Shane Jones wants to buy a one-way bus ticket to Toowoomba,Queensland.The ticket costs $142,but Mr.Jones has only $80.If Shane puts the money in an account that pays 9% interest compounded monthly,how many months must Shane wait until he has $142 (round to the nearest month)?
A)73 months
B)75 months
C)77 months
D)79 months

Question

Quizplus · Accepted Answer

First, we need to use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the ending amount, P is the principal (starting amount), r is the interest rate, n is the number of times interest is compounded per year, and t is the number of years.
We know that P = $80, A = $142, r = 0.09 (9%), n = 12 (monthly compounding), and we want to solve for t.
Plugging these values into the formula, we get:
$142 = $80(1 + 0.09/12)^(12t)
Simplifying, we get:
1.775 = (1.0075)^12t
Taking the natural logarithm of both sides, we get:
ln(1.775) = ln(1.0075)^12t
Using the rule of logarithms that says ln(a^b) = b*ln(a), we can simplify further:
ln(1.775) = 12t*ln(1.0075)
Dividing both sides by 12ln(1.0075), we get:
t = ln(1.775)/(12ln(1.0075))
Using a calculator, we get:
t ≈ 76.8 months
Rounding to the nearest month, we get:
t ≈ 77 months
Therefore, Shane would need to wait 77 months (about 6 and a half years) to have enough money to buy the bus ticket.

Shane Jones Wants to Buy a One-Way Bus Ticket to Toowoomba,Queensland.The