f $3800 is invested in a savings account for which interest is compounded quarterly, and if the $3800 turns into $4300 in 2 years, what is the interest rate of the savings account?
6.22
P = Po(1+r)^n.
r = Quarterly % rate expressed as a decimal.
n = 4Comp./yr. * 2yrs = 8 Compounding periods.
P = 3800*(1+r)^8 = 4300,
(1+r)^8 = 4300/3800 = 1.132,
8*Log(1+r) = Log1.132,
Log(1+r) = Log1.132/8 = 0.00673,
1+r = 10^(0.00673),
r = 10^(0.00673) - 1 = 0.0156,
APR = 4 * 0.0156 = 0.0625 = 6.25%.
To find the interest rate of the savings account, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = Amount after time t
P = Principal (initial amount)
r = Annual interest rate (as a decimal)
n = Number of times interest is compounded per year
t = Number of years
In this case, we have:
A = $4300
P = $3800
t = 2 years
n = 4 (compounded quarterly)
We can rearrange the formula to solve for r:
r = ( (A/P)^(1/nt) ) - 1
Substituting the given values into the formula:
r = ( ($4300/$3800)^(1/(4*2)) ) - 1
Calculating:
r = (1.131579)^0.125 - 1
r ≈ 1.027 - 1
r ≈ 0.027 or 2.7%
Therefore, the interest rate of the savings account is approximately 2.7%.
To find the interest rate of the savings account, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = The future value of the investment
P = The principal (initial investment)
r = The annual interest rate (in decimal form)
n = The number of times interest is compounded per year
t = The number of years
In this case, the principal (P) is $3800, the future value (A) is $4300, the time (t) is 2 years, and the interest is compounded quarterly, so n = 4.
We need to solve for r.
4300 = 3800(1 + r/4)^(4*2)
To simplify the equation, let's divide both sides by 3800:
4300 / 3800 = (1 + r/4)^(8)
1.1316 ≈ (1 + r/4)^8
Now, take the eighth root of both sides to isolate (1 + r/4):
(1.1316)^(1/8) ≈ 1 + r/4
1.0277 ≈ 1 + r/4
Subtract 1 from both sides to isolate r/4:
0.0277 ≈ r/4
Now, multiply both sides by 4 to solve for r:
0.1110 ≈ r
So, the interest rate of the savings account is approximately 11.10%.
To get this answer, you can follow these steps:
1. Calculate the future value (A) by using the formula A = P(1 + r/n)^(nt), where P = $3800, n = 4, t = 2, and solve for A.
2. Divide the future value (A) by the principal (P) to get the value of (1 + r/4)^8.
3. Take the eighth root of the value obtained in step 2 to get the value of (1 + r/4).
4. Subtract 1 from the value obtained in step 3 to get the value of r/4.
5. Multiply the value obtained in step 4 by 4 to get the final value of r, which represents the interest rate.