I went through this a few years ago. I bought my house for $270k and put 10% down, so I owed $243k. The house appraised for $275k. I paid around $2000 extra towards the principal each month (thanks roommates!) until I owed $216k, which was 80% of the original loan (and <80% of the original appraised value). I contacted the mortgage company and requested that they remove the PMI. They required that I get a broker's price opinion (BPO), which is a quicker and cheaper version of an appraisal. The purpose was to check if the house was still worth at least $270k. The BPO cost $150. It said that the house was worth $250k. Bullshit. No way the house dropped $25k in value in a year. I tried to argue that point with the mortgage company, and they told me I could pay $400-500 for an appraisal to override the BPO.
I had no guarantee the appraisal would be any better. Instead, I borrowed $15k from my parents (we're very open about money and after watching me attack the mortgage, they knew I could pay them back quickly) and put it towards the mortgage. I then owed $200k, which was 80% of the BPO value of $250k. This time, my request to remove PMI was successful. If I had not been able to get a quick $15k loan, I would have continued to pay extra towards the principal until I owed $200k. That would have taken 8 months. The BPO was valid for 1 year.
The math usually heavily favors paying off PMI. It is essentially interest on the balance of your mortgage above 80% of the original loan. So in your case, 80% of the original loan is $215600. You are paying PMI on 240000-215600 = 24400. It's costing you $128 per month to secure $24400. That's an interest rate of 6.3%... right now. As you continue to make payments, that interest rate gets higher because you're paying the same PMI to secure a smaller amount of money.
If/when you're ready to remove PMI, contact your mortgage company to determine exactly how they evaluate your request. Understand how long an appraisal/BPO is valid, in case it comes back lower than you expect. They should send you their policies in writing.