Compute the compound quarterly amount after 1 year for $100 invested at 12% interest compounded quarterly. What simple interest rate will yield the same amount in 2 years?
P = Po(1+r)^n.
r = (12%/4) / 100% = 0.03 = Quarterly %
rate expressed as a decimal.
n = 4comp./yr * 1yr = 4 Compounding periods.
a. P = 100(1.03)^4 = $112.55.
b. P = Po + Po*r*t.
r = APR expressed as a decimal.
t = 2 Years.
100 + 100*r*2 = 112.55.
100*r*2 = 112.55 - 100 = 12.55.
r = 12.55 / 200 = 0.06275 = 6.275%.
To compute the compound quarterly amount after 1 year for $100 invested at 12% interest compounded quarterly, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
P = the principal amount (initial investment)
r = annual interest rate (in decimal form)
n = number of times interest is compounded per year
t = number of years
In this case, P = $100, r = 12% = 0.12 (convert to decimal), n = 4 (quarterly compounded), and t = 1 year. Plugging in these values into the formula:
A = 100 * (1 + 0.12/4)^(4*1)
Now, let's simplify the formula before calculating:
A = 100 * (1 + 0.03)^4
A = 100 * (1.03)^4
A = 100 * 1.1255
A ≈ $112.55
Therefore, after 1 year, the compound quarterly amount will be approximately $112.55.
Now, let's determine the simple interest rate that will yield the same amount in 2 years.
We can use the formula for simple interest:
A = P(1 + rt)
Where:
A = the final compound quarterly amount ($112.55)
P = the principal amount ($100)
r = the simple interest rate (in decimal form)
t = the number of years (2)
We need to solve for r. Plugging in the given values into the formula:
112.55 = 100 * (1 + r * 2)
Now, let's simplify the equation:
112.55 = 100 + 200r
Rearranging the equation:
200r = 112.55 - 100
200r = 12.55
Dividing both sides by 200:
r = 12.55 / 200
r ≈ 0.06275
Therefore, a simple interest rate of approximately 6.275% will yield the same amount in 2 years.