An investment of $1000 was made in a certain account and ear

HI anon1,I think you miss the point " compound interest" , the formula must be

1000* (1+r)^12 = 4000 (1)
1000* (1+r)^x =8000 (2) ( x : is the number of years we must find)

(1): --> 1+r= (12 root of 4)
(2)--> (1+r)^x=( 12 root of 4)^x = 8
---> 4^(x/12) =8
2^(x/6)=2^3
x/6=3 >>>x=18

hope this help ^^

I'm sorry, I'm losing you.

I'm getting

(1) (1+r)^12 = 4
(2) (1+r)^x = 8

I don't know what goes on after this... not understanding your work

But why can't we just find out what the r first and then work backwards from the solutions to find the amount of years necessary to get an interest of 7000 / total of 8000

(1): --> 1+r = √(12&4) (get 12 root of 4) then apply this to (2)
(2)--> (1+r)^x=√(12&4) ^x = 8
---> 4^(x/12) =8
2^(x/6)=2^3
x/6=3 >>>x=18

if you find r it hard for you to calculate !

Hey Vips,

I tried to figure out

from 1000 to 4000 = 4 times
from 4000 to 8000 = 2 times.

But couldn't

You mean 1000 to 4000 = 4*1000
4000 to 8000 = 2 *2000

If so how u took first as 1000 and next as 2000 ..... Or any other logic?

Can you brief it plzzz... !!!

I used a logical approach to solve this:
In 12 years, the investment doubled twice. Therefore, in the first 6 years, the investment doubled from 1000 to 2000, and in the next 6 years the investment doubled again from 2000 to 4000. Since the rate of return remains the same, the investment will take another 6 years to double from 4000 to 8000.
Total time is 12+6=18

Bunuel following that same logic: if I put 1000 into an account with 100% interest in year 1 I will get 2000, year 2 4000 year 3 8000. So I increased my investment eight fold in three years. Now if I want my investment to go to double to 16,000 it would take one more year. But according to what has been said above 3/4 would be the answer. Are you counting the initial deposit as year 1?

At the rate of 100% (*2) the investment increases 8 times in 3 years.
In 3 years the investment increases 2*2*2 = 8 times (from $1,000 to $8,000). The doubling period is 1 year.
Thus, at the same rate compounded annually, it would need additional 3/3 = 1 year to double (from $8,000 to $16,000).
Therefore, 3+1=4 years are needed $1,000 to increase to $16,000.

We can solve this another way-

The amount 1000 quadrupled in 12 years- that means the investment increased by 1/3rd (4/12) each year.

Thus to increase the investment by 2 more times ( 1/3 a year + 1/3+ 1/3+ 1/3 + 1/3 + 1/3) it will take 6 years.

12+ 6 = 18 years!

The math is not that bad if you just use the exponential notation for roots. We have:

1000 (1+r) ^12 = 4000 ----> (1+r) = 4 ^(1/12) = 2 ^ (1/6)

1000 (1+r) ^x = 8000 ----> (1+r) = 8 ^ (1/x) = 2 ^ (3/x)

Find x:

1 / 6 = 3 / x -----> x = 18

Can solve it like this...

Taking simple numbers: Invested $1 and it grew $4 in 12 years
4 = (1+R)^12

Taking square root both the side, we get: 2 = (1+R)^6

To solve for $8:
Multiply both equation we get: 4*2 = (1+R)^(12+6)
8 = (1+R)^18

Answer: 18 years

start=1000
year1= 1000x
year2=1000x^2
...
year 12=1000x^12

so 1000x^12=4000

x^12=4

1000(x^12)^?=8000

(x^12)^?=8

since x^12=4, 4^?=8

2^2y=2^3

2y=3 so y=3/2

1000(x^12)^(3/2)=8000

Using the power property, we obtain 1000(x^18)=8000

so 18 years

Not the best way but made an educated guess by estimating with simple interest:
-- If it took 12 years to get from 1000 to 4000, it should take considerably less than +12 to get to 8000 since we will be increasing the amount of interest added each year.
-- If we used simple interest the answer would be 28 years (3000/12 so $1000 per 4 years).

E) is definitely too big
D) is out too because 24 is too close to 28 (compounding should make the amount of time much less)
A) is definitely too small (no way to make up that much in only 4 years even with compounding)

Between B and C, I thought B was more likely to be correct since it's closer to 12 and 20 is exactly 8 away from 12 and 28.

\(1000* \frac{(100+r)}{100}^{12} = 4000\)
let 100+r/100 be t
t^12=4
t= 4^1/12

to find:
1000*t^n=8000
substituting value of t
4^n/12=8
2^2n/12=2^3
n/6=3
n=18

Let me give a simplified approach to such questions

use percentages to solve this ...

Increase of 300% (from 1k to 4k) in 12 Years....

Next 12 Years, again 300%.. which comes out to be 16k, but we need 8k.... so divide time/2 ... ie. 6 more Years...

Total = 12 Years + 6 Years = 18 Years

We're told : \(1,000 (1 + \frac{r}{100})^{12} = 4,000\)

= \( (1 + \frac{r}{100})^{12} = 4\)

= \( (1 + \frac{r}{100})^{6} = 2\)

= \( (1 + \frac{r}{100})^{18} = 2^3 = 8\)

After 18 years, the initial investment would have increased to $8,000.

Bunuel Could you explain why you took 3/3? Based on the original question posted by someone else, I also arrived at 3/4.
for 8 times -> it took 3 years. to double -> it takes 3/4th of year using proportions.

I thought you followed the same pattern with original question:
for 4 times -> it took 12 years. To double -> it takes 6 years using proportions.

where are we missing ?

The tricky thing here is that the method folks are (correctly) using to determine doubling periods does NOT automatically tell us the annual rate of change. When we're dealing with exponential growth, it's not correct to say that something that quadruples in 12 years is growing by 4/12 = 1/3 of the initial investment per year. If that were the case, the growth would not be exponential.

The helpful thing about doubling periods is that they give us a nice stable number to work with. Once our exponential growth leads to a doubling, that same time span will keep producing another doubling. So we're working specifically with number of DOUBLINGS, not total growth. In the original example, there were 2 doublings and we just needed one, so it would take 1/2 as long. In the follow-up question, there were 3 doublings and we just wanted one, so it would take 1/3 as long.

If we don't have a clean number like 2 or 3, you can see the trouble we get into. What if our investment has grown by 60% after 3 years? Has it grown by 20% each year? No, that's just the average growth. We've multiplied by some factor 3 separate times to reach 1.6, so x^3 = 1.6. That's not going to be too fun without a calculator, although you can predict that the annual rate will be a bit less than 20% or the compounding would cause the growth to exceed 60%.