How can equations solve these two word problems?

[ExamFever]

Homework Statement

P1
We put 4000 USD per year into a savings bank at the beginning of the year for 10 years. After 10 years we take out 4000 USD each year. How much money will there be left in the account by the end of twenty years, if the rate of interest is 5%?

P2
A train departed from city A to town B 180 km away at 9 in the morning. Tom arrived at the railway station late, therefore he weant from A to B by car. He arrived at B the same time as the train. The average speed of the car was 40 km/hours more than the average speed of the train. When started Tom from A?

Homework Equations

The Attempt at a Solution

These are two word problems I have difficulty with.

P1
I can calculate this year by year, add up the total and give an answer to the question. However, I feel this is not the way they will want me to do it. Is there any way to put this into formula? I have tried several things including a geometric sequence but it does not seem to add any more value then the line-by-line approach.

P2
I actually sent in this problem to some "Dr. Math" website that helps with math problems. I tolld them the following:


V = velocity of the train
V+40 = velocity of the car

9+(180/V)=x+(180/(V+40)) - two variables here, can't solve x

This is the closest I come to breaking the problem down into an
(solveable) equation.

I've tried other options but I always keep having two variables.

They sent me a reply saying that this one could not be solved without more information. Just wanted to check with the pro's here whether that is correct.

Thanks!

 

[Borek]

Have you checked how much money there will be after first 10 years? Doesn't look to me like you can draw money for the next 10 years at $4000 per year.

I agree with Dr. Whoever Math - there is not enough information to give a unique answer. It can be expressed as a function of train speed only (which you have almost done already).

 

[Mentallic]

P1) Borek, if you add $4000 per annum for 10 years with interest on it, and then withdraw $4000 per annum for the next 10 years, you will still have money left over since you deposited and then withdrew the same amount, but interest was earned on money you left in the bank.

Your hunch was correct, you need to solve this in a more elegant way rather than just by calculating it year by year.

Set up this problem in such a way that you can find the pattern happening each year,

Y₁ = Year 1 = 4000
Y₂ = Y¹(1.05)+4000 = (4000)(1.05)+4000 = 4000(1+1.05)
Y₃ = Y₂(1.05)+4000 = (4000(1+1.05)(1.05)+4000 = 4000(1+1.05+1.05²)
...
Y₁₀ = ?
Y₁₁ = Y₁₀(1.05)-4000
...
Y₂₀ = ?

Can you find the pattern to finally get Y₂₀? From there you would use a geometric series formula to simplify and solve it.

P2) Dr. Math is correct, you can't solve this problem without knowing more information, such as what time the train and car get to B or what time the car leaves A. What you have is a linear relationship between the two variables (the way I solved the problem, I ended up with time of car and time of train variables, and you had velocity and time of something instead, they're not much different really) then if you take say V=50, then you will get some value for x, and if you use V=60 you will get some other value for x. Both values of V and x satisfy the conditions in the problem, so you can't narrow down what the precise values are supposed to be without further information.

 

[Borek]

P1) Borek, if you add $4000 per annum for 10 years with interest on it

Sorry, misread the question, missed "per annum" part, somehow I thought you start with just $4k

 

[Mentallic]

Haha that's a relief, because I felt kind of silly explaining such a logical concept