Compound Interest. A man invest $5000 in an account that pays 8.5% interest per year , compounded quarterly.
A. Find the amount after 3 years
B. How long will it take for the i investment to double ?
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Compound Interest. A man invest $5000 in an account that pays 8.5% interest per year , compounded quarterly.
A. Find the amount after 3 years
B. How long will it take for the i investment to double ?
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Step-by-step explanation:
A. To find the amount after 3 years, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final amount
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
In this case, the principal amount (P) is $5000, the annual interest rate (r) is 8.5% or 0.085 as a decimal, and the interest is compounded quarterly, so the number of times compounded per year (n) is 4. The number of years (t) is 3.
Plugging in these values into the formula, we get:
A = 5000(1 + 0.085/4)^(4*3)
A = 5000(1 + 0.02125)^(12)
A = 5000(1.02125)^(12)
A ≈ $6,061.68
Therefore, the amount after 3 years will be approximately $6,061.68.
B. To find out how long it will take for the investment to double, we can rearrange the formula for compound interest:
A = P(1 + r/n)^(nt)
We want to find the value of t when the final amount (A) is double the principal amount (P). So, A = 2P.
2P = P(1 + r/n)^(nt)
Dividing both sides by P, we get:
2 = (1 + r/n)^(nt)
Taking the natural logarithm of both sides, we have:
ln(2) = nt ln(1 + r/n)
Solving for t, we get:
t = ln(2) / (n ln(1 + r/n))
Plugging in the values, with P = $5000, r = 8.5% or 0.085 as a decimal, and n = 4, we can calculate:
t = ln(2) / (4 ln(1 + 0.085/4))
t ≈ 8.16 years
Therefore, it will take approximately 8.16 years for the investment to double.