Question

35) Juan deposits $750 into an account. He was told that his balance will double every 8.5 years. Virginia deposits $500 into a separate account. Her money will triple, every 10 years. After a certain number of years they will both have the same amount in their accounts. How many years will it take for this to happen?

Answer 1

We will have to solve for the rate of both accounts.
We'll use this complicated formula:
log(1 + rate) = {log(total) -log(Principal)} ÷ Years

One account doubles the money every 8.5 years:
(We'll make total = 2 and principal = 1)
log(1 + rate) = {log(2) -log(1)} ÷ 8.5
log(1 + rate) = 0.30102999566 / 8.5
log(1 + rate) = 0.0354152936
10^0.0354152936 = 1.0849639136
rate = 8.49639136

The other account triples the money every 10 years:
(We'll make total = 3 and principal = 1)
log(1 + rate) = {log(3) -log(1)} ÷ 10
log(1 + rate) = 0.47712125472 / 10
log(1 + rate) = 0.047712125472
10^0.047712125472 = 1.116123174
rate = 11.6123174

Okay, NOW we have to calculate when will $750 invested at 8.49639136 interest equal $500 invested at interest 11.6123174?

That seems difficult to solve exactly because we have 2 unknowns:
We don't know the AMOUNT of money when one account equals the other and we don't know the TIME it will take.

Amount 1 = 750 * (1.0849639136)^years
Amount 2 = 500 * (1.116123174)^years

I don't know how to solve for those equations when Amount 1 = Amount 2.
However, I was able (by trial and error) to determine a precise answer.
In 14.32005 years, both accounts will equal $2,411.07.