Difference between revisions of "2023 AMC 8 Problems/Problem 14"
m (→Solution 1) |
m (→Solution 2) |
||
| Line 14: | Line 14: | ||
==Solution 2== | ==Solution 2== | ||
| − | The value of his entire stamp collect is <math>8</math> dollars. To make <math>\$7.10</math> with stamps, he should remove <math>90</math> cents worth of stamps with as few stamps as possible. To do this, he should start by removing as many <math>25</math> cent stamps as possible | + | The value of his entire stamp collect is <math>8</math> dollars. To make <math>\$7.10</math> with stamps, he should remove <math>90</math> cents worth of stamps with as few stamps as possible. To do this, he should start by removing as many <math>25</math> cent stamps as possible as they have the greatest denomination. He can remove at most <math>3</math> of these stamps. He still has to remove <math>90-25\cdot3=15</math> cents worth of stamps. This can be done with one <math>5</math> and <math>10</math> cent stamp. In total, he has <math>20\cdot3=60</math> stamps in his entire collect. As a result, the maximum number of stamps he can use is <math>20\cdot3-5=\boxed{\textbf{(E)}\ 55}</math>. |
~pianoboy | ~pianoboy | ||
Revision as of 02:08, 19 February 2023
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
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 | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
