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They’re both the same distance from Earth when they meet.

Maybe I read it wrong. I don't see how math is necessary at all. If the two ships "cross" each other it means they're at the same point in space right? Or like right next to each other? So they're both the same distance from earth aren't they? Am I about to get woooshed? Lol

Edit: Upon re reading the question I realize it actually says "which one is the closet to Earth."

I'm afraid I don't know the answer to that one...

Some simplifying assumptions.

Let the distance from Earth to Mars be equal to 1, and assume it does not change. Let the direction from Earth to Mars be the positive direction.

Assume that the rockets travel at a constant velocity.

The displacement of the rockets can be represented with the lines

S_B(t) = (1/150)(t-30) = (1/150)t - (1/5)
S_A(t) = (-1/200)t + 1

Where t is time in days since rocket A took off. Notice rocket A has a negative slope (negative velocity) since it is moving from Mars to Earth. Rocket A has an initial position of 1, since it starts at Mars. Rocket B has a horizontal shift to the right of 30 days, representing it taking off later.

The rockets cross where these lines intersect. So

(1/150)t - (1/5) = (-1/200)t + 1
((1/150)+(1/200))t = 1 + (1/5)
(7/600)t = (6/5)
t = 720/7 ~= 103

So the rockets cross approximately 103 days after rocket A took off. The position at that time is

S_A(720/7) = (-1/200)(720/7) + 1
    =(-18/35) + 1 = 17/35 ~= 0.49

So when they cross, they are about 49% of the way from Earth to Mars. Just closer to Earth than Mars.

When they cross each other, which one is the closet to the earth ?

This is why you read the while question before trying to answer it. When the cross, they are both the same distance from Earth.

Edit: I didn't read the entirety of the problem, but in any case, this should help you state almost anything regarding the simple math. Note that in actuality, I don't think there would be a true meeting place due to orbital paths, but if you treat it as a linear "train" problem, this is how I would do it.

This may not be the simplest, but here's an easy way to just use lots of substitution and basic algebra.

Let t = time in days to meet

Let a = speed (not velocity) of rocket A

Let b = speed (not velocity) of rocket B

1 = 200 * a

1 = 150 * b

200a = 150b

a = (3/4)b

1 = (t * a) + (t - 30) * b

Substitute for a

1 = (3/4)bt + bt - 30b = (7/4)b - 30b

Recall that 1 = 150 * b and set these equal

150b = (7/4 * t - 30) b

Divide by b

150 = 1.75t - 30

1.75t = 180

t ~ 103 days

At 103 days, the ships will meet, and since it's over half the time it takes for rocket A to reach Earth, the meeting point will be closer to Earth.

(Edit: late at night, math brain no work)