Question

you have $3,750 to deposit. if you deposit the money in a savings account at your local bank, you will earn 1.75% annual interest and will be able to make atm withdrawals at your bank’s atms. if you deposit the money in an online savings account, you will earn 3.25% interest, but you will be charged $4 every time you make an atm withdrawal. assuming that your atm withdrawals do not reduce the amount of interest you earn, roughly how often must you make an atm withdrawal for the local savings account to be a better deal than the online savings account?
a. thirteen times per year
b. once per month
c. every three weeks
d. every two months

please select the best answer from the choices provided

○ a
○ b

Explanation:

Step1: Calculate annual interest for local bank

Annual interest for local bank: $3750\times1.75\% = 3750\times0.0175 = 65.625$ dollars.

Step2: Calculate annual interest for online bank

Annual interest for online bank: $3750\times3.25\% = 3750\times0.0325 = 121.875$ dollars.

Step3: Find the difference in interest

Difference in interest: $121.875 - 65.625 = 56.25$ dollars. This is the maximum total ATM fees we can have for online bank to be worse than local bank.

Step4: Calculate number of ATM withdrawals

Each ATM withdrawal costs $4$ dollars. Let $n$ be the number of withdrawals. We need $4n > 56.25$. Solving for $n$: $n > \frac{56.25}{4} = 14.0625$. Wait, but wait, maybe I mixed up. Wait, local bank has lower interest but no fees, online has higher interest but fees. So to make local better, the fees from online should eat up the extra interest. So extra interest from online is $3.25\% - 1.75\% = 1.5\%$ of $3750$. So $3750\times0.015 = 56.25$ dollars. So the total ATM fees (which is $4$ per withdrawal) should be more than $56.25$ for online to be worse. So number of withdrawals $n$ where $4n > 56.25$ → $n > 14.06$. So we need more than 14 withdrawals a year. Now let's check the options:

- Option a: 13 times per year. 13×4=52 < 56.25. Not enough. Wait, maybe I made a mistake. Wait, no: local bank's total return is interest (65.625) + 0 fees. Online bank's total return is interest (121.875) - 4n. We want local > online: 65.625 > 121.875 - 4n → 4n > 121.875 - 65.625 → 4n > 56.25 → n > 14.06. So we need more than 14 withdrawals. Wait, the options:

a. Thirteen times per year: 13×4=52. 121.875 - 52=69.875 > 65.625. So online is still better. Wait, that can't be. Wait, maybe I flipped. Wait, local bank's net is interest (65.625) + 0. Online's net is interest (121.875) - 4n. We want local > online: 65.625 > 121.875 - 4n → 4n > 121.875 - 65.625 → 4n > 56.25 → n > 14.06. So n must be more than 14. But the options:

Wait, maybe the question is how often, so let's check the frequency:

Option b: Once per month → 12 times a year. 12×4=48. 121.875 - 48=73.875 > 65.625. Online better.

Option c: Every three weeks. There are about 52 weeks in a year, so 52/3 ≈17.33 times a year. 17.33×4≈69.33. 121.875 - 69.33≈52.54 < 65.625. So local is better. Wait, but 17.33 is more than 14.06.

Option d: Every two months → 6 times a year. 6×4=24. 121.875 -24=97.875 >65.625. Online better.

Wait, but the original options: maybe I miscalculated. Wait, the initial deposit is $3750$. Let's recalculate the interest difference:

Interest rate difference: 3.25% - 1.75% = 1.5% per year. So 1.5% of 3750 is 56.25, as before. So the total ATM fees must exceed 56.25 for online to be worse. So number of withdrawals n: 4n >56.25 → n>14.06. So we need more than 14 withdrawals a year.

Now check the options:

a. 13 times: 13×4=52 <56.25 → online still better (121.875 -52=69.875 >65.625).

b. Once per month: 12 times → 12×4=48 <56.25 → online better.

c. Every three weeks: 52 weeks /3 ≈17.33 times →17.33×4≈69.32 >56.25 → online net:121.875 -69.32≈52.55 <65.625 → local better.

d. Every two months: 6 times →6×4=24 <56.25 → online better.

But the options given in the radio buttons are A and B, but the original options are a, b, c, d. Wait, maybe the user's image cut off, but from the text, the options are a (13 times), b (once per month), c (every three weeks), d (every two months). So the correct answer should be c? But the radio buttons are A and B. Wait, maybe a typo. Wait, maybe I made a mistake. Wait, let's re-express:

We want local bank (no fees, lower interest) to be better than online (fees, higher interest). So:

Interest from local: I_local = 3750×0.0175 = 65.625

Interest from online: I_online = 3750×0.0325 = 121.875

Net for online: I_online - 4n

We want I_local > I_online - 4n → 65.625 > 121.875 - 4n → 4n > 121.875 - 65.625 → 4n > 56.25 → n > 14.06

So number of withdrawals per year must be more than 14.

Now, let's check the frequency:

- 13 times a year: n=13 <14 → no.

- Once per month: 12 times <14 → no.

- Every three weeks: 52/3 ≈17.33 times >14 → yes.

- Every two months: 6 times <14 → no.

So the correct option is c (every three weeks), but in the radio buttons, maybe the options are mislabeled. But according to the text options, the answer is c. But the user's radio buttons show A and B, maybe a mistake. But based on calculation, the answer is c (every three weeks). But since the user's options in radio are A and B, maybe the original question's options were different. Wait, maybe I miscalculated the interest difference. Wait, 3.25% -1.75% is 1.5% of 3750: 3750×0.015=56.25. So 56.25/4=14.06. So more than 14 withdrawals. So 13 is less, 12 is less, 17 is more, 6 is less. So the answer is c. But the radio buttons are A (13 times) and B (once per month). Maybe the user's image has a typo. But based on the calculation, the correct answer is c (every three weeks), but if we have to choose from A and B, maybe there's a mistake. Wait, maybe I flipped the interest. Wait, no: local bank has 1.75%, online has 3.25%. So online has higher interest, but fees. So to make local better, fees must eat up the extra interest. So yes, the calculation is correct.

Answer:

c. Every three weeks (But if only A and B are options, there might be a mistake. However, based on the text options, the answer is c.)