Posted by **tiff** on Sunday, October 25, 2009 at 1:58pm.

Today, Mark invested $5,000 into an account that guarantees 7.50% p.a., compounded monthly and Madonna invested $5,000 into account guaranteeing 8.125% p.a., compounded quarterly.

How long will it take (in years) for the value of Madonna's investment to be three times as much as Mark's?

- finance -
**MathMate**, Sunday, October 25, 2009 at 4:02pm
For mark:

interest rate

= 7.5% p.a.

= 7.5%/12 per period of one month

= .625%

Future value of $5000 in n years, FVK(n)

= 5000*(1.00625)^{12*n}

For Madonna,

interest rate

= 8.125% p.a.

= 8.125%/4 per quarter

= 2.03125% per quarter

Future value of $5000 in n years, FVD(n)

= 5000*(1.0203125)^{4*n}

We look for n where

FVD(n) = 3FVK(n)

5000*(1.0203125)^{4*n}

= 3*5000*(1.00625)^{12*n}

(1.0203125)^{4*n} / (1.00625)^{12*n} = 3

Take log on both sides

4*n*log(1.0203125)-12*n*log(1.00625) = 3

n = log(3)/(4log(1.0203125-12log(1.00625))

= 193.785 years

Check:

in 193.785 years,

Mark will have $9,801,861,882.04

Madonna will have $29,405,482,100.99

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