i need help with these problems below !
mike deposited $500 for 9 months at 8%, compounded quarterly.
a. how many times was interest added to mike's account?
b. what percent interest was added each time?
c. what was the balance in mike's account at the end of 9 months?
sara deposited $1400 for 1 year at 10%, compounded semiannually.
a. how many times was interest added to sara's account?
b. what percent interest was added each time?
c. what was the balance in sara's account at the end of one year?
i don't understand how to do these problems so please explain & show work.
Sure, I'd be happy to help you with these problems! Let's break them down one by one.
For Mike:
a. To find out how many times interest was added to Mike's account, we need to know the compounding period. In this case, interest is compounded quarterly, which means it is added every 3 months within the 9-month period. Therefore, there are 9 months / 3 months = 3 compounding periods.
b. The interest rate is given as 8%, compounded quarterly. Since there are 4 quarters in a year, we can calculate the interest added each time as 8% / 4 = 2%.
c. To calculate the balance at the end of 9 months, we use the formula for compound interest: A = P(1 + r/n)^(nt), where A is the final amount, P is the principal (initial deposit), r is the interest rate in decimal form, n is the number of compounding periods per year, and t is the time in years.
Substituting the values into the formula, we have A = 500(1 + 0.08/4)^(4*9/12) = 500(1 + 0.02)^3 = 500(1.02)^3 ≈ $530.60.
For Sara:
a. The interest is compounded semiannually, which means it is added every 6 months within the 1-year period. Therefore, there are 1 year / 6 months = 2 compounding periods.
b. The interest rate is given as 10%, compounded semiannually. Since there are 2 semiannual periods in a year, we can calculate the interest added each time as 10% / 2 = 5%.
c. Using the same formula as before, A = 1400(1 + 0.10/2)^(2*1/1) = 1400(1 + 0.05)^2 = 1400(1.05)^2 ≈ $1547.
So, at the end of 9 months, Mike's account balance is approximately $530.60, and at the end of 1 year, Sara's account balance is approximately $1547.
I hope this helps! Let me know if you have any further questions.
Sure! Let's start with Mike's problem first.
a. To find out how many times interest was added to Mike's account, we need to know how many quarters are there in 9 months. Since there are 4 quarters in a year, we can calculate the number of times interest was added by dividing the number of months by 3 (rounding up if necessary).
Number of times interest was added = 9 months / 3 months (per quarter) = 3 quarters
b. Now let's find the percent interest added each time. The interest rate is given as 8% per year and compounded quarterly. To find the interest rate per quarter, we divide the annual interest rate by the number of quarters in a year.
Percent interest added each time = 8% / 4 quarters = 2% per quarter
c. To find the balance in Mike's account at the end of 9 months, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
where A is the final amount, P is the principal (initial deposit), r is the interest rate as a decimal, n is the number of times interest is compounded per year, and t is the time in years.
In this case, we have:
P = $500
r = 8% or 0.08 (as a decimal)
n = 4 (quarters per year)
t = 9 months / 12 months (per year) = 0.75 (as a decimal)
Plugging in the values:
A = 500(1 + 0.08/4)^(4*0.75)
= 500(1 + 0.02)^3
= 500(1.02)^3
= 500(1.0612)
= $530.60
Therefore, the balance in Mike's account at the end of 9 months is approximately $530.60.
Now let's move on to Sara's problem.
a. To find out how many times interest was added to Sara's account, we need to know how many six-month periods are there in 1 year.
Since each six-month period is considered a semiannual period, there are 2 semiannual periods in 1 year.
b. To find the percent interest added each time, the interest rate is given as 10% per year and compounded semiannually. Similar to Mike's problem, we divide the annual interest rate by the number of semiannual periods in a year.
Percent interest added each time = 10% / 2 semiannual periods = 5% per semiannual period
c. To find the balance in Sara's account at the end of 1 year, we can use the same compound interest formula as before.
In this case, we have:
P = $1400
r = 10% or 0.10 (as a decimal)
n = 2 (semiannual periods per year)
t = 1 (year)
Plugging in the values:
A = 1400(1 + 0.10/2)^(2*1)
= 1400(1 + 0.05)^2
= 1400(1.05)^2
= 1400(1.1025)
= $1543.50
Therefore, the balance in Sara's account at the end of 1 year is $1543.50.
I hope this explanation and step-by-step breakdown help you understand how to solve these types of problems.
Sure, I'll be happy to help you with these problems. Let's start with Mike's account first.
To find the number of times interest was added to Mike's account, we need to determine the compounding period. In this case, it is stated that the interest is compounded quarterly, which means it is added every 3 months.
a. To calculate the number of times interest was added, we need to divide the total time period (9 months) by the compounding period (3 months): 9 months / 3 months = 3.
So, interest was added 3 times to Mike's account.
b. To find the percent interest added each time, we need to divide the annual interest rate (8%) by the number of compounding periods per year. Since the interest is compounded quarterly, we have 4 compounding periods per year.
So, the percent interest added each time is: 8% / 4 = 2%.
c. To find the balance in Mike's account at the end of 9 months, we need to use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the final balance
P = the principal amount (initial deposit)
r = the annual interest rate (as a decimal)
n = the number of times interest is compounded per year
t = the time period in years
Applying the values to the formula:
A = 500(1 + 0.08/4)^(4*9/12)
A = 500(1 + 0.02)^(36/12)
A = 500(1.02)^3
A ≈ $530.60
Therefore, the balance in Mike's account at the end of 9 months is approximately $530.60.
Now, let's move on to Sarah's account.
a. Similar to Mike's case, we need to find the number of times interest was added. The interest is compounded semiannually, meaning it is added every 6 months.
So, for Sarah's account, the time period is 1 year (12 months), and the compounding period is 6 months.
To calculate the number of times interest was added: 12 months / 6 months = 2.
Thus, interest was added 2 times to Sarah's account.
b. To find the percent interest added each time, we need to divide the annual interest rate (10%) by the number of compounding periods per year. Since the interest is compounded semiannually, there are 2 compounding periods per year.
Thus, the percent interest added each time is: 10% / 2 = 5%.
c. Employing the compound interest formula as in Mike's case:
A = P(1 + r/n)^(nt)
A = 1400(1 + 0.10/2)^(2*1/1)
A = 1400(1 + 0.05)^2
A ≈ $1545.00
Therefore, the balance in Sarah's account at the end of one year is approximately $1545.00.
I hope this clarifies the process and the steps to solve these problems. If you have any further questions, feel free to ask!