip takes one hour, so the following equation is true (time=distance/speed):

Whilst Romeo slept, Juliet walked, oblivious to the fact, covering some distance. The distance she traveled, in half an hour, is ½J, so we reduce distance left to walk by this. We know they meet at 10:21, which is 51 minutes after Romeo leaves (10:21 subtract 09:30). This gives us a second equation:

Substituting D from first equation, we can eliminate it.

Clearly Romeo has long legs, walking over twice as quickly as Juliet!

On a regular day:

And what we want to find out it the time it will take Romeo to walk the distance himself, which is just D/R

Romeo should leave his house at 7:34:17 am (and maybe bring flowers).