Compound Interest Question
James has 10,000 to invest. If he invest money in a savings account that pays 4% APR, calculate the following:
a) how long will it take to have a total of 15,000 if interest is compounded quarterly (3months)?
b)how much will he have after 7yrs if interest is compounded yearly? monthly?
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To calculate compound interest, we use the formula:
A = P(1 + r/n)^(nt)
where:
A = the future value of the investment
P = the principal amount (initial investment)
r = annual interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
a) To calculate how long it will take to have a total of 15,000 if interest is compounded quarterly, we need to find the value of t.
Let's plug in the given information into the compound interest formula:
15,000 = 10,000(1 + 0.04/4)^(4t)
Divide both sides of the equation by 10,000:
1.5 = (1 + 0.01)^(4t)
Take the logarithm of both sides of the equation:
log(1.5) = log((1 + 0.01)^(4t))
Using logarithm properties, we can bring down the exponent:
log(1.5) = 4t * log(1.01)
Divide both sides of the equation by 4 * log(1.01):
t = log(1.5) / (4 * log(1.01))
Using a calculator, we find:
t ≈ 3.5 years
So, it will take approximately 3.5 years for the investment to grow to $15,000 if interest is compounded quarterly.
b) To calculate how much James will have after 7 years if interest is compounded yearly and monthly, we can use the compound interest formula with the respective compounding periods.
For yearly compounding:
A = 10,000(1 + 0.04/1)^(1 * 7)
A = 10,000(1 + 0.04)^7
Using a calculator, we find:
A ≈ $13,987.10
So, James will have approximately $13,987.10 after 7 years if interest is compounded yearly.
For monthly compounding:
A = 10,000(1 + 0.04/12)^(12 * 7)
A = 10,000(1 + 0.003333)^84
Using a calculator, we find:
A ≈ $13,997.99
So, James will have approximately $13,997.99 after 7 years if interest is compounded monthly.