Homework Help: Compound Interest Systems of Equations Problem

1. Apr 16, 2015

ConstantineO

1. The problem statement, all variables and given/known data
Romeo was given a gift of $10,000 when he turned 16. He invested it at 3% per annum. Three years later, Juliet was given$10,000, which she invested at 5% per annum. When will the two amounts be equal in value?

2. Relevant equations
Compound Interest Formula
Total = Capital(1+interest)^Years
t = c(1+i)^n
3. The attempt at a solution
Deciding how to create my formula is where things get fuzzy. The way I initially attempted this was to add a 3 year head start onto Romeo's interest formula. This is what results (Used wolfram to save time rewriting):

This eventually results in "4.61103." However, this is not the answer at the back of my textbook. They decided to opt for a different formula where the time is subtracted from Juliet's interest function. This is shown here:

This results in the correct answer of 7.61103. What I'm wondering, why do I have to subtract 3 years from Juliet's compound interest formula, and why does adding 3 years to Romeo's compound interest formula produce the wrong answer?

2. Apr 16, 2015

BvU

So clearly the book starts counting at the beginning of the story: when Romeo brings his money to the bank.
That means Julia's interest clock is at x-3 when Romeo's is at x.

3. Apr 16, 2015

SammyS

Staff Emeritus
What is Romeo's age when then two accounts have equal value?

4. Apr 16, 2015

ConstantineO

Is this always the convention when dealing with problems like this? From my experience with physics courses, I assumed that time shares a relation with the total amount of money generated by compound interest. The interest and initial capital is thus fixed, and you are only left with a relation between time and the total sum of money. If you were to treat time as a vector which you could progress backwards and forwards on, each snapshot through time depending on which way you were progressing would net either a larger or smaller amount of the total. I don't understand why the book is forcing me to start at the beginning of the story. The story is like pages of a book which can be turned towards or away from the end of the novel. The content of the novel does not change and its contents are already written.

I don't understand what Romeo's age has to do with this? Is this some kind of statement that alludes to time only progressing forward or something?

5. Apr 16, 2015

BvU

Nothing so fancy. The book is sloppy by not stating when it wants the clock (calendar in this case) to show t = 0. From the book answer it appears that it assumes t = 0 when Romeo turns 16. The exercise could equally well have assumed t = 0 when Julia receives the ten grand. If I were grading this, I'd have to allow both answers, certainly when the calculation steps are shown.

6. Apr 16, 2015

ConstantineO

I actually just figured it out while discussing this question with a friend just a few moments before you posted. I took the delta of the two times from both of the ways of solving this question and realized that this is nothing more than a semantics issue of where you measure time from. This question is poorly setup, and I am going to voice my irritation to the person marking.

7. Apr 16, 2015

Ray Vickson

A more serious problem is that when compounding annually, the answer is "never". At Romeo's 23rd birthday, Romeo's account contains more money than Juliet's, while at his 24th birthday, Juliet's account contains more than Romeo's. Unless there is something like continuous compounding, the two amounts never match exactly; that is, unless you can withdraw your money part-way through a year and earn a part-year's interest, you could never find a time where the two withdrawals are equal. (However, I would not raise this issue with your teacher if I were you; just "go with the flow".)

Also, instead of expressing irritation, it would be wiser for you to say that you will give two answers (depending on where to start measuring time) and point out (nicely) that the question is a bit ambiguous.

8. Apr 16, 2015

SammyS

Staff Emeritus
RGV makes an excellent point. This is a short-coming of many problems of this type.

It's often the case that for any fraction of a single compounding period that's "left over", simple interest is computed for that extra amount of time. With this method, find the value of the account at x = 4, then see if the values can be equal at any time in the 4th year using simple interest.

(Edited slightly.)

Last edited: Apr 16, 2015
9. Apr 16, 2015

ConstantineO

I don't wish to sound like an imbecile, but I have no clue what that means. How would I set up a systems of equation if I subbed in 4 for both x's? I believe I would be left with only one unknown for either equation, or am I misunderstanding you? Please clarify what you mean by using x=4.

10. Apr 16, 2015

SammyS

Staff Emeritus
When x = 4, (7years for Romeo, 4 years for Juliet), what is the value in each in each account?

Start from that point for each & using simple interest, and see how long it takes for the values to be equal.

11. Apr 16, 2015

ConstantineO

I am not familiar with the simple interest formula, so I googled it. I am operating under the assumption that it is different than the compound interest formula.

I am to assume this is correct?
Simple interest Formula
I = "the grown money"
p = the initial capital
r = interest rate per year
t = t
(I) = (p)(r)(t)

I think I know what you're getting at, so let me make an attempt.

Romeo Compound Interest Calculation Total
Ar = 10,000(1.03)^4+3
Ar = 10,000(1.03)^7
Ar = 12,298.73865

Julie Compound Interest Calculation Total
Aj = 10,000(1.05)^4
Aj = 12,155.0625

Romeo Simple Interest Calculation
(Ir) = (12,298.73865)(0.03)(t)
(Ir) = (368.94)(t)

Juliet Simple Interest Calculation
(Ir) = (12,155.0625)(0.05)(t)
(Ir) = 607.753125(t)

Equate the two to equal each other.

(368.94)(t) = 607.753125(t)
(368.94)(t) -607.753125(t) = 0
t(368.94-607.753125)=0

This doesnt work though...

12. Apr 16, 2015

Ray Vickson

It's not that complicated. Let $R_t, J_t$ be the amounts in Romeo's and Juliet's accounts at time $t$ (in years, measured from Romeo's 16th birthday). We have:
$$\begin{array}{ccc} & t=7 & t=8 \\ R_t & 12298.74 & 12667.70 \\ J_t & 12155.06 & 12762.82 \end{array}$$
You can draw two straight-line plots, one going from x = 0 at (7,R7) to x = 1 at (8,R8) and another from x = 0 at (7,J7) to x = 1 at (8,J8). Where the two lines cross is the point where simple interest calculations (for the partial year) would give you equality of monetary values. This will be close to (but not exactly equal to) the value x = 0.611---that is, for time 7.611 ---that your previous calculation would have produced. (The difference is due to the slight curvature in the money vs.time graph over the year, due to continuous compounding over the year, as compared with the straight line obtained from simple interest accumulation over the year.)

13. Apr 16, 2015

ConstantineO

Could you show me an example of this because I am simply not understanding what you're saying, or can you show me how to equate the two together using algebra?

14. Apr 17, 2015

ConstantineO

This is getting out of hand, so I am going to try to be as clear and as concise as possible. I know how to calculate the correct answer for this question, and I realize where I first made my mistake. I am now wondering what everyone is talking about when they mention using simple interest. I have literally 0 experience doing any simple interest calculations, and I have no idea what sammyS is talking about here.

If anyone is holding information back because they believe an attempt has not been made and it would violate forum rules, please stop. I understand how to do the book's question using the books method, and I know and where the discrepancy between the 4.61103 and 7.61103 came from. I have made a wholehearted attempt so please for sake of my sanity just give me an example or step by step breakdown. I learn through examples, I look at the process involved, and I create a model in my head how everything interacts and relates with each other.

What I am seeking is an explanation and a simple example of what this simple interest method is for calculating the 0.61103 year after the exact 7 years has passed.

What I am understanding so far.
- First use the compound interest formula to create outputs for both Romeo's and Juliet's accounts after being influenced by exactly 7 years of interest.
- This is done by assigning "4" to the variable x
- Get said products, in this case being:
Romeo's Total After Exactly 7 Years = 12,298.73865
Juliet's Total After Exactly 7 Years = 12,155.0625
- Since you know that the two amounts are still not equal, you must calculate the remaining time between the two using the Simple interest formula
- This is what I am having problems with and what I would appreciate having an example for. Preferably shown algebraically, so I am not plotting nearly straight lines in Desmos.

Not to sound contentious, but when you try to find the point of intersect of two nearly straight lines with a slope that is in hundredths of a unit, it is just plainly byzantine. I am probably doing this wrong, but graphically representing each simple interest formula and trying to find where they meet is insane.

Last edited: Apr 17, 2015
15. Apr 17, 2015

Ray Vickson

How could I have made my explanation any simpler? There was no hidden information or holding back of anything. I said: just draw two straight-line graphs for Romeo and Juliet, where the lines connect their (time,money) points at the start and end of the final year.

The true graphs of time vs. money will be curved, because of compound interest, but over a short period such as one year the "curvature" will be small, and the graph will look almost like a straight line; replacing the curve by a straight line ---only for that single year---would give you the "simple" interest schedule for that year. It will be almost the same as the compound interest schedule, because the curvature is small over short times such as a year.

If you don't believe me, just draw the graphs of money vs. time for Romeo and for Juliet. You will see that they curve up, but are almost straight over short periods. The only way to really understand it is to sit down and do it.

16. Apr 17, 2015

ConstantineO

I am going to assume the two lines would be plotted as
y=(12155.0625)(0.05)(x)
and
y=(12298.73865)(0.03)(x)

If that's the case, I don't understand how any human being is supposed to find the point of intersect graphically like this.

I did sit down and do it. I have been sitting down and ripping my hair out doing this for sometime.

Edit - In the event that anyone else would like to question how committed I am to "sitting down and doing it." I've had enough of that nonsense from teachers with superiority issues and over bloated opinions that like to question how hard I am making an attempt.

Last edited: Apr 17, 2015
17. Apr 17, 2015

Ray Vickson

As long as you persist in trying to plot the wrong thing you will need to keep pulling out your hair. Go back and read what I wrote in #12. Do the plot exactly as I indicated there.

Better still plot the two curves y = 10000 * (1.03)^x and y = 10000 * (1.05)^(x-3) for x = from 3 to 8. They are curved, aren't they? Don't they look almost straight over the shorter interval 7 \leq x \leq 8? in each case if you were to plot a straight line from (7,y(7)) to (8,y(8)) it would look very close to the actual curve (x,y(x)) for x between 7 and 8. The "simple interest" effect between 7 and 8 would be the straight line, while the "compound interest" effect would be the curve. They both pass through the same points at 7 and 8, but they differ in between.

18. Apr 17, 2015

ConstantineO

As I have stated before.
I was completely uncertain of what to plot. I am being told to use the simple interest formula, but the two formulas you have provided are compound interest formulas:
y = 10000 * (1.03)^x
y = 10000 * (1.05)^(x-3)
I plotted these long before I even got to this whole simple interest business when trying to figure out the original discrepancy between 7.61103 and 4.61103. I am fully aware that the compound interest lines are nearly straight. I don't see how the average rate of change between x =7 and x =8 is going to help me discover the remaining time of 0.61103 years. I am still unclear of what to plot? Do I plot the compound interest based functions for Romeo and Juliet, or do I plot the Simple interest functions? I have done both and I am not seeing anything that is giving me any kind of better understanding. Is this operation you describe too difficult represent algebraically?

I am getting far too irritated by this and far too frustrated and am quickly losing the desire to continue.

19. Apr 17, 2015

ConstantineO

I was on the right track here. There is no need for this graphing nonsense. I forgot a single number which was my "+ 3" in my simple interest formula. I was a fool not to recognize that the lack of a constant would render my simple interest system of equation unsolvable. The lack of anyone speaking up about this is worrisome. In the future, when referencing the slope of a secant line between the range of two x values of a curve, please write in the notation of 7 < x < 8. I had absolutely no clue what you were talking about.

So let's try that last bit again with a fixed formula.

Romeo Simple Interest Calculation
(Ir) = (12,298.73865)(0.03)(t+3)
(Ir) = (368.94)(t+3)

Juliet Simple Interest Calculation
(Ir) = (12,155.0625)(0.05)(t)
(Ir) = 607.753125(t)

Now the Systems of Equation
(607.753125)(t) = (368.94)(t+3)
(607.753125)(t) = (368.94)(t)+ 1106.82
t(607.753125-368.94) = 1106.82
t(238.813125) = (1106.82)
t = (1106.82)/(238.813125)
t = 4.634669891
Not too far away from 4.61103. I wonder why...

Let's try the simplified interest formula that measures time from Julias deposit date.

Romeo Simple Interest Calculation
(Ir) = (12,298.73865)(0.03)(t)
(Ir) = (368.94)(t)

Juliet Simple Interest Calculation
(Ir) = (12,155.0625)(0.05)(t-3)
(Ir) = 607.753125(t-3)

607.753125(t-3)= (368.94)(t)
(607.753125)(t) - 1823.259375 = (368.94)(t)
(607.753125)(t) - (368.94)(t) =1823.259375
t(607.753125 - 368.94) = 1823.259375
t(238.813125) = 1823.259375
t = 7.634.634669891
Not too far away from 7.61103. Surprise surprise....

Look Ma! No graphs.
7.634.634669891 - 4.634669891 = 3 just like 7.61103 - 4.61103 =3. I wonder if they're related?
I don't think I am wrong by stating that the time after the 7 year mark would actually be 0.634669891 of a year instead of 0.61103 if fractional compound interest is indeed calculated with simplified interest.

We can take things a step further and look at the secant line's slope of the curve that models interest from Romeo's deposit date during 7 < x < 8. Let's look at the average rate of change for both Romeo's and Juliet's account.

Romeo Compound Interest AROC: 7 < x < 8

While x = 4
Ar = 10,000(1.03)^4+3
Ar = 10,000(1.03)^7
Ar = 12,298.73865
x = 4
y =12,298.73865

While x =5
Ar = 10,000(1.03)^5+3
Ar = 10,000(1.03)^8
Ar = 12,667.70081
x = 5
y = 12,667.70081

Romeo's AROC
= (12,667.70081 - 12,298.73865) / (5 -4)
= 368.96216

Juliet's Compound Interest AROC 7 < x < 8

While x = 4
Aj = 10,000(1.05)^4
Aj = 12,155.0625
x = 4
y =12,155.0625

While x = 5
Aj = 10,000(1.05)^5
Aj = 12,762.81563
x = 5
y = 12,762.81563

Juliet's AROC
= (12,762.81563 - 12,155.0625) / (5-4)
=607.7531

After that huge bunch of calculations we now have slopes that we can create linear relations with.

Romeo's Linear Relation
y = (368.96216)(x + 3)

Juliet's Linear Relation
y = 607.7531(x)

Let's see what the Solution is for these in equations when put into a system.

(368.96216)(x + 3) = 607.7531(x)
(368.96216)(x) + 1106.88648 = 607.7531(x)
607.7531(x) -(368.96216)(x) = 1106.88648
x(607.7531-368.96216) = 1106.88648
x(238.783884) = 1106.88648
x = 4.635515854

Notice a pattern?

Time Measured from Romeo's Deposit Compound Calculation Product
x = 4.61103
Time Measured from Romeo's Deposit Simplified Interest Product
x = 4.634669891
Time Measured from Romeo's Deposit AROC: 7 < x< 8 Product
x = 4.635515854

I think its safe to say that they matched sometime in the range of 0.635515854 - 0.634669891- 0.61103 of the start of the 8th year. I think I made my point. I don't need any graphs to know how to rock, and I certainly showed that I sat down and did the work.

20. Apr 17, 2015

Ray Vickson

Of course it can be done without graphs. Graphs can be of help in setting up the algebraic equations that must, eventually, be solved without graphs. However, you seemed to not understand how simple interest works (your claim, not mine) and to see that a graphical representation can sometimes be helpful---not always, just sometimes. If it was not helpful to you, fine. And, of course I suggested plotting the compound-interest graphs, but I guess you missed out the part where I said that during a short period, such as 1 year, the graph looks almost straight, and that a straight line replacement for the graph (but ONLY in that single year) gives you the simple-interest effect within the year.

Your writeup is almost incomprehensible to me, but your final answer seems OK. You say you still do not know why it is different from the original answer, and that is precisely what I was speaking of when I mentioned graphs: you could see right away why there is a difference.

Last edited: Apr 17, 2015
21. Apr 17, 2015

ConstantineO

I showed the difference when I computed the average rate of change of the non-linear exponential function during the interval 7<x<8. Although the slope is negligible it has a significant enough effect to alter the x value by hundredths of a unit. Of course the linear function based on simplified interest is going to differ due to it being well "linear." I had no clue what any of this meant:

I had no idea what to graph or what any of that statement meant so, I calculated three different ways of finding x to a reasonable certainty in my post above while borrowing some information from my initial post.

It was calculated by using the original compound interest formula,by modifying the simple interest formula and creating a systems of equations, and by interpolating the value of x using the secant line that is created by calculating the average rate of change between 7<x<8 and using the two gained slopes to create two new linear equations for use in a final system of equations.

Time Measured from Romeo's Deposit Compound Calculation Product (Initial Calculation from first post)
x = 4.61103
Time Measured from Romeo's Deposit Simplified Interest Product ( A modified and fixed version of my initial simple interest calculation and likely the most accurate)
x = 4.634669891
Time Measured from Romeo's Deposit AROC: 7 < x< 8 Product (Third method using intervals and average rate of change during (7<x<8) which was more or less done to numerically and logically support my modified simple interest calculation "A really messy way of checking my answer")
x = 4.635515854

22. Apr 17, 2015

Ray Vickson

If you draw the straight line from the point (7,12298.74) to (8,12667.70) you will get the simple-interest balance throughout year 7--8 for Romeo. Similarly for Juliet. That is what I wrote, but used symbols instead of numbers.

Of course, this amounts to using the linear function $R_t = 12298.74\, (1 + .03\,(t-7))$ for $7 \leq t \leq 8$. So, the simple-interest equality time for Romeo and Juliet is the solution of the linear equation
$$12298.70\,(1 + .03 \, (t-7)) = 12155.086\, (1 + 0.5 \, (t-7)),$$
while the compound-interest equality time is given by the solution of the nonlinear equation
$$12298.70 \, (1.03)^{t-7} = 12155.06 \, (1.05)^{t-7}.$$
The two solutions will be only slightly different because for $7 \leq t \leq 8$ the nonlinear function is almost linear---has small curvature over short intervals. Basicallly, we would be replacing geometric growth by arithmetic growth.

23. Apr 17, 2015

ConstantineO

Why have you arranged these formulas like this? A google search for me nets that the simple interest formula is

Interest = (Principal) (Rate) ( Time)

I am going to assume that.
Principal = 12298.74
Rate = 0.03
Time would = t-7

Why is there 1 + the Rate? Mathematically speaking it won't do much since you will only be subtracting either the Interest or in your case the Interest and the Principal combined from each other? This statement is unsolvable without one of the lines being shifted on the x-axis. As it stands, this is utterly useless due to both lines sharing the exact same solution of 7 to make them equal to each other. You've ultimately recited and made the exact same mistake as I did initially when I stated:

You cannot just equate Romeo's and Juliet's linear simplified interest lines like this. The logic even by graphing standards does not make sense. This statement gets killed off as soon as I sub a 7 in for either since it would instantly devolve into multiplying by 0. This algebraically speaking is nonsense, and they would both share a point at origin.

12298.70(1+.03(tâ7))=12155.086(1+0.5(tâ7))

The same here.

Now if you graph each function and see where the points intersect, yes you will find the answer but the above here has no basis in algebraic meaning. The new compound interest formula here is completely unnecessary since we can take the original two functions as described in the huge load of wolfram images and create a systems of equation from them that produces the same answer and doesn't require using intervals.

24. Apr 17, 2015

BvU

There is 1 + Rate * time, because when, as you write,
Interest = (Principal) (Rate) ( Time)
then amount is
Principal + Interest = (Principal) + (Principal) (Rate) ( Time) = (Principal) [ 1 + (Rate) ( Time) ]

25. Apr 17, 2015

ConstantineO

Oh I realize that. I am just wondering why he decided to write it like that when you are just creating a ratio in the step that follows. I was just wondering if there was any certain reason that they opted to do this.