Problem (Algebra):
Emma goes to the store to buy apples and peaches. She buys five of each, hands the shopkeeper one $\$ 5$ bill, but then has to give the shopkeeper another; she gets back some change. Jonah goes to the same store, buys 2 apples and 12 peaches, and tries to pay with a single $\$ 10$ bill. But that's not enough, so Jonah has to give the shopkeeper another $\$ 10$ bill, and also gets some change. Finally, Helen goes to the same store to buy 25 peaches. Assuming that the price in cents of each fruit is an integer, compute the least amount of money, in cents, that Helen can expect to pay.

Solution:
Let $a$ be the price of one apple and $p$ be the price of one peach, in cents. The first transaction shows that $500<5 a+5 p<1000$, hence $100<a+p<200$. The second transaction shows that $1000<2 a+12 p<2000$, so $500<a+6 p<1000$. Subtracting the inequalities yields $300<5 p<900$, so $60<p<180$. Therefore the price of 25 peaches is at least $25 \cdot 61=\mathbf{1 5 2 5}$ cents.

Final answer: 1525