Investing Math Problem

Question

Miguel decided to invest some of his money in an account gaining 3% interest compounded semiannually. He ultimately would like to purchase a $60,000 car.

a. How much would he have to invest initially have the necessary money in 4 years? Round your answer to the nearest whole dollar.

b. Miguel realized he only has $20,000 to invest. If he invests all $20,000 in the same amount described above, how long would it take for him to reach the $60,000 he needs? Round to nearest whole year.

Answer 1

Hello Alexia,

The compound interest formula is A = P (1+ r/n)^nt with A = final amount, P = principal (initial amount invested), r = interest rate, n = number of compounding periods per year, t = time

a. You're given A = 60000, r = 0.03, n = 2 (because semi-annual), t = 4. You need to find P, the initial investment amount

60000 = P (1+ 0.03/2)^2*4 ---> 60000 = P (1 + 0.015)⁸ ---> 60000 = P (1.015)⁸

From there, you should be able to solve for P and round to the nearest dollar.

b. This time around, you're given: P = 20000 and need to find t, the time it'd take for Miguel's investment to turn into $60,000. n =2 as in part a; r = 0.03

60000 = 20000 (1+ 0.03/2)^2*t ---> 60000 = 20000 (1.015)^2t

Divide both sides by 20000: 3 = (1.015)^2t

Take natural log of both sides: ln 3 = ln (1.015)^2t

Use power rule for logarithms to simplify right-hand side: ln 3 = 2t ln(1.015)

solve for t:

ln 3

t = ___________

2*ln(1.015)

From there, all you have to do is plug it into your calculator and round to nearest whole year.