2023 AMC 8 Problems/Problem 14
Contents
Problem
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.)
Solution 1
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
Solution 2
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
.
~pianoboy
~MathFun1000 (Rewrote for clarity and formatting)
Animated Video Solution
~Star League (https://starleague.us)
Video Solution by Magic Square
https://youtu.be/-N46BeEKaCQ?t=4280
Video Solution by Interstigation
https://youtu.be/1bA7fD7Lg54?t=1078
Video Solution (CREATIVE THINKING!!!)
~Education, the Study of Everything
Video Solution by harungurcan
https://www.youtube.com/watch?v=VqN7c5U5o98&t=449s
See Also
2023 AMC 8 (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | ||
All AJHSME/AMC 8 Problems and Solutions |
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