Question

Kevin invests $100 in an account that triples every 2 years. Anna invests $1000 in an account that doubles every 3 years. How many years will their accounts be equal in value? Assuming that no deposits or withdrawals were made.


Anyone? Please please please help me :) :3

Answer 1

Try writing both of them as a function, then find the point where they intercept.

Answer 2

Let's look at Kevin first

Answer 3

He starts with 100, then 2 years later he has 300, and 2 years later he has 900 (the 300 gets tripled, not the 100)

Answer 4

So can we find a function that has such behavior?

Answer 5

@zimmah but there's a formula for this right? Is it effective to use formulas such has, A = P(1 + r/n)^nt and A=Pe^rt ??

Answer 6

ohhh wait, i see, so it's not the 100 that gets tripled.

Answer 7

Yes it is for the first 2 years.

Answer 8

But then, once you have 300, the 300 will get tripled 2 years later.

Answer 9

You can use a simular equation

Answer 10

To the one you typed.

Answer 11

yeah?

Answer 12

So how will i put those into an equation? :)

Answer 13

I have to equate both of the equations to find t right? t = time :)

Answer 14

100(1+2)^(t/2) I believe, try it and see if it gives the values we would expect.

Answer 15

@zimmah you used A = P(1+r/n)^nt ?? :)

Answer 16

Yes, check if the one I created is correct, the try to make one for Anna, than if you have both you ca type an = sign bet gg them and solve for 0

Answer 17

@zimmah what confuses me is the r. What's the r? Haha.

Answer 18

do we use r as constant?

Answer 19

The r is the rate in % of increase.

Answer 20

ahhh okay :) so r is 3 for kevin?

Answer 21

wait, i still dont get it. haha pardon meee

Answer 22

Well, not exactly

Answer 23

oh coz r is %?

Answer 24

Let me start my computer any I will expai

Answer 25

okay i'll wait :)

Answer 26

ok, here we go

Answer 27

looking at the increase from year 0 to year 2, kevin has 100, and it increase from 100 to 300.

Answer 28

that's a 300% increase.

Answer 29

so if you write it in the form 100(1+r) than r would be 2, because 1+2=3 (note that it are percentages, and 3 actually means 300%

Answer 30

you understand thus far?

Answer 31

Okaay i do :)

Answer 32

alright, now notch it up a step.

Answer 33

we can't simply say that after 4 years (2 periods of 2 years) his money will increase with 600%. can you see why?

Answer 34

because it's not the 100 that triples :)

Answer 35

indeed

Answer 36

it's the 300 :)

Answer 37

so we have to make an equation that makes a 'copy' of itself, that sort of 'remembers' the value it has in between two years and than uses a copy of itself with the new value and repeats it for n amount of times where n is the amount of periods you need.

Answer 38

this may sound hard, but once i show you, it will probably be clear.

Answer 39

\[100(1+2)=300\]
\[300(1+2)=900\]
900(1+2)=2700

Answer 40

you see the (1+2) is the thing that always stays constant, and it what is causing the tripling.

Answer 41

so we want to 'copy' that part of the formula n amount of times. (where n is the amount of periods we want to know)

Answer 42

what's the mathematical symbol for making n amount of copies of itself as factors?

Answer 43

if i want for example 3 copies of 2 that multiply with each other, how could i write it?

Answer 44

so 2*2*2, how can i write it mathematically?

Answer 45

2^3??

Answer 46

yes

Answer 47

so, since every two years our the percentage part of our equation needs to make a copy of itself, we can use ^(t/2) as our factor.

Answer 48

\[\Large \left( 1+2 \right)^{\frac{ t }{ 2 }}\] works like a copying machine, it will print you t/2 times (1+2)

Answer 49

and we do the same for Anna?

Answer 50

yes, try it yourself, so we can see if you can put what you learned to practise.

Answer 51

\[100(3)^{t/2} = 1000(2)^{t/3}\]

uhmmm, like this? :)

Answer 52

yes

Answer 53

so im gonna use natural log to solve it right? :D

Answer 54

and use my scientific calculator haha..

Answer 55

uhm, i got 7.23 years.

Answer 56

\[\large 100(3)^{\frac{ t }{ 2 }}-1000(2)^{\frac{ t }{ 3 }}=0\]

Answer 57

or you can use that indeed lol

Answer 58

i got the same

Answer 59

@zimmah Thank you!! :)

Answer 60

you're welcome

Answer 61

am i right or what but i can see from the equation that it has a relation to exponential growth formula :) @zimmah

Answer 62

yes, it does.

Answer 63

it's a variation of it.

Answer 64

ohhhhh, i see :)

Answer 65

well thanks for making me realize everything LOL

Answer 66

it's ok, i like it when people truly understand what is going on instead of just copying formulas without knowing what the formula does, people enjoy it a lot more when they know what is going on.

Answer 67

i highly agree with you, sir! haha :)