Can anybody help me with this problem?
When will a payment of 300,000 galleons cancel Mr. Weasley’s liabilities to Gringott’s Bank of 100,000 galleons due after 2 years and 150,000 galleons due after 5 years if money is worth at an effective rate of 12%?
I'm stuck at the given.
The combined Present Value $A$ of the two payment commitments is given by $$A=100000(1.12)^{-2}+150000(1.12)^{-5}.$$ We want to find the time $T$ such that the PV of a payment of $300000$ at time $T$ is equal to $A$. So we want to solve the equation $$300000(1.12)^{-T}=A.$$ Equivalently, we want to solve $$(1.12)^T=\frac{300000}{A}.$$ Take the logarithm of both sides.