Jodi and Jim take a walk in the park. They both start out on their left foot. Jodi's step is only 2/3 of Jim's. At which step for Jodi and which step for Jim will they be ready to take a "left foot step" together again?
Try this way of looking at the problem. Draw a sequence of lines each 2/3 inch long starting on the margin of your notepad and alternately labelling them left, right. Draw another sequence of lines just below that, starting at the same margin, each 1 inch long and alternately labelling them left, right. You should see that each LR pair in 4/3 inch long on one row and 2 inches long on the other. You want to know when they next align: in other words, when a mutiple of 4/3 is next equal to a multiple of 2. You'll find that 3 x (4/3) = 2 x (2) so the two are next in step after three double (LR) steps of one and two of the other.
a)It could be that Jodi's and Jim's steps are synchronized--that is, they step on their left and right feet at exactly the same time. It is only that Jodi's steps are 2/3 as long as Jim's.
b)This maybe is what you mean.
b.1) When Jim's left foot touches the ground again, Jodi's left foot is still on air, and still 1/3 short of the way to the ground.
And then, after the 2nd touching on the ground of Jim's left foot, the left foot of Jodi is (1/3 +1/3) = 2/3 of the way.
And then, on the 3rd touching on the ground of Jim's left foot, the left foot of Jodi is (2/3 +1/3) = 1 of the way, or it still on the ground.
Count those steps.
Jim's left foot makes 3 touchings on the ground.
Jodi's left foot makes 2 touchings.
Jim's and Jodi's left feet are again on the ground at the same time.
m = number of times the left foot of Jim touches the ground.
d = number of times the left foot of Jodi touches the ground.
d = (2/3)m
When m = 1, d = (2/3)(1) = 2/3
when m = 2, d = (2/3)(2) = 4/3
when m = 3, d = (2/3)(3) = 6/3 = 2 ---------answer.
If by step you mean "left-right" = one step; "right-left" = another step; or, "left-right-left" is two steps, then double the numbers found above because the numbers above count left-right-left as one step only.
6 steps for Jim while 4 steps by Jodi and their left feet are both on the ground, at the same time, again. -------answer.