`r = 9%, t = 25` Find the principal P that must be invested at a rate r, compounded monthly, so that $1,000,000 will be available for...

Thursday, July 7, 2016

The formula in compounding interest is

$\displaystyle{A}={P}{{\left({1}+\frac{{r}}{{n}}\right)}}^{{{n}\cdot{t}}}$

where

A is the accumulated amount

P is the principal

r is the annual rate

n is the number of compounding periods in a year, and

t is the number of years.

Plugging in the given values, the formula becomes:

$\displaystyle{1000000}={P}{{\left({1}+\frac{{0.09}}{{n}}\right)}}^{{{n}\cdot{25}}}$

Since the r is compounded monthly, the value of n is 12.

$\displaystyle{1000000}={P}{{\left({1}+\frac{{0.09}}{{12}}\right)}}^{{{12}\cdot{25}}}$

The right side of the equation simplifies to

$\displaystyle{1000000}={P}{{\left({1.0075}\right)}}^{{300}}$

Isolating the P, it becomes

$\displaystyle\frac{{1000000}}{{{1.0075}}^{{300}}}={P}{{\left({1.0075}\right)}}^{{300}}$

$\displaystyle{106287.83}={P}$

Therefore, the principal amount that should be invested is $106,287.83 .

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