Rio invested 200 dollar at an annual interest rate of r% compounded an

Question

Rio invested 200 dollar at an annual interest rate of r% compounded annually, and the value of the investment becomes 1,800 dollars at the end of 10 years. If 100 dollars is invested at the same rate of interest compounded annually, after how many years will the investment of 100 became 2700 dollars ?

A. 10
B. 12
C. 15
D. 20
E. 24

A. 10
B. 12
C. 15
D. 20
E. 24

Bunuel · Answer

Rio invested 200 dollar at an annual interest rate of r% compounded annually, and the value of the investment becomes 1,800 dollars at the end of 10 years:
 
 200(1+\frac{r}{100})^{10}=1800

(1+\frac{r}{100})^{10}=9

(1+\frac{r}{100})=\sqrt {9} 
 
If 100 dollars is invested at the same rate of interest compounded annually, after how many years will the investment of 100 became 2700 dollars:
 
 100(1+\frac{r}{100})^{n}=2700

100(\sqrt {9})^{n}=2700

(\sqrt {9})^{n}=27  
 
Taking to the 10th power:
 
 9^n=27^{10}

3^{2n}=3^{30}

2n=30

n = 15 .
 
Answer: C.

tgsankar10 · Answer

First Investment:

200*(1+r)^{10}=1800

(1+r)^{10}=9

(1+r)=9^{(\frac{1}{10})}

Second Investment:

100(1+r)^n=2700

(1+r)^n=27

9^{(\frac{n}{10})}=27

3^{2*(\frac{n}{10})}=3^3

Equating the powers:

\frac{2n}{10}=3

n=\frac{30}{2}

n=15

Answer: C

Kinshook · Answer

S2930

Given: Rio invested 200 dollar at an annual interest rate of r% compounded annually, and the value of the investment becomes 1,800 dollars at the end of 10 years.

Asked: If 100 dollars is invested at the same rate of interest compounded annually, after how many years will the investment of 100 became 2700 dollars ?
Let the number of years required be n and rate of interest be r annually.

Amount/Principal in 10 years = (1+r%)^10 =  \frac{1800}{200} = 9 = 3^2 
Amount/Principal in 1 years = (1+r%)=  9^{1/10} = 3^{1/5}

ln (1+r%) =  \frac{ln 3 }{ 5}

Amount/Principal in n years = (1+r%)^n =  2700/100 = 27 = 3^3 times

n ln(1+r/%) = 3 ln 3

n ln(1+r/%)/ln (1+r/%) =  n = 3 ln 3 / (\frac{ln 3}{5}) = 15  years

IMO C

pappal · Answer

200(1+ r/100)^10=1800
(1+ r/100)^10=9 or (1+ r/100)^5=3------------------i
let in t yrs. the value of 100 becomes 2700
so, 100(1+ r/100)^t=2700 
(1+ r/100)^t=3^3 or (1+ r/100)^t/3=3----------------ii
from i and ii 
t/3=5 or t=15 yrs.

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