Bank interest plans. I tried to use the formula this time.

[Femme_physics]

Homework Statement

In a certain bank they offer 2 saving plans

Plan A gives yearly interest of 5%, and you can invest in this plan only in units of full year.

Plan B gives a biyearly interest of 10% and you can invest in this porgram in units of 2 full years.

A man has decided to invest his money in plan B for 4 years, and at the end of this period to invest one full year at plan A all the money he's saved on plan B.Would he have saved more money if he had invested all his money to a period of 5 years on plan A. Explain.

Homework Equations

eumyang has shown me an interesting formula to solve such questions

I think you have to use the compound interest formula
A(t) = A_0 \left(1 + \frac{r}{n} \right)^{nt}
where
A₀ = the principal
t = time in years
n = number of compounding periods per year (monthly: n = 12; quarterly: n = 4...)
r = interest rate expressed as a decimal

The Attempt at a Solution

 

[Mark44]

It would be really nice for several reasons if you posted your work inline rather than as a scanned image.

 

[Femme_physics]

It would be really nice for several reasons if you posted your work inline rather than as a scanned image.

Really? It just saves me a lot of time since I'd have to

A) Retype everything to PC that's already written on paper
B) Learn how to use LaTex

I figured if I can scan it I can save some energy.

Is this better? So you won't have to click

http://img191.imageshack.us/img191/6947/bankers.jpg

 

[Mentallic]

For idea #1, since plan B is invested biyearly, and

n = number of compounding periods per year (monthly: n = 12; quarterly: n = 4...)

then n should instead be...?

And you haven't considered the part where it says

and at the end of this period to invest one full year at plan A all the money he's saved on plan B.

 

[Femme_physics]

then n should instead be...?

0.5?

And you haven't considered the part where it says

I thought I did-- that's why I called the result of the first 4 years "Ao1"!

 

[I like Serena]

0.5?

Yes!
So in idea#1 you should use 0.5 everywhere where you now used 2.

I thought I did-- that's why I called the result of the first 4 years "Ao1"!

Actually you did, but you also added a A₀₁ too many in idea#1.


Furthermore in idea#2 you used n=2 which is not right. What should n be?

 

[Mark44]

Really? It just saves me a lot of time since I'd have to

A) Retype everything to PC that's already written on paper
B) Learn how to use LaTex

I figured if I can scan it I can save some energy.

Is this better? So you won't have to click

http://img191.imageshack.us/img191/6947/bankers.jpg

This is better than having to open a thumbnail image, but it's not as good as having the contents inline with your post. For one thing, if you have a mistake, we can't insert a line right under the mistake.

I concede the time it would take to type in the formulas and equations, but you have already typed in a fair amount of explanation, and to type in your work would only be a little bit more.

None of what you have written requires the use of LaTeX. You can make exponents by clicking the X² button, and subscripts by clicking the X₂ button. These buttons are on the extended menu that is available after you click the Go Advanced button, just below the text entry area.

For example, here is the first line under your Idea #1. No LaTeX was used here.
A₀ = (1 + 0.10/2)^2.4 = A₀₁ = 1.477

(At least, that's what I think you wrote. I'm not clear on what A₀₁ is supposed to mean, assuming that's what it says.

 

[Femme_physics]

Yes!
So in idea#1 you should use 0.5 everywhere where you now used 2.



Actually you did, but you also added a A₀₁ too many in idea#1.

Hmm. I see now!

Furthermore in idea#2 you used n=2 which is not right. What should n be?

I uploaded the wrong scan! CRAP! I did fix it before posting here. Yes, I know, it should be equal 1!




What about now?

http://img861.imageshack.us/i/correctlast.jpg/

 

[ideasrule]

There are two potential problems here. First, that compound interest formula is only approximate, not exact. That would be OK if we weren't trying to compare small differences between two nearly-identical investment plans, which we're doing here.

Second, in that formula, "r" is the yearly interest rate. For plan B, you're given the biyearly interest rate, which is not "r". You can't just divide it by 2, either; that only works for simple interest, and we're dealing with compound interest.

Instead of using the formula, think about it this way. For plan B, after 2 years, you have (1+0.1)*A0 where A0 is what you had before. After 4 years, you have 1.1^2*A0. Now you take that 1.1^2*A0 dollars and invest it in plan A for a year. What are you going to have at the end?

 

[Femme_physics]

There are two potential problems here. First, that compound interest formula is only approximate, not exact. That would be OK if we weren't trying to compare small differences between two nearly-identical investment plans, which we're doing here.

Second, in that formula, "r" is the yearly interest rate. For plan B, you're given the biyearly interest rate, which is not "r". You can't just divide it by 2, either; that only works for simple interest, and we're dealing with compound interest.

Instead of using the formula, think about it this way. For plan B, after 2 years, you have (1+0.1)*A0 where A0 is what you had before. After 4 years, you have 1.1^2*A0. Now you take that 1.1^2*A0 dollars and invest it in plan A for a year. What are you going to have at the end?

I see you posted this 3 minutes after my latest post. Could I have corrected some of the mistakes you've mentioned there already?

 

[ideasrule]

I don't think you did. Your answer for "idea 2" is correct, but for idea 1, you performed (1+0.2)^2. It should really be (1+0.1)^2, because your balance increases by 10% every two years, and you're waiting for 4 years.

 

[Femme_physics]

but it's r/n

r is the "interest rate expressed as a decimal" = 0.10
and n, we agreed it's 0.5

With that I get

0.10/0.5

So what changes from these 2?

 

[I like Serena]

I see you posted this 3 minutes after my latest post. Could I have corrected some of the mistakes you've mentioned there already?

In the formula you have given, "r" has not been defined entirely properly.

What it should say is:
r = interest rate per year expressed as a decimal

Since bank B gives a biyearly interest of 10%, it has r=0.05 (and not r=0.10).

 

[I like Serena]

but it's r/n

r is the "interest rate expressed as a decimal" = 0.10
and n, we agreed it's 0.5

With that I get

0.10/0.5

So what changes from these 2?

It changes from

0.10/0.5

to

0.05/0.5

 

[Femme_physics]

What it should say is:
r = interest rate per year expressed as a decimal

Since bank B gives a biyearly interest of 10%, it has r=0.05 (and not r=0.10).

Ohh...thanks! That's the fix I needed :D

I like I like Serena!

Thank you everyone :) And sorry for the lame pun, ILS.
Got the right answers!

 

[I like Serena]

Ohh...thanks! That's the fix I needed :D

I like I like Serena!

Thank you everyone :) And sorry for the lame pun, ILS.

Got the right answers!

That's quite all right, I've used a variant of that pun for ages as my signature