# Vitamins for the Mind – Solutions

The 4:00 Meeting – Solution

Let Tt be time it takes Tom to walk half the distance between their
original places.

Let Tb be time it takes Bill to walk half the distance between their
original places.

A) Bill leaves at 4:00 – 3Tb.
Tom arrives 11 minutes later. That’s 4:00 -3Tb +11.
Tom walks back to the center: 4:00 -3Tb +11 + Tt.
At which time it’s 4:00: 4:00 -3Tb +11 + Tt = 4:00.
The 4 o’clock’s cancel:

Tt -3Tb + 11 = 0.

B) Similarly,                           Tb -3Tt + 15 = 0.

C) Solving, we get Tt = 7 minutes, and Tb = 6 minutes.
So Tom leaves at 3:39 and Bill leaves at 3:42.

Alice Forgets Her Purse – Solution

Let Ve be the rate of the down-moving escalator in steps per second.
Let Va be the rate of Alice in steps per second as she goes down the escalator.
Let L be the number of steps on the surface of the escalator.

A) Alice takes 50 steps going down and 125 steps going up, so because her rate
going up is five times as fast, the ratio of her down and up times is

50     125         50    25          50     Va         2
—- / ——   =   –/ —-   =   –— x —-   =   —-
Va     5Va         Va    Va         Va     25         1

So if Alice takes 50 steps going down, she takes 25 steps going up.

B) As Alice goes down, 50Va steps disappear go over the bottom, so the
difference, L – 50Va, must be 50 because she arrived at the bottom in 50 steps.

We have L – 50Va = 50,  so:
50Va = L – 50.

As Alice goes up, 25Va steps come over the top, so the sum, L + 25Va, must be
125 because she arrived at the top in 125 steps. We have L + 25Va = 125, so:

25Va = 125 – L.

C) Dividing the equations in A) and B), we have:

50Va             2         L – 50
———-   =   —-    = ———
25Va             1        125 – L

Solving:   250 – 2L = L – 50;    L = 300;    and, finally,

L = 100 steps on the surface of the escalator.