One account doubles every 2 years. A second account triples every 3 years.

Question

One account doubles every 2 years. A second account triples every 3 years. Assuming the accounts start with the same amount of money, which account is growing more rapidly?

Answer 1

Account 1 is doubling every 2 years.

So you have 0, 2, 4, 6, .... being the interval of doubling

Let X be the initial amount. Then in account 1 we can write the amount as X*(2^n/2) where n is an even number

Account 2 is tripling every 3 years

So you have 0, 3, 6, 9, 12.... being the intervals of tripling

With X being the initial amount, then account 2 we can write the amount at any time as X*(3^m/3) where m is an integer divisible by 3

Because n is increasing by 2 and m is going up by 3, the common periods where n=m is for numbers divisible by 6 (2*3)

So at 6 years, Account 1 is X*(2³) = 8X

At the same 6 years Account 2 is X*(3²) = 9X

9X > 8X for the same time period so Account 2 is growing faster than Account 1

Answer 2

in 6 years the 1st account went from $100 to $200 in 2 years, to $400 in 4 years to $800 in 6 years

while the 2nd acount went from $100 to $300 in 3 years, then to $900 in 6 years

the 2nd account grows faster

3 = e^r3

ln3 = 3r

r = (ln3)/3 = annual growth rate of B

2 = e^r2

ln2 = 2r

r = (ln2)/2 = growth rate of B

(ln3)/3 > (ln2)/2

.366 > .347

36.6% > 34.7%