Astronaut in space, calculating velocity etc

1. The problem statement, all variables and given/known data
Hey!
I´m stuck on a physics problem. I had to translate this problem into English, so please excuse any grammar mistakes!
An astronaut is on a mission in space. He is currently outside his spaceship which is standing still. The astronauts lifeline breaks, and he starts floating away from the spaceship with a velocity of 0,10 m/s. In an attempt to save himself and return back to his spaceship, the astronaut gets rid of a tool weighing 12 kg. The astronaut weighs 90 kg.
a) At what speed and in which direction should the tool be dispatched so that the astronaut can return to the spaceship with a velocity of 0,05 m/s?
b) What average force does the astronaut need to use if if takes 0,5 s to dispatch the tool, and the astronaut then returns to the spaceship with a velocity of 0,05 m/s?

2. Relevant equations
Newton II: F=m*a
Newton III: F1=F2

3. The attempt at a solution

a) I know that the answer is supposed to be 1,225 m/s, but I did not get this answer when attempting to solve the question.
I tried to solve the question in the following way:
Astronauts velocity before getting rid of the tool =0,1 m/s
Astronauts velocity after getting rid of the tool = 0,05 m/s
Astronauts weight before getting rid of the tool = 90 + 12 kg = 102 kg = m1
Astronauts weight after getting rid of the tool = 90 kg=m2
Which means that Δv1 = (0,05-0,1) m/s = -0,05 m/s
From Newton II we know that m1a1=m2a2 and hence m1Δv1=m2Δv2
I plugged the above numbers into the equation m1Δv1 = m2Δv2 and solved for Δv2 and got the following answer: -0,056 m/s
Where did I go wrong??

b) Since I already know that the answer to question a) was supposed to be 1,225 m/s, I tried solving for the tools acceleration in order to then use Newton II´s formula F=m*a and Newton III´s law that F1=F2.

ΔVtool/Δtime = acceleration tool
acceleration tool = 1,225 m/s/ 0,5 s = 2,45 (m/s^2)
F= m*a = 12 kg * 2,45 (m/s^2) = 29,4 N
Which is also wrong! The answer is supposed to be 27 N

I urgently need your help! Thanks!
Astrid

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