Solution:- As per Given Problem
Principal = $1750
Rate of interest (i) = 2.3 % =23/100 compounded annually
Required amount = $2400
To find time =n (years)
By using compound interest formula
[tex]A=P(1+i)^n\\\text{we get,}\\\Rightarrow 2400=1750(1+\frac{2.3}{100})^n\\\Rightarrow\frac{2400}{1750}=(1+0.023)^n\\\Rightarrow1.3714=(1.023)^n\\\text{taking log on both sides,we get}\\log(1.3714)=log((1.023)^n)\\\Rightarrow0.3158=n(log(1.023))\\\Rightarrow0.3158=n(0.0227)\\\Rightarrow\ n=13.911\approx 14[/tex]
So, it will take around 14 years to earn enough money to go on the trip.
Total amount required for vacation = $2400
Amount I am having = $1750
invested rate of interest compounded annually = 2.3%
Compound interest formula for Accumulated amount is
A = P ( 1 +i )ⁿ
where A = $2400 , $1750 , i = 2.3%
⇒$2400 = $1750 ( 1 + 2.3/100)ⁿ
⇒ 2400/1750 = (100 + 2.3/100)ⁿ
⇒ 2400/1750 = ( 102.3/100)ⁿ
⇒ 2400/1750 = (1023/1000)ⁿ
⇒ 1.3714 = (1.023)ⁿ
1.023^ 14 = 1.374
approx equal to 1.3714
so we can say , that need to wait for 14 years to go on vacation.
lets recheck our answer by using n = 14
A = P ( 1 + 2.3%)∧14
A = 1750 × ( 1 + 2.3/100) ∧14
A = 1750 × ( 1.023)∧14
A = 2406.66 nearly equal to our required amount .
Hence it will take 14 years to earn enough money to go on the trip
