3. Let's say you have two bank cards for making purchases: a debit card and a credit card. Today, you decided to buy airline tickets for 20,000 rubles. If you pay for the purchase with a credit card (the credit limit allows it), you will have to return the money to the bank in $\mathrm{N}$ days to avoid going beyond the grace period, during which you can repay the credit for free. In this case, the bank will pay a cashback of $0.5 \%$ of the purchase amount after 1 month. If you pay for the purchase with a debit card (there is more than enough money on the card), you will receive a cashback of $1 \%$ of the purchase amount after 1 month. It is known that the annual interest rate on the average monthly balance of funds on the debit card is $6 \%$ per year (for simplicity, assume that each month has 30 days, interest on the card is paid at the end of each month, and interest earned on the balance is not capitalized). Determine the smallest number of days $\mathrm{N}$, under equal conditions, for which it is more profitable to pay for this purchase of airline tickets with a credit card. (15 points)
# Solution:

When paying by credit card, the amount of 20,000 rubles will be on your debit card for $\mathrm{N}$ days, which will earn you $\frac{6 \mathrm{~N}}{100 \cdot 12 \cdot 30} \cdot 20000$ rubles in interest on the remaining funds.

You will also receive $20000 \times 0.005 = 100$ rubles in cashback.

When paying by debit card, you will receive a cashback of 200 rubles after 1 month.

For it to be more profitable to pay for this purchase with a debit card, the inequality $\frac{6 N}{100 \cdot 12 \cdot 30} \cdot 20000 + 100 > 200$ must be satisfied.

This inequality holds if $\mathrm{N} > 30$. Therefore, the minimum number of days in the grace period, during which it is more profitable to pay for this purchase with a credit card, is 31 days.

Moscow Schoolchildren's Olympiad in Financial Literacy. 2017-2018 academic year. Final stage. $10-11$ grades. Variant 1

## Criteria:

Correctly accounted for the fact that interest is accrued on the average monthly balance of funds - up to 4 points.

Correctly calculated the cashback when paying with a credit card - up to 3 points.

Correctly calculated the cashback when paying with a debit card - up to 3 points.

Correctly formulated the inequality to find the required number of days - up to 3 points.

Obtained a correct and justified answer - up to 2 points