If the moon were 5x smaller, how much weaker would the gravitational force between it and Earth be?

It would be 20% of the current gravitational force.

Using Newton's Universal Gravitation: F = Gm1m2 / r^2 Where

F = Gravitational force

G = Universal Gravitational Constant

m1 = Mass of one object (Earth)

m2 = Mass of second object (Moon)

r = Distance between center of objects

First, let's find the mass of the moon. I'm going to assume that it just means a 5th of the mass.

Real Mass of the Moon = 7.347E+22 kg

New Mass = (7.347E+22 kg) / 5

New Mass = 1.4694E+22 kg

Now you find the distance between Earth and Moon, center to center.

Radius Earth = 6,371,000 m

Radius Moon = 1,737,000 m (I'm assuming that the mass is 5th smaller not the whole thing)

Distance between the two points (look it up) = 384,403,000 m

Total = 46,548,300 m

So now:

F = unknown

G = 6.6726E-11 N-m^2/kg^2

m1 = 5.974E+24 kg

m2 = 1.4694E+22 kg

r = 46,548,300 m

Solve!

F = { (6.6726E-11 N-m^2/kg^2) * (5.974E+24E) * (1.4694E+22) } / (46,548,300 m)^2

F = { 5.857E+12 N-m^2 } / (2.167E+15 m^2)

F = 2.703E+21 N

But we need to compare, so fill in the same equation but with the mass of the real moon (7.347E+22 kg)

F = { (6.6726E-11 N-m^2/kg^2) * (5.974E+24E) * (7.347E+22 kg) } / (46,548,300 m)^2

F = { 2.928E+37 N-m^2 } / (2.166E+15 m^2)

F = 1.35E+22 N

So... divide and it's a difference of 0.2 times the gravitational force!