Difference between revisions of "2023 AMC 8 Problems/Problem 14"
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Most stamps make <math>7.10.</math> You have 20 of each coin, nickles, dimes and quarters. | Most stamps make <math>7.10.</math> You have 20 of each coin, nickles, dimes and quarters. | ||
| − | If we want to have the most amount of stamps we have to have the most amount of smaller value coins. We can use 20 nickels and 20 dimes to bring our total cost to <math>7.10 - 3.00 = 4.10</math>. However when we try to use quarters the 25 cents don’t fit evenly, so we have to give back 15 cents in order to make the quarter amount 4.25 the most efficient way to do this is give back a dime and a nickel to have 38 coins used so far. Now we just use <math>\frac{425}{25} = 17</math> quarters to get a grand total of <math>38 + 17 = \boxed{\ | + | If we want to have the most amount of stamps we have to have the most amount of smaller value coins. We can use 20 nickels and 20 dimes to bring our total cost to <math>7.10 - 3.00 = 4.10</math>. However when we try to use quarters the 25 cents don’t fit evenly, so we have to give back 15 cents in order to make the quarter amount 4.25 the most efficient way to do this is give back a dime and a nickel to have 38 coins used so far. Now we just use <math>\frac{425}{25} = 17</math> quarters to get a grand total of <math>38 + 17 = \boxed{\textbf{(E)}\ 55}</math>. |
~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat | ~apex304, SohumUttamchandani, wuwang2002, TaeKim, Cxrupptedpat | ||
==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, 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>. | + | 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 01:21, 26 January 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
Most stamps make 
You have 20 of each coin, nickles, dimes and quarters.
If we want to have the most amount of stamps we have to have the most amount of smaller value coins. We can use 20 nickels and 20 dimes to bring our total cost to 
. However when we try to use quarters the 25 cents don’t fit evenly, so we have to give back 15 cents in order to make the quarter amount 4.25 the most efficient way to do this is give back a dime and a nickel to have 38 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
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. 
