`r = 8%, t = 35` Find the principal P that must be invested at a rate r, compounded monthly, so that $1,000,000 will be available for...

Sunday, October 23, 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 values A = 1000000, r = 0.08, and t = 35, the formula becomes:

$\displaystyle{1000000}={P}{{\left({1}+\frac{{0.08}}{{n}}\right)}}^{{{n}\cdot{35}}}$

Since the r is compounded monthly, then n=12.

$\displaystyle{1000000}={P}{{\left({1}+\frac{{0.08}}{{12}}\right)}}^{{{12}\cdot{35}}}$

Isolating the P, it yields

$\displaystyle\frac{{1000000}}{{{\left({1}+\frac{{0.08}}{{12}}\right)}}^{{{12}\cdot{35}}}}={P}$

$\displaystyle{61377.75}={P}$

Therefore, the principal amount that should be invested is $61,377.75 .

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