Question

You’re trying to save to buy a new $180,000 Ferrari. You have $29,000 today that can be invested at your bank. The bank pays 3.6 percent annual interest on its accounts.

How long will it be before you have enough to buy the car? (Do not round intermediate calculations and round your answer to 2 decimal places, e.g., 32.16.)

Length of Time in years?

Answer 1

Required to save $180,000 means it is maturity amount and amount invested $29,000 and bank pay interest @3.6% p.a. and will become $180,000 after t years and bank paid interest every year means we received interest on interest means compounded annual so we can use compound interest formula to calculate time.

A = P (1+r)^t

A = Matured Amount

P = Orincipal Amount

r = Interest rate

t = Time /Years of investment

$180,000 = $29,000 (1+0.036)^t

(1+0.036)^t = $180,000 / $29,000

(1+0.036)^t = 6.2069

(1.036)^t = 6.2069

Here we take t = 50

So,

(1.036)⁵⁰ = 5.8612 which is less than 6.2069 so now we take t = 52

(1.036)⁵² = 6.2908

From above we understand that t falls between 50-52 so we take 51

So,

(1.036)⁵¹ = 6.0722

so t falls in 51-52

now interpolate it

t = Base year + [{( Value at t required) - (Value at t = 51)} / {(Value at t = 52) - (Value at t =51)}]

t = 51 Years + {(6.2069 - 6.0722) / (6.2908 - 6.0722)} x 1 Year

t= 51 Years + (0.13470.2186) x 1 year

t = 51 Years + 0.6161 Year

t = 51.6161 Years or say 51.62 Years Approx.

So if interest compound annually then approx 51.62 Years of length time.