I am trying to independently solve physics problems and I am attempting to solve Chapter 5 problem 4-4 on page 140 from the book "Feynman's Tips on Physics".

The problem is as follows:

A painter weighing 180 lbs working from a "bosun's" chair hung down the side of a tall building desires to move in a hurry. He pulls down on a fall rope with such a force that he presses against the chair with only a force of 100 lb. The chair itself weighs 30 lb.

(a) What is the acceleration of the painter and the chair?

(b) What is the total force supported by the pulley?

The diagram is as follows:

My Work

(a) The question states the weights of the painter and the chair are 180 lbs and 30 lbs respectively. I used this idea to perform the following step:

$$\text {Total mass of the painter and the chair} = \left(\frac{(180 + 30) \rm lbs}{g}\right) $$

He exerts a downward force of 100 lb on the chair. His net motion will be upwards.

I think the 100 lbs force the person exerts on the chair is transferred to the rope he is pulling on.

But that is just the string he is pulling on. The diagram shows that only one string is attached to the bosun chair. That string will have an upwards force of (100 + 180 + 30) lbs (as shown in the picture). This way, one string will have a 100 lbs force and other a force of (100 + 180 + 30) lbs. I don't know if this is even possible.

How is the 100 lbs downwards force affecting the overall system (the painter and the bosun's chair)?

EDIT:

Steeven has pointed out to me that an upwards force of 100 lbs is exerted on the rope as a result of the painter pulling on it (rope). The questions states that the painter pulls down on the rope (i.e. exerts a downwards force on the rope). I can understand that by Newton's third law, an equal and opposite force as a result of pulling is produced (as Steeven has said).

However, would that not cause the force due to pulling to get cancelled out?