Intersection between exponential models?

[Capricorn1997]

Homework Statement

Here is a math problem I would to make sure I have understood correctly.

a)

Billy goes to be bank. He deposite 500 dollars in his account. The bank offers a interest of 2 percent a year.

Sally goes to the bank 4 years later.She deposite 250 dollars at the same bank. But get an interest of 2.5 % a year.

write a exponential models for how much money Sally and Billy has in the bank.

b) How many years will pass before Sally has more money in her account than Billy?

Homework Equations

[/B]
fv = pv(1+i)^n

where fv is future value, pv is the present value and i is interest and n number of years.

The Attempt at a Solution

a) Billy: u(n) = 500 * (1.02)^n , Where n is the number of years Billy keeps his money in his account.

Sally: u(n) = 250*(1.025)^(n+4), Where n is the number of years Sally keeps his money in his account.

b) How many years will pass before Sally has more money in her account than Billy?

Calculating this isn't equal to solving:

500 * (1.02)^n = 250*(1.025)^(n+4) ??

if I solve this equation using Maple I get the result n = 121.55

But can that be right?

 

[haruspex]

Sally: u(n) = 250*(1.025)^(n+4), Where n is the number of years Billy keeps his money in his account.

Who invested first?

 

[Capricorn1997]

Who invested first?

Sally.

 

[haruspex]

Sally.

Right, so check the use of n in Sally's formula.

 

[Capricorn1997]

Right, so check the use of n in Sally's formula.

If Billy went to the bank 4 years before Sally, then he is 4 years ahead of her ?

Thusly the correct answer for a) is ?

a) Billy: u(n) = 500 * (1.02)^(n+4) , Where n is the number of years Billy keeps his money in his account.
Sally: u(n) = 250*(1.025)^(n), Where n is the number of years Sally keeps his money in his account.

How ever if I in b) solve the equation.

Billy: u(n) = 500 * (1.02)^(n+4) = 250*(1.025)^n in Maple that gives me the result n = 157.95 ?

 

[haruspex]

Your equation for Billy was right the first time. It's the Sally equation that's wrong.
It would be better to keep n having the same meaning both cases, the number of years Billy has had money in the bank.

 

[Capricorn1997]

Your equation for Billy was right the first time. It's the Sally equation that's wrong.
It would be better to keep n having the same meaning both cases, the number of years Billy has had money in the bank.

Okay,
when I beg your pardon. Where does the 4 come into play in Sally's equation ? if its as n+4 in ?

u(n) = 250*(1.025)^(n+4),

 

[haruspex]

Okay,
when I beg your pardon. Where does the 4 come into play in Sally's equation ? if its as n+4 in ?

u(n) = 250*(1.025)^(n+4),

When Billy has been earning interest for five years (n=5), how long has Sally been earning interest?

 

[haruspex]

Sally.

No, sorry, that's wrong.

 

[Capricorn1997]

No, sorry, that's wrong.

If Sally's money has been earning interest for 1 year, then Billy's has been earning interest for 4+1 years.

Meaning that Billy is 4+n years ahead of Sally. While Sally is at n+1 years ?

If that not true, then I am quitting math for good :(

 

[Ray Vickson]

If Sally's money has been earning interest for 1 year, then Billy's has been earning interest for 4+1 years.

Meaning that Billy is 4+n years ahead of Sally. While Sally is at n+1 years ?

If that not true, then I am quitting math for good :(

Don't quit math for good. Just go take a walk or go for a run to help you calm down. After you have relaxed a bit, sit down and think carefully before writing down formulas.

If you have trouble jumping to the general formula right away, look at a few initial cases. Starting at time 0 (when Billy puts in his money), what is Billy's balance after 1 year has passed? Answer = (1.02) 500. What is Sally's balance then? Answer = 0 (she has not put in money yet). After 2 years have passed, what are Billy's and Sally's balances? Billy has (1.02)^2 500, Sally has 0. After 3 years have passed, Billy has (1.02)^3, Sally has 0. After 4 years, Billy has (1.02)^4 500, Sally has (1.025) 250. After 5 years Billy has (1.02)^5 500, Sally has (1.025)^2 250.

Keep going like that: after n years, Billy has __________ and Sally has _________ .

BTW: do not be surprised if it takes a long time for Sally to catch up to Billy; she starts with a much smaller balance than Billy and her rate of interest is only slightly higher that Billy's.

 

[haruspex]

If Sally's money has been earning interest for 1 year, then Billy's has been earning interest for 4+1 years.

Yes.

Meaning that Billy is 4+n years ahead of Sally.

No, 4 years ahead.
To reduce confusion, let's stop using 'n' and use B for the number of years for which Billy has been earning interest and S for the number of years Sally has been earning interest. We need to find the relationship between these two variables at any given time.
When B=5, what is S?
When B = 6, what is S?
Can you generalise that?