Help - all-cash acquisition impact on equity value

Company A: EV = 100, market cap = 80, EBITDA = 10, net income = 4, EV/EBITDA = 10x, P/E = 20x Company B: EV = 40, market cap = 40, EBITDA = 8, net income = 2, EV/EBITDA = 5x, P/E = 20x

Say company A acquires B using 100% cash. What are the combined EBITDA and P/E multiples?

I understand that combined EV/EBITDA is 10x, because A's new EV = 100 + 40 = 140 as it has 40 less in cash, so net debt increases by 40, since EV adds debt and subtracts cash, less cash means higher EV. So total combined EV = 180. Total EBITDA = 10 + 8 = 18, so combined EV/EBITDA = 10x still, same as A before.

But what I don't understand is why combined P/E is stated to be 20x still. This is how I calculated: A's new market cap = 80 - 40 = 40. I reasoned that market cap includes cash (while EV excludes cash). So lower cash balance should mean lower market cap. Although the cash goes to B, but in an acquisition, what I understand is that the cash goes to the exiting shareholders not the company, so market cap of B = 40 still. Thus combined market cap = 80. Total net income is 6. So combined P/E = 80/6 = 13.3x.

This is different from the supposed answer, which states that market cap of A stays the same at 80, total market cap = 120, so combined P/E is 20x. But I don't understand why market cap of A stays the same?? Shouldn't there be some impact from the cash spent... Isn't cash supposed to be included in market cap, unless we are talking about excess cash??

Would appreciate any help!