Jenny deposits $867 into an account earning 5.309%, compounded weekly.
(a) What will be her account balance in 14 weeks?
(b) How long will it take for the account balance to grow to $1,500?
.05309/52 = .001021
so every week multiply by 1.001021
after n weeks
x = 867 (1.001021)^n
if n= 14
x = 867 (1.001021)^14
= 867* 1.014388
= 879.48
1500 =867 (1.001021)^n
1.73 = (1.001021)^n
ln 1.73 = n ln 1.001021
etc
(a) Well, with that interest rate, Jenny's money will be growing faster than a Chia Pet on steroids! To calculate her account balance in 14 weeks, we can use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final account balance
P = the initial deposit
r = the interest rate (in decimal form)
n = the number of times the interest is compounded per year
t = the number of years
So, plugging in the values, we have:
P = $867
r = 5.309% = 0.05309
n = 52 (weekly compounding)
t = 14 weeks / 52 weeks/year
Now, let me put on my mathemagician hat and calculate that for you...
A = $867 * (1 + 0.05309/52)^(52 * (14/52))
After doing some fancy calculations, the answer is approximately $906.58. So, Jenny will have a balance of around $906.58 in her account after 14 weeks. I bet she can't wait to see her money grow faster than a kid with an ice cream cone on a hot summer day!
(b) Ah, now we have a challenge to work with! Let's find out how long it will take for Jenny's account balance to grow to $1,500. This time, we'll need to rearrange the compound interest formula a bit to solve for time (t). Here's the modified formula:
t = ln(A/P) / (n * ln(1 + r/n))
Where:
A = the final account balance ($1,500)
P = the initial deposit ($867)
r = the interest rate (0.05309)
n = the number of times the interest is compounded per year (52)
Alright, let me grab my calculator and crunch those numbers...
t = ln(1500/867) / (52 * ln(1 + 0.05309/52))
After some mathematical magic, the answer is approximately 7.87 years. So, it will take around 7.87 years for Jenny's account balance to grow to $1,500. Time flies when you're waiting for money to grow, huh?
To calculate Jenny's account balance in 14 weeks, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future account balance
P = the principal amount (initial deposit)
r = the annual interest rate (expressed as a decimal)
n = the number of times interest is compounded per year
t = the number of years
In this case:
P = $867
r = 5.309% = 0.05309
n = 52 (since interest is compounded weekly, there are 52 weeks in a year)
t = 14/52 = 0.26923 years (14 weeks divided by 52 weeks in a year)
(a) What will be her account balance in 14 weeks?
A = 867(1 + 0.05309/52)^(52*0.26923)
Calculating the above expression will give us the account balance after 14 weeks.
(b) To find out how long it will take for the account balance to grow to $1,500, we need to rearrange the compound interest formula:
A = P(1 + r/n)^(nt)
Rearranging for t, we have:
t = (log(A/P))/(n * log(1 + r/n))
We can plug in the values for A ($1,500), P ($867), r (0.05309), and n (52).
Calculating the above expression will give us the time it takes for the account balance to reach $1,500.
To answer these questions, we need to use the compound interest formula:
A = P(1 + r/n)^(nt)
Where:
A = the final account balance,
P = the principal amount (initial deposit),
r = the annual interest rate (expressed as a decimal),
n = the number of times the interest is compounded per year, and
t = the number of years.
Let's calculate the answers step by step.
(a) What will be her account balance in 14 weeks?
Step 1: Convert the annual interest rate into its decimal form:
The annual interest rate is 5.309%. We convert it to a decimal by dividing by 100:
r = 5.309 / 100 = 0.05309
Step 2: Determine the number of times the interest is compounded per year:
The question states that the interest is compounded weekly, which means n = 52 (total number of weeks in a year).
Step 3: Calculate the account balance using the compound interest formula:
A = 867 * (1 + 0.05309/52)^(52*14)
= 867 * (1 + 0.001025)^(728)
Now, you can use a calculator or a computer program to simplify the calculation:
A ≈ 867 * (1.001025)^728
A ≈ 867 * 1.560747 *(Approximated to nearest cent)*
A ≈ $1,353.11
Therefore, Jenny's account balance in 14 weeks will be approximately $1,353.11.
(b) How long will it take for the account balance to grow to $1,500?
We rearrange the compound interest formula to solve for t:
t = (log(A/P)) / (log(1 + r/n))
Step 1: Convert the annual interest rate into its decimal form:
The annual interest rate is 5.309%. We convert it to a decimal by dividing by 100:
r = 5.309 / 100 = 0.05309
Step 2: Determine the number of times the interest is compounded per year:
The question states that the interest is compounded weekly, which means n = 52 (total number of weeks in a year).
Step 3: Plug in the known values into the formula to solve for t:
t = (log(1500/867)) / (log(1 + 0.05309/52))
= (log(1.7280023)) / (log(1.001025))
Using a logarithm calculator or computer program:
t ≈ (0.23805091) / (0.00024869) *(Approximated to the nearest week)*
t ≈ 957.09
Therefore, it will take approximately 957 weeks for Jenny's account balance to grow to $1,500.