Buy out scenario and bringing on an equity investor

A company has $180,000 invested into it. In order to buy out a partner I need $88,000. My current manager wants to invest in the company and I am encouraging it in order to keep a long-term employee on board with an attitude of running the business like an owner. I am for the most part an absentee owner and oversee the big-picture strategy and vision, but I am also the one who started the company and was granted the right to franchise so I stay involved and would never leave the company. My question is, can I determine the equity share I give the manager myself or does it have to be an exact percentage of the company value? Ideally, I would want to get $80,000 from my manager and grant her 40% equity, the manager would also continue to draw their salary. Could I offer her less than 40% since she will continue to draw her salary?

When you sell shares in the company to this manager, you can sell them at any price you want. You can sell 1% for $80,000 or 99% for $80,000. The only rule is that you both have to agree to the price. When you agree to the price, that implies the valuation. For example if you agree to sell 40% of the company for $80,000, that implies that the whole company is worth $200,000.

You said "in order to buy out a partner" you need $88,000. I'm not sure how you got to this, but I presume this means that the partner wants $88,000 for her shares. You didn't tell us what percentage of the shares the partner has. That might also be implying a certain valuation. If the $88,000 would buy 44% of the company, that also implies a $200,000 valuation. This would be consistent with selling 40% of the company for $80,000.

You also said the company has "$180,000 invested into it." That does not mean anything, because you did not tell us what percentage of the equity was purchased for $180,000. I'm guessing it was 100%. In that case, the company was initially worth $180,000, but since that was presumably a while ago, it may be worth more (or a lot more) now. Valuing companies is an art as much as a science.