Jenny deposits $867 into an account earning 5.309%, compounded weekly.

Question

Create an appealing image showing a banking concept. The image should feature a young female character, possibly of Hispanic descent, smartly dressed and sitting in front of a computer, depicting a financial planning scenario. A vision of her deposit represented by a bundle of dollar bills totaling $867 is next to her. Just above this bundle of bills, illustrate an indication of weekly time passing along with the percentage growth value. Also, convey the concept of the account balance growing to $1,500 symbolically. The vision should not contain any text.

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?

Answer 1

.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

Answer 2

(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?

Answer 3

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.

Answer 4

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.