# Thread: Problem Solving - Shore Lights

1. ## Problem Solving - Shore Lights

An observer on a boat on Port Phillip Bay at night can see three different shore lights. The red one winks every 6 seconds, the white one every 10 seconds and the green one every 14 seconds

a. How often do the red and green lights wink on together?
b. How often do the green and white lights wink on together?
c. How often do the red and white lights wink on together?
d. How often do all three lights wink on together?

The observer notes that at midnight the red and white lights wink together. Six seconds later the green, and obviously the red, light wink on.

e. When will they all next wink together?
f. When will this next occur exactly on the hour?

2. Make a timeline and then notice the pattern. I would show you mine but it's so complex that it won't work.

3. Hello, Mr`Fantasy!

An observer on a boat on Port Phillip Bay at night can see 3 different shore lights.
The red one winks every 6 seconds, the white every 10 seconds and the green every 14 seconds.

a. How often do the red and green lights wink on together?
b. How often do the green and white lights wink on together?
c. How often do the red and white lights wink on together?
d. How often do all three lights wink on together?

The observer notes that at midnight the red and white lights wink together.
Six seconds later the green, and obviously the red, light wink on.

e. When will they all next wink together?
f. When will this next occur exactly on the hour?

We are dealing with the Least Common Multiple (LCM).

(a) Red (6 seconds) and Green (14 seconds)
. . .They wink together every: $LCM(6,14) = 42$ seconds.

(b) Green (14 seconds) and White (10 seconds)
. . .They wink together every: $LCM(14,10) = 70$ seconds.

(c) Red (6 seconds) and White (10 seconds)
. . .They wink together every: $LCM(6,10) = 30$ seconds.

(d) Red (6 seconds), White (10 seconds), Green (14 seconds)
. . .They wink together every: $LCM(6,10,14) = 210$ seconds.

At midnight, the Red and White wink together.
. . Six seconds later, the Green winks.

Red and White wink together every 30 seconds.
They wink together at $30m$ seconds after midnight for some integer $m$.

Green winks every 14 seconds.
It winks at $6 + 14n$ seconds after midnight for some integer $n$.

When are two times equal?
. . $30m \:=\:6 + 14n\quad\Rightarrow\quad n \:=\:\frac{3(5m-1)}{7}$ .[1]

Since $n$ is an integer, $5m-1$ is a multiple of 7.
The first time this happens is $m = 10$. .Then: . $n = 21$.

(e) They wink together $3m = 300$ seconds after midnight . . . at 12:05 a.m.

The next time [1] gives integral $n$ is when $m = 17\,\:n = 36$.

(f) They wink together $3m = 510$ seconds after midnight . . . at 12:08½ a.m.