sample gravity problem: distance to the moon/earth balance point

How far would you have to go toward the moon so that the pulls of gravity of both Earth and Moon are equal? M_earth=6E²⁴Kg, M_moon=7.35E²²Kg, separation of Earth and moon = 3.844E⁸meters
There are two forces in question here... the gravitational force of the Earth and the gravitational force of the moon.
We set these two equal to each other and solve for "r" and "R".

F_earth= Gm_objectm_earth/r²_to
earthF_moon= Gm_objectm_moon/R²_to
moon

set the two equal to each other...

Gm_objectm_earth/r²_to
earth=Gm_objectm_moon/R²_to
moon

Divide everything by "G" and "m_object" cleans things up a bit...

m_earth/r²_to
earth=m_moon/R²_to
moon

We already know the distance from Earth to the moon = 384400000m, so

R+r = 384400000

So if we isolate either "r" or "R", we can insert that value into the other equation...

r = sqrt(R²m_earth/m_moon) goesinto R+r = 384400000 for "r"...

R + sqrt(R²m_earth/m_moon) = 384400000

Square everything to get rid of those unsightly parantheses...

R² + R²m_earth/m_moon=1.478E¹⁷

^{Fill in the masses of Earth
and moon...}

R²+ R²(6.0E²⁴ /7.35E²² kg) = 1.47E¹⁷

82.63R² =1.47E¹⁷

R = 4.22E⁷meters from the moon
and
r = 3.42E⁸meters from Earth

If you were asked what % of the way you'd have to go to the moon...

3.42E⁸/3.844E⁸= 90% of the way there