SOLUTION: you deposit $2000 in an account that pays 2% annual interest compounded quarterly. How long will it take for the balance to reach $2400? I have having trouble completing this p

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Question 76388: you deposit $2000 in an account that pays 2% annual interest compounded quarterly. How long will it take for the balance to reach $2400?
I have having trouble completing this problem, please help with the steps.
Tx, H
Found 2 solutions by stanbon, Edwin McCravy:
Answer by stanbon(75887) About Me  (Show Source):
You can put this solution on YOUR website!
you deposit $2000 in an account that pays 2% annual interest compounded quarterly. How long will it take for the balance to reach $2400?
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Formula: A = P(1 + r/n)^(nt)
2400 = 2000(1 + 0.02/4)^(4t)
1.2 = 1.005^(4t)
Take the log of both sides to get:
log(1.2) = 4tlog(1.005)
4t = log(1.2)/log(1.005)
4t=36.56
t=9.14 years.
==============
Cheers,
Stan H.

Answer by Edwin McCravy(20056) About Me  (Show Source):
You can put this solution on YOUR website!

you deposit $2000 in an account that pays 2% annual
interest compounded quarterly.  How long will it 
take for the balance to reach $2400?

I have having trouble completing this problem, 
please help with the steps.

Tx, H


A+=+P%281+%2B+r%2Fn%29%5Ent

where

A = final amount
P = beginning amount
r = annual interest rate expressed as a decimal
n = number of times a year interest is compounded
t = number of years.

In this problem

A = $2400
P = $2000
r = .02
n = 4 (quarterly)
t = unknown

Substituting:

2400+=+2000%281+%2B+.02%2F4%29%5E%284t%29

2400+=+2000%281+%2B+.005%29%5E%284t%29

2400+=+2000%281.005%29%5E%284t%29

Divide both sides by 2000

2400%2F2000+=+%282000%281.005%29%5E%284t%29%29%2F2000

1.2+=+1.005%5E%284t%29

take the natural log of both sides:

ln%281.2%29+=+ln%281.005%5E%284t%29%29

Use the rule of logarthims ln%28X%5EY%29=Y%2Aln%28X%29 on the
right hand side:

ln%281.2%29+=+4tln%281.005%29

Divide both sides by 4ln%281.005%29

ln%281.2%29%2F%284ln%281.005%29%29+=+%284tln%281.005%29%29%2F%284ln%281.005%29%29+

ln%281.2%29%2F%284ln%281.005%29%29+=+t

Get your calculator and work out the left sides;

9.13884909=t

At the end of 9 years, the balance will be $2393.36
which is short of $2400 by $6.64, then on the next
compounding 3 months later, the balance will be 
$2408.34, which is over $2400 by $8.34.  The balance
won't ever reach $2400 exactly, but the first time 
it will exceed $2400 will be 9 years and 3 months.

Edwin