Difference between revisions of "2015 AMC 10A Problems/Problem 13"
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<math> \textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7 </math> | <math> \textbf{(A) }3\qquad\textbf{(B) }4\qquad\textbf{(C) }5\qquad\textbf{(D) }6\qquad\textbf{(E) }7 </math> | ||
| − | ==Solution== | + | ==Solution 1== |
| − | Let Claudia have <math>x</math> 5-cent coins and <math>12-x</math> 10-cent coins. It is easily observed that any multiple of 5 between 5 and <math>5x + 10(12 - x) = 120 - 5x</math> inclusive can be obtained by a combination of coins. Thus, <math>24 - x = 17</math> combinations can be made, so <math>x = 7</math>. But the answer is not 7, because we are asked for the number of 10-cent coins, which is 12 - 7 = \boxed{5}. <math>\textbf{(C)}</math> | + | Let Claudia have <math>x</math> 5-cent coins and <math>\left( 12 - x \right)</math> 10-cent coins. It is easily observed that any multiple of <math>5</math> between <math>5</math> and <math>5x + 10(12 - x) = 120 - 5x</math> inclusive can be obtained by a combination of coins. Thus, <math>24 - x = 17</math> combinations can be made, so <math>x = 7</math>. But the answer is not <math>7,</math> because we are asked for the number of 10-cent coins, which is <math>12 - 7 = \boxed{\textbf{(C) } 5}</math> |
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| + | ==Solution 2== | ||
| + | Since the coins are 5-cent and 10-cent, all possible values that can be made will be multiples of <math>5.</math> To have exactly <math>17</math> different multiples of <math>5,</math> we will need to make up to <math>85</math> cents. If all twelve coins were 5-cent coins, we will have <math>60</math> cents possible. Each trade of a 5-cent coin for a 10-cent coin will gain <math>5</math> cents, and as we need to gain <math>25</math> cents, the answer is | ||
| + | <math>\boxed{\textbf{(C) } 5}</math> | ||
| + | |||
| + | ==Solution 3 (Quick Insight)== | ||
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| + | Notice that for every <math>d</math> dimes, any multiple of <math>5</math> less than or equal to <math>10d + 5(12-d)</math> is a valid arrangement. Since there are <math>17</math> in our case, we have <math>10d + 5(12-d)=17 \cdot 5 \Rightarrow d=5</math>. Therefore, the answer is <math>\boxed{\textbf{(C) } 5}</math>. | ||
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| + | ~MrThinker | ||
| + | |||
| + | == Solution 4 == | ||
| + | |||
| + | Dividing by 5cents to reduce clutter: | ||
| + | |||
| + | |||
| + | <math>D </math> double coins and <math>S </math> single coins can reach any value between <math>1</math> and <math>2D + S</math>. Set <math>2D + S = 17 </math> and <math>D + S = 12</math>. | ||
| + | |||
| + | Subtract to get <math>D = \boxed{\textbf{(C) } 5}</math>. | ||
| + | |||
| + | ~oinava | ||
| + | |||
| + | ==Video Solution== | ||
| + | https://youtu.be/F2iyhLzmCB8 | ||
| + | |||
| + | ~savannahsolver | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2015|ab=A|num-b=12|num-a=14}} | {{AMC10 box|year=2015|ab=A|num-b=12|num-a=14}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
| + | |||
| + | [[Category:Introductory Number Theory Problems]] | ||
Revision as of 14:36, 11 July 2024
Contents
Problem 13
Claudia has 12 coins, each of which is a 5-cent coin or a 10-cent coin. There are exactly 17 different values that can be obtained as combinations of one or more of her coins. How many 10-cent coins does Claudia have?
Solution 1
Let Claudia have 
5-cent coins and 
10-cent coins. It is easily observed that any multiple of 
between and 
inclusive can be obtained by a combination of coins. Thus, 
combinations can be made, so 
. But the answer is not 
because we are asked for the number of 10-cent coins, which is 
Solution 2
Since the coins are 5-cent and 10-cent, all possible values that can be made will be multiples of 
To have exactly 
different multiples of 
we will need to make up to 
cents. If all twelve coins were 5-cent coins, we will have 
cents possible. Each trade of a 5-cent coin for a 10-cent coin will gain cents, and as we need to gain 
cents, the answer is
Solution 3 (Quick Insight)
Notice that for every 
dimes, any multiple of less than or equal to 
is a valid arrangement. Since there are in our case, we have 
. Therefore, the answer is .
~MrThinker
Solution 4
Dividing by 5cents to reduce clutter:
double coins and 
single coins can reach any value between 
and 
. Set 
and 
.
Subtract to get 
.
~oinava
Video Solution
~savannahsolver
See Also
| 2015 AMC 10A (Problems • Answer Key • Resources) | ||
| Preceded by Problem 12 |
Followed by Problem 14 | |
| 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 AMC 10 Problems and Solutions | ||
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. 
