2023 AMC 8 Problems/Problem 14
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 -cent, -cent, and -cent stamps, with exactly of each type. What is the greatest number of stamps Nicolas can use to make exactly in postage? (Note: The amount corresponds to dollars and cents. One dollar is worth cents.)
Let's use the most stamps to make We have of each stamp, -cent (like nickels), -cent (like dimes), and -cent (like quarters).
If we want to have the highest number of stamps, we have to have the highest number of the smaller value stamps (like the coins above). We can use nickels and dimes to bring our total cost to . However, when we try to use quarters, the cents don’t fit evenly, so we have to give back cents in order to make the quarter amount . The most efficient way to do this is to give back a -cent (dime) stamp and a -cent (nickel) stamp to have stamps (coins) used so far. Now, we just use quarters to get a grand total of .
~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat
The value of his entire stamp collect is dollars. To make with stamps, he should remove cents worth of stamps with as few stamps as possible. To do this, he should start by removing as many cent stamps as possible as they have the greatest denomination. He can remove at most of these stamps. He still has to remove cents worth of stamps. This can be done with one and cent stamp. In total, he has stamps in his entire collect. As a result, the maximum number of stamps he can use is .
~MathFun1000 (Rewrote for clarity and formatting)
Animated Video Solution
~Star League (https://starleague.us)
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~Education, the Study of Everything
Video Solution by harungurcan
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