SOLUTION: Steven invests $2000 into a GIC which will double every five years, wheras Dana invests $1000 into a GIC which will triple evry three years. How many years will pass before they ha

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Question 130665: Steven invests $2000 into a GIC which will double every five years, wheras Dana invests $1000 into a GIC which will triple evry three years. How many years will pass before they have the same amount of money?
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!
Steven invests $2000 into a GIC which will double every five years, wheras Dana invests $1000 into a GIC which will triple evry three years. How many years will pass before they have the same amount of money?


The rule is:

P+=+A%281%2Br%2Fn%29%5E%28nt%29

We will assume the interest accrues once yearly, so n = 1

and the formula becomes

A+=+P%281%2Br%2F1%29%5E%281t%29

A+=+P%281%2Br%29%5Et

So for Steven, P = 2000, so the amount A, in t years, is

A+=+2000%281%2Br%29%5Et

Steven's GIC doubles every five years,
So we substitute t = 5, and A = 4000 (double 2000),
so we can solve for r:

4000+=+2000%281%2Br%29%5E5

Divide both sides by 2000

2=%281%2Br%29%5E5

Take the 5th root of both sides:

root%285%2C2%29+=+root%285%2C%281%2Br%29%5E5%29+

Simplifying the right side:

root%285%2C2%29+=+1%2Br+

Solving for r:

root%285%2C2%29+-+1+=+r

Substituting that into

A+=+2000%281%2Br%29%5Et

A+=+2000%281%2Broot%285%2C2%29+-+1%29%5Et

or

A+=+2000%28root%285%2C2%29%29%5Et

This is the formula for Steven's amount A in t years.

---------------------------------------------

Now we do exactly the same thing with Dana's
GIC.

A+=+P%281%2Br%29%5Et

Now for Dana, P = 1000, so the amount A, in t years, is

A+=+1000%281%2Br%29%5Et

DANA's GIC triples every three years,
So we substitute t = 3, and A = 3000 (triple 1000),
so we can solve for r:

3000+=+1000%281%2Br%29%5E3

Divide both sides by 1000

3=%281%2Br%29%5E3

Take the cube root of both sides:

root%283%2C3%29+=+root%283%2C%281%2Br%29%5E3%29+

Simplifying the right side:

root%283%2C3%29+=+1%2Br+

Solving for r:

+root%283%2C3%29+-+1+=+r

Substituting that into

A+=+1000%281%2Br%29%5Et

A+=+1000%281%2Broot%283%2C3%29+-+1%29%5Et

or

+A+=+1000%28+root%283%2C3%29+%29%5Et

This is the formula for Dana's amount A in t years.

-------------------------------

So to find out when they have an equal amount, we

set their two A's equal, and solve for t:

+A+=+2000%28+root%285%2C2%29+%29%5Et+ and A+=+1000%28+root%283%2C3%29+%29%5Et

2000%28root%285%2C2%29%29%5Et+=+1000%28root%283%2C3%29%29%5Et

Divide both sides by 1000

2%28root%285%2C2%29%29%5Et+=+%28root%283%2C3%29%29%5Et

Change the roots to fractional exponentials of their radicands:

+2+%28+2%5E%281%2F5%29+%29%5Et+ = +%283%5E%281%2F3%29%29%5Et++

Multiply inner exponents by outer exponents

2%282%5E%28t%2F5%29+%29+ = 3%5E%28t%2F3%29

Write the first 2 as 2%5E1

2%5E1%282%5E%28t%2F5%29+%29+ = 3%5E%28t%2F3%29

Change the exponent 1 to 5%2F5 so you 
can add exponents on the left:

2%5E%285%2F5%29%282%5E%28t%2F5%29+%29+ = 3%5E%28t%2F3%29

Add exponents:

+2%5E%285%2F5%2Bt%2F5%29++ = 3%5E%28t%2F3%29

2%5E%28%285%2Bt%29%2F5%29++ = 3%5E%28t%2F3%29

2%5E%28%285%2Bt%29%2F5+%29++ = 3%5E%28t%2F3%29

Raise both sides to the 15th power

++2%5E%28+%285%2Bt%29%2F5+%29%5E15+%29++++ = +%28+3%5E%28t%2F3%29+%29%5E15++

Multiply inner exponents by outer exponents:

++2%5E%28++15%285%2Bt%29+%29%2F5+%29 = +3%5E%28%2815t%29%2F3%29

Cancel 5 into 15 on left and 3 into 15 on the right:

2%5E%283%285%2Bt%29%29 = 3%5E%285t%29

2%5E%2815%2B3t%29 = 3%5E%285t%29

Take the log (either log10 or ln) of both sides

log%28%282%5E%2815%2B3t%29%29%29 = log%28%283%5E%285t%29%29%29

Use the rule of logs +log%28b%2Cx%5EN%29+=+N%2Alog%28b%2Cx%29

+%2815%2B3t%29log%282%29+ = +%285t%29log%283%29+

Replace log%282%29 by A and replace log%283%29 by B

+%2815%2B3t%29A+ = +%285t%29B+

+A%2815%2B3t%29 = B%285t%29

15A%2B3At = 5Bt%29

Isolate the terms in t on the right

15A = 5Bt-3At

Factor out t on the right:

15A = t%285B-3A%29

Divide both sides by 5B-3A

%2815A%29%2F%285B-3A%29 = t

Now replace A by log%282%29 
and replace B by log%283%29

t = %2815log%282%29%29%2F%285log%283%29-3log%282%29%29  

Use calculator to get right side

t = 3.045801234

So the answer is just after 3 years.

Edwin