# Problem 4. (8 points)

Lena receives 50,000 rubles per month and spends 45,000 rubles per month. She gets her salary on the 6th of each month. Lena has: a deposit in the bank $\mathrm{X}$ with a $1\%$ monthly interest rate with monthly capitalization and the possibility of topping up, but no withdrawals allowed, with the calculation day being the 6th of each month; a debit card, on which 6% of the minimum balance for the month is credited on the 6th of each month at an annual rate of $8\%$; a credit card with a payment due date of the 7th of each month and a limit of 100,000 rubles, with a $3\%$ fee for cash withdrawals or transfers.

Lena can pay all her expenses using her bank card. Describe the optimal strategy for Lena and calculate the annual income she can earn from her cooperation with the bank (in rubles).
Solution:

1) Of all the bank's products, the deposit brings Lena the highest income. She can use $50-45=5$ thousand rubles for savings, depositing them immediately after receiving her salary. Over a year, this will bring Lena $5000 * \frac{1.01 * (1.01^{12}-1)}{1.01-1} - 5000 * 12 \approx 4000$.
2) The remaining 45 thousand rubles per month cannot be deposited by Lena, as she will spend them during the month after receiving her salary, and the deposit does not allow withdrawals. Therefore, Lena needs to use the second most profitable instrument - a debit card with interest on the balance.
3) If Lena simply deposits the money into a debit card and uses it for payments, she will not gain any additional benefit, because by the end of the month, after receiving her salary, the balance on the card will be zero, and interest is calculated on the minimum balance.
4) To maximize her income, it is profitable for Lena to pay with a credit card and keep her salary on a debit card. This way, the minimum balance on the debit card will always be 45 thousand rubles, so over a year, this will bring Lena $45000 * 0.08 = 860$ rubles.

Let's describe Lena's strategy in detail: On the 6th of each month, after receiving her salary, she should deposit 5000 rubles into the deposit and 45000 rubles into the debit card. Then, on the 7th, pay off the credit card debt. All expenses during the month should be made using the credit card. Thus, Lena's annual income from cooperation with the bank will be $4000 + 860 = 4860$ rubles.

Criteria: description of the correct strategy - 4 points, calculation of income from the debit card - 2 points, correct formula for deposit income - 2 points.