regina deposits 3500 in a savings account that pays 7 1/2 % interest , compounded semiannually. how much interest does the account earn in the first 6 months?
3500 * .075/2
1400
To calculate the amount of interest earned in the first 6 months, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = final amount
P = principal amount
r = annual interest rate (as a decimal)
n = number of times interest is compounded per year
t = time in years
Given:
P = $3500
r = 7.5% (expressed as 0.075)
n = 2 (semiannually compounded)
t = 0.5 years (6 months)
Substituting the values into the formula:
A = 3500(1 + 0.075/2)^(2 * 0.5)
Simplifying the formula:
A = 3500(1.0375)^(1)
Calculating the value inside the parentheses:
A = 3500(1.0375)
A = 3631.25
To find the interest earned, subtract the principal amount:
Interest = A - P
Interest = 3631.25 - 3500
Interest = $131.25
Therefore, the account earns $131.25 in interest in the first 6 months.
To calculate the interest earned by the savings account in the first 6 months, we can use the formula:
Interest = Principal * (1 + (interest rate / n))^(n * t) - Principal
Where:
- Principal is the initial deposit amount, which is $3500 in this case.
- Interest rate is given as 7 1/2 %, which can be converted to a decimal as 0.075.
- n is the number of times interest is compounded per year. In this case, it is compounded semiannually, so n = 2.
- t is the time in years, which for the first 6 months is 0.5 years.
Let's now substitute the values into the formula and calculate the interest:
Interest = 3500 * (1 + (0.075 / 2))^(2 * 0.5) - 3500
First, divide the interest rate by the number of times compounded per year:
0.075 / 2 = 0.0375
Next, calculate the value inside the parentheses:
1 + 0.0375 = 1.0375
Raise the value by the power of the number of times compounded per year multiplied by the time in years:
(1.0375)^(2 * 0.5) = 1.0193
Finally, calculate the interest earned:
3500 * (1.0193) - 3500 = 34.76
Therefore, the account earns $34.76 in interest during the first 6 months.