You have 700 dollars in your bank account. Suppose your money is compounded every month at a rate of 0.5 percent per month.
(a) How much do you have after t years.
(b) How much do you have after 100 months
(a) The value after t years is modeled using the monthly interest rate:
(Value After t Months) = (Initial Value)*(Rate)^(12 Months * t)
Q(t) = Q(0)*(1.005)^(12*t) => Q(t) = 700*(1.005)^(12t)
(b) To find the value after 100 months, modify the formula from part a to;
Q(t) = 700*(1.005)^(t), as we're dealing with months now, not years. Once that is done, plug in 100 for t in the new formula:
Q(100) = 700*(1.005)^(100) = 1152.66794
amount = 700(1.005)^(12t)
amount = 700(1.005)^100
= 1152.67
To calculate the amount of money you have after a certain period of time, we will use the formula for compound interest:
A = P(1 + r)^n
Where:
A = the final amount of money
P = the initial amount of money
r = the interest rate (in decimal form)
n = the number of compounding periods
(a) To find out how much money you have after t years:
Since the interest rate is given monthly at a rate of 0.5 percent, we need to convert it to a decimal by dividing it by 100: r = 0.005.
The number of compounding periods will be the number of months in t years: n = 12t.
Applying this formula:
A = $700 * (1 + 0.005)^(12t)
(b) To determine the amount of money you have after 100 months:
Since we are given the number of months, we don't need to convert the interest rate or calculate the number of compounding periods. We can directly apply the formula:
A = $700 * (1 + 0.005)^100
To calculate the amount of money you will have after a certain number of years or months with compound interest, you can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment/amount of money
P = the principal amount (initial amount you have)
r = the annual interest rate (in decimal form)
n = the number of times that interest is compounded per year
t = the number of years/periods
In this case, we have an interest rate of 0.5% per month, compounded monthly. Let's calculate the answers to the given questions:
(a) How much do you have after t years?
To find out the amount of money you will have after t years, you can use the formula:
A = 700(1 + 0.005/12)^(12*t)
Let's say you want to calculate how much you will have after 5 years. Plug in the values:
A = 700(1 + 0.005/12)^(12*5)
A = 700(1 + 0.0004167)^(60)
A ≈ 700(1.0004167)^(60)
A ≈ 700 * 1.0253184
A ≈ 717.72
After 5 years, you would have approximately $717.72 in your bank account.
(b) How much do you have after 100 months?
To find out the amount of money you will have after 100 months, you can use the formula:
A = 700(1 + 0.005/12)^(12*100)
Plug in the values:
A = 700(1 + 0.0004167)^(1200)
A ≈ 700(1.0004167)^(1200)
A ≈ 700 * 1.748252
A ≈ 1223.776
After 100 months, you would have approximately $1223.78 in your bank account.