Peter and his wife want to buy a house, and they are planning to save $500 each month for the down payment.
The table shows how much interest Peter and his wife can earn if they invest $500 each month with a 5.5% annual interest rate, compounded monthly.
If the goal is to save at least $25,000 for the down payment, how long should Peter and his wife invest $500 each month? How long would it take to save $25,000 if they were to save $500 each month with no interest?
(1 point)
Peter and his wife should invest $500 monthly at 5.5% annual interest for years. With no
interest, it would take them months to save up $25,000.
To find out how long it will take Peter and his wife to save $25,000 with a 5.5% interest rate, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = Final amount
P = Principal amount (the amount they are saving each month)
r = Annual interest rate (in decimal form)
n = Number of times the interest is compounded per year
t = Number of years
In this case, P = $500, r = 0.055 (5.5% annual interest rate in decimal form), and n = 12 (compounded monthly).
Let's plug in the values:
$25,000 = $500(1 + 0.055/12)^(12t)
Simplifying:
50 = (1 + 0.00458)^(12t)
Taking the logarithm of both sides:
log(50) = log((1 + 0.00458)^(12t))
Using the power rule of logarithms:
log(50) = 12t * log(1 + 0.00458)
Simplifying further:
t = log(50) / (12 * log(1.00458))
Using a calculator, we can find that t is approximately 6.74 years.
Therefore, Peter and his wife should invest $500 monthly at a 5.5% annual interest rate for approximately 6.74 years to save $25,000 for the down payment.
If they were to save $500 each month with no interest, it would take them $25,000 / $500 = 50 months to save up $25,000.