Series of Spring Training Games, the Yankees scored 35 runs. The Dodgers
 and Tigers, respectively, tallied half and a third as many as the Reds. The
 Reds' scores were as much below the Yankees' as the Dodgers' were above the
 Tigers'. How many runs did each team score?

So 
Y=35
D=R/2
T=R/3
Y-R=D-T

multiply equations 2 and 3 by 2 and 3, respectively.
2D=R
3T=R

See that?  3T=R=2D, therefore, 3T=2D.

So
Y=35
3T=2D=R
35-R=D-T

Let's mutate the second equation to our liking.
R=3T=2D
T=2/3D=R/3
D=R/2=1.5T

Now, take D and T in the equation 35-R=D-T and replace them with their equivalents of R.
35-R=R/2-R/3

We're more than halfway done :D
Now do R/2-R/3=3R/6-2R/6=(3R-2R)/6=R/6.

Thence,
35-R=R/6.

Now do +R on both sides.
35=R/6+R
35=7R/6

times 6 both sides
210=7R

divide both by 7
30=R.

The Reds tallied 30.

Back to this:
D=R/2
T=R/3
Replace R's with 30:
D=30/2=15
T=30/3=10.

The answers are:
Yankees--35
Reds-----30
Dodgers--15
Tigers---10

Hope that helped!