Question

Nicolas is planning to send a package to his friend Anton, who is a stamp collector. To pay for the postage, Nicolas would like to cover the package with a large number of stamps. Suppose he has a collection of $5$ -cent, $10$ -cent, and $25$ -cent stamps, with exactly $20$ of each type. What is the greatest number of stamps Nicolas can use to make exactly $\$7.10$ in postage?

(Note: The amount $\$7.10$ corresponds to $7$ dollars and $10$ cents. One dollar is worth $100$ cents.)

$45$

$46$

$51$

$54$

$55$

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Written solution:

Note that we want to get $7 \cdot 100 + 10 = 710$ cents. Let us try to see if it is possible to use up all of the $5$ -cent and $10$ -cent stamps.

All of these two types of stamps combined would be worth $20(5 + 10) = 20 \cdot 15 = 300$ cents. We then would need $710 - 300 = 410$ cents, which cannot be created with just $25$ -cent stamps.

We can, however, make $425$ cents with $25$ -cent stamps. Using only $19$ each of $5$ and $10$ -cent stamps would total $300 - 5 - 10 = 285$ cents. This means we would then need $710 - 285 = 425$ cents. This can be achieved with $425 \div 25 = 17$ $25$ -cent stamps.

This lets us use $\begin{align*} 2 \cdot 19 + 17 &= 38 + 17 \\ &= 55 \end{align*}$ stamps.

Thus, E is the correct answer.

2023 AMC 8 Problem 14

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Problem 14 in Other Years