Question

You and your best friend want to take a vacation to Peru.  You have done some research and discovered that it will cost $2400 for the plane tickets, all-inclusive hotel and resort, and souvenirs.  You have already saved $1750.  If you invest this money in a savings account with a 2.3% interest rate compounded annually, how long will it take to earn enough money to go on the trip?  Use the compound interest formula A = P (1 + i)n, where A is the accumulated amount, P is the principal, i is the interest rate per year, and n is the number of years. Round your final answer to the nearest tenth

Answer 1

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

Answer 2

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

`A=P(1+i)^n\\\text{we get,}\\\Rightarrow 2400=1750(1+(2.3)/(100))^n\\\Rightarrow(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`

So, it will take around 14 years to earn enough money to go on the trip.