The interesting question is never the numerator. It is what the lender divides by, because the same property has several defensible values and the rules do not agree on which one to use.
On a purchase, the denominator is the lower of the contract price and the appraised value. That single convention has one consequence worth planning for. If the appraisal lands below the agreed price, the lender sizes the loan against the appraisal, so the ratio rises and the gap has to be closed with cash, a renegotiation, or an exit under an appraisal contingency. Nothing about the purchase price changed.
On a refinance, the denominator is a fresh appraisal. This is the one route by which rising prices help, and it is written into the statute: for a refinance of the borrower's principal residence, "original value" means the appraised value the lender relied on to approve the refinance (12 USC 4901(12)).
For mortgage insurance cancellation, the denominator is frozen. The Homeowners Protection Act measures against the value at origination rather than against a current appraisal, which is why appreciation alone does not retire private mortgage insurance on the loan you already have.
Several thresholds sit on the same scale, and holding them as one list is more useful than meeting them one page at a time.
Two of those behave differently under prepayment, which is worth knowing before paying a mortgage down for the purpose. Extra principal brings the 80 percent request forward, because the borrower may elect to have that date measured on actual payments. It does nothing at all for the 78 percent automatic termination, which follows the initial schedule "irrespective of the outstanding balance" (12 USC 4901(18)).
The ratio is also how negative equity gets described. On an auto loan the collateral usually depreciates faster than the balance amortizes, so a loan that starts near 100 percent of the vehicle's value can spend its early years above it. That is the same measurement applied to a different asset, and it is why a long term on a depreciating asset behaves so unlike a long term on a house.