Question

A student has an internship in their senior year whereby they were able to save some money (call this variable SAVE). They deposit this money into the bank with an interest rate of 5% per year. How long will it take for them to have their bank account double if the interest is compounded:

a) Annually.
b) Quarterly.
c) Monthly,
d) Daily
Group 1: your SAVE amount is $5000
Group 2 2: your SAVE amount is $5500
Group 3: your SAVE amount is $6000
Group 4: your SAVE amount is $6500
Group 5: your SAVE amount is $7000
Group 6: your SAVE amount is $7500

Answer 1

a) Annually: Approximately
`\(t \approx (\log(2))/(\log(1 + r))\) years, where \(r = 0.05\).`

b) Quarterly: Approximately
`\(t \approx (\log(2))/(\log(1 + (r)/(4)))\) years.`

c) Monthly: Approximately
`(t \approx (\log(2))/(\log(1 + (r)/(12)))\) years.`

d) Daily: Approximately
`\(t \approx (\log(2))/(\log(1 + (r)/(365)))\) years.`

Step-by-step explanation:

In compound interest calculations, the formula to find the time (\(t\)) it takes for an amount to double is given by the compound interest formula:

`\[A = P \left(1 + (r)/(n)\right)^(nt)\]`

where:

`- \(A\)` is the final amount,

`- \(P\)` is the principal amount (initial deposit),

`- \(r\)` is the annual interest rate (as a decimal),

`- \(n\)` is the number of times interest is compounded per year,

`- \(t\)` is the time in years.

To find the time it takes to double the amount
`(\(A = 2P\))`, we rearrange the formula:

`\[2 = \left(1 + (r)/(n)\right)^(nt)\]`

Solving for \(t\) in each scenario gives the formulas mentioned in the final answer.

For each group, you can substitute the given
`\(SAVE\)` amount as the principal (P) in the respective formula to find the time it takes for that specific amount to double under the given compounding frequency.

Answer 2

a) Annually: Approximately 14 years.

b) Quarterly: Approximately 13 years.

c) Monthly: Approximately 12 years.

d) Daily: Approximately 11 years.

Step-by-step explanation:

In compound interest calculations, the formula for the future value of an investment is given by (FV = P(1 +
`(r)/(n))^(nt)`), where:

FV is the future value of the investment,

P is the principal amount (initial deposit),

r is the annual interest rate (decimal),

n is the number of times that interest is compounded per year, and

t is the number of years.

For this scenario, we want to find t when FV is twice the principal 2P. Given P as the SAVE amount and an annual interest rate of 5%, we can substitute the values into the formula for each group:

a) Annually (compounded once per year): 2P = P(1 + 0.05)^t), solving for (t) gives t approx 14 years.

b) Quarterly (compounded four times per year): 2P = P(1 +
`(0.05)/(4))^(4t)`), solving for t gives t approx 13 years.

c) Monthly (compounded twelve times per year): 2P = P(1 +
`(0.05)/(12))^(12t)`), solving for t gives t approx 12 years.

d) Daily (compounded 365 times per year): 2P = P(1 +
`(0.05)/(365))^(365t)`), solving for t gives t approx 11 years.

Therefore, it will take approximately 14 years for the bank account to double with annual compounding, 13 years with quarterly compounding, 12 years with monthly compounding, and 11 years with daily compounding.