[ Arithmetic. Mental calculation, etc.]

At the Flower City Exchange, 1 lemon and 1 banana can be exchanged for 2 oranges and 23 cherries, while 3 lemons can be exchanged for 2 bananas, 2 oranges, and 14 cherries. Which is more valuable: a lemon or a banana?

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Let the cost of one fruit be: lemon $L$, banana $B$, orange $A$, cherry $V$. We can reason in different ways.

The first method. From the problem statement, we get two equations that must be satisfied simultaneously: $L+B=$ $2 A+23 V ; 3 L=2 B+2 A+14 V$. Multiply both sides of the second equation by $2: 6 L=4 B+4 A+28 V$ Using the first equation, we get: $4 A+28 V>2 A+23 V=L+B$. Therefore, $6 L>4 B+L+B$, from which $L>B$.

The second method. Suppose that $L \leq B$; then $2 L \leq 2 B$. Then, since 3 lemons can be exchanged for 2 bananas, 2 oranges, and 14 cherries, we have $L \geq 2 A+14 V$. Using now the fact that 1 lemon and 1 banana can be exchanged for 2 oranges and 23 cherries, we get: $B \leq 9 V \leq 14 V+2 A \leq L$, which means $B<L$. This contradicts the initial assumption, so the lemon is more expensive than the banana.