Our Slovak grandmother was shopping in a store that only had apples, bananas, and pears. Apples were 50 cents each, pears were 60 cents each, and bananas were cheaper than pears. The grandmother bought five pieces of fruit, including exactly one banana, and paid 2 euros and 75 cents.

How many cents could one banana cost? Determine all possibilities. (K. Jasenčáková)

Hint. Start by calculating with $s$ fruits whose prices you know.
In addition to bananas, grandma bought apples and pears, whose prices per piece we know. There were a total of four pieces of apples and pears. Let's go through the individual possibilities and determine the cost of the apples and pears. Subtract from the total cost and from there determine the price of the banana, which we will compare with the price of the pear:

- Four apples and no pears cost $4 \cdot 50=200$ cents; one banana would cost $275-200=75$ cents, which is more than the price of a pear.
- Three apples and one pear cost $3 \cdot 50+1 \cdot 60=210$ cents; one banana would cost $275-210=65$ cents, which is more than the price of a pear.
- Two apples and two pears cost $2 \cdot 50+2 \cdot 60=220$ cents; one banana would cost $275-$ $-220=55$ cents, which is less than the price of a pear.
- One apple and three pears cost $1 \cdot 50+3 \cdot 60=230$ cents; one banana would cost $275-$ $-230=45$ cents, which is less than the price of a pear.
- No apples and four pears cost $4 \cdot 60=240$ cents; one banana would cost $275-240=35$ cents, which is less than the price of a pear.

The banana could cost 35, 45, or 55 cents.

Note. A pear is 10 cents more expensive than an apple, and by exactly that much the price of the banana must decrease if we replace an apple with a pear in the considered purchase. With this observation, the previous discussion can be significantly simplified.