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**Prove It** The way a perpetuity works is that every time the interest is calculated, it is paid out, so the balance never changes.

Essentially what you want here is for the interest every month (which is going to be the payout) to be \$52 000.

As we are looking at what happens every month, a simple interest calculation is all that's needed.

$\displaystyle \begin{align*} I &= P\,R \\ P &= \frac{I}{R} \\ P &= \frac{52\,000}{\frac{14.4}{1200}} \textrm{ (do you understand why we need to divide by 12?)} \\ &= 4\,333\,333.33 \end{align*}$

So at the time that the library is opened, we need \$4 333 333.33 invested in the perpetuity so it will pay out \$52 000 each month.

However, we have 8 years to earn that amount. So what we want to do is work out how much money to invest NOW, so that in 8 years there is \$4 333 333.33 invested. I'll assume that there is only the initial investment and no further payments are made. This needs a compound interest calculation.

$\displaystyle \begin{align*} A &= P\,\left( 1 + R \right) ^n \\ 4\,333\,333.33 &= P\,\left( 1 + \frac{14.4}{1200} \right) ^{ 96 } \\ P &= \frac{4\,333\,333.33}{\left( 1 + \frac{14.4}{1200} \right) ^{96}} \\ P &= 1\,378\,773.40 \end{align*}$

So \$1 378 773.40 needs to be invested now so that in 8 years there is enough money in order to support the library in perpetuity.