8th grade math opened ended question?

Question

Suppose that you invest $10,000 for 5 years.

* If the compound interest rate is doubled, does the amount accumulated also double at the end of 5 years? Defend your answer with an actual illustration.

Answer 1

This is eighth grade? You must really be smart!

The formula for continuously compounded interest is A = Pe^(rt)

A = final amount, P = starting amount, r = rate as a decimal, t = years

The amount accumulated is the interest which is A - P

Try for r= 0.05 and r = 0.10

A = 10000e^(.05 x 5) = 12840.25 = 2840.25 interest

A = 10000 e^(.10 x 5) = 16487.21 = 6487.21 interest

You can see it more than doubled

Answer 2

The answer is no, unless the interest rate is the uninteresting zero percent.

A REALLY easy example is this: suppose the interest rate is 100%.
You invest 10,000, and after one year it is worth 20,000
After two years it is worth 40,000.

If instead the rate were 200%, then after 1 year it is worth 30,000 and after 2, 90,000. MORE than twice the accumulated amount at the 100% interest rate.
This of course can be extended to 5 years, and the answer will be even more dramatic.

Answer 3

okay.. the interest rate is not given, so let's use 0.1 and 0.2,
equation we using is Ce^rt,
C is the constan, r is the inteset rate, t is the time
plug in the number, you will find the answer.
if you really want to know how the equation work. here we go!
some calculus & differntial equation math here.

dQ/Dt = .1Q
dQ = .1Q x Dt
dQ/Q = .1Dt intergrat both parts
ln(Q) = .1t + c use e to cancial ln
e^ln(.1Q) = ce^.1t
Q = Ce^.1t
C is the $ start with. t is the time you in year.
.1 is the interest rate. and this is not 8th grade question
if rate = .1
Q = 10,000e^(.1 x 5)
Q = 10,000e^.5
Q = 16487.2interest is 6487.2, double is 12974.4
if rate = .2
Q = 10,000e^(.2 x 5)
Q = 10,000e^1
Q = 27182.8interest is 7182.8
so they are not equal.

Answer 4

no.itwill not double
let the rate be 10% compounded annually
A10=10000(110/110)^5
A20=10000(120/100)^5
it is easily seen that10000[(120/100)^5-1] is not twice 10000[(110/100)^5-1] and so the interest will not be double

Answer 5

Say you get an interest rate of 10%.

Then you would have (1.1 for 100% + 10%):

1 10,000 * 1.1 = 10,100
2 10,100 * 1.1 = 11,110
3 11,110 * 1.1 = 12,221
4 12,221 * 1.1 = 13,443.10
5. 13,443.10 * 1.1 = 14,787.41

So your interest is $14,784.41 - $10,000 = $4,787.41.

Now say you get an interest rate of 20%.

Then you would have (1.2 for 100% + 20%):

1 10,000 * 1.2 = 12,000
2 12,000 * 1.2 = 14,400
3 14,400 * 1.2 = 17,280
4 17,280 * 1.2 = 20,736
5. 20,736 * 1.2 = 24,883.20

So your interest is $24,883.20 - $10,000 = $14,883.20

That's almost three times as much interest as when the rate was 10%.

Answer 6

I dont think it would.
$10,000 for 5 years

Assume your interest rate is $2/year.
$10,000+($2)(5)
$10,000+$10= $10,010

Double your interest rate: $4
$10,000+($4)(5)
$10,000+$20
$10,020
Your amount did NOT double.

Answer 7

No the acumulated amount will not double, yah that is kinda advanced, i remember doing linear equations and algebra in grade 8.