Difference between revisions of "2004 AMC 12A Problems/Problem 11"
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{{duplicate|[[2004 AMC 12A Problems|2004 AMC 12A #11]] and [[2004 AMC 10A Problems/Problem 14|2004 AMC 10A #14]]}} | {{duplicate|[[2004 AMC 12A Problems|2004 AMC 12A #11]] and [[2004 AMC 10A Problems/Problem 14|2004 AMC 10A #14]]}} | ||
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== Problem == | == Problem == | ||
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is <math>20</math> cents. If she had one more quarter, the average value would be <math>21</math> cents. How many dimes does she have in her purse? | The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is <math>20</math> cents. If she had one more quarter, the average value would be <math>21</math> cents. How many dimes does she have in her purse? | ||
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<math>\text {(A)}\ 0 \qquad \text {(B)}\ 1 \qquad \text {(C)}\ 2 \qquad \text {(D)}\ 3\qquad \text {(E)}\ 4</math> | <math>\text {(A)}\ 0 \qquad \text {(B)}\ 1 \qquad \text {(C)}\ 2 \qquad \text {(D)}\ 3\qquad \text {(E)}\ 4</math> | ||
− | == Solution 1 == | + | ==Solutions== |
+ | === Solution 1 === | ||
Let the total value, in cents, of the coins Paula has originally be <math>v</math>, and the number of coins she has be <math>n</math>. Then <math>\frac{v}{n}=20\Longrightarrow v=20n</math> and <math>\frac{v+25}{n+1}=21</math>. Substituting yields: <math>20n+25=21(n+1),</math> so <math>n=4</math>, <math>v = 80.</math> Then, we see that the only way Paula can satisfy this rule is if she had <math>3</math> quarters and <math>1</math> nickel in her purse. Thus, she has <math>\boxed{\mathrm{(A)}\ 0}</math> dimes. | Let the total value, in cents, of the coins Paula has originally be <math>v</math>, and the number of coins she has be <math>n</math>. Then <math>\frac{v}{n}=20\Longrightarrow v=20n</math> and <math>\frac{v+25}{n+1}=21</math>. Substituting yields: <math>20n+25=21(n+1),</math> so <math>n=4</math>, <math>v = 80.</math> Then, we see that the only way Paula can satisfy this rule is if she had <math>3</math> quarters and <math>1</math> nickel in her purse. Thus, she has <math>\boxed{\mathrm{(A)}\ 0}</math> dimes. | ||
− | == Solution 2 == | + | === Solution 2 === |
If the new coin was worth <math>20</math> cents, adding it would not change the mean. The additional <math>5</math> cents raise the mean by <math>1</math>, thus the new number of coins must be <math>5</math>. Therefore there were <math>4</math> coins worth a total of <math>4\times20=80</math> cents. As in the previous solution, we conclude that the only way to get <math>80</math> cents using <math>4</math> coins is <math>25+25+25+5</math>. Thus, having three quarters, one nickel, and no dimes <math>\boxed{\mathrm{(A)}\ 0}.</math> | If the new coin was worth <math>20</math> cents, adding it would not change the mean. The additional <math>5</math> cents raise the mean by <math>1</math>, thus the new number of coins must be <math>5</math>. Therefore there were <math>4</math> coins worth a total of <math>4\times20=80</math> cents. As in the previous solution, we conclude that the only way to get <math>80</math> cents using <math>4</math> coins is <math>25+25+25+5</math>. Thus, having three quarters, one nickel, and no dimes <math>\boxed{\mathrm{(A)}\ 0}.</math> | ||
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+ | ==Video Solution== | ||
+ | https://youtu.be/9Q463wjzCzQ | ||
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+ | Education, the Study of Everything | ||
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+ | == Video Solution by OmegaLearn== | ||
+ | https://youtu.be/HISL2-N5NVg?t=1649 | ||
+ | |||
+ | ~ pi_is_3.14 | ||
== See also == | == See also == | ||
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[[Category:Introductory Algebra Problems]] | [[Category:Introductory Algebra Problems]] | ||
{{MAA Notice}} | {{MAA Notice}} | ||
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+ | Thank you for reading these solutions by the users of AoPS! |
Latest revision as of 17:39, 31 December 2022
- The following problem is from both the 2004 AMC 12A #11 and 2004 AMC 10A #14, so both problems redirect to this page.
Contents
Problem
The average value of all the pennies, nickels, dimes, and quarters in Paula's purse is cents. If she had one more quarter, the average value would be
cents. How many dimes does she have in her purse?
Solutions
Solution 1
Let the total value, in cents, of the coins Paula has originally be , and the number of coins she has be
. Then
and
. Substituting yields:
so
,
Then, we see that the only way Paula can satisfy this rule is if she had
quarters and
nickel in her purse. Thus, she has
dimes.
Solution 2
If the new coin was worth cents, adding it would not change the mean. The additional
cents raise the mean by
, thus the new number of coins must be
. Therefore there were
coins worth a total of
cents. As in the previous solution, we conclude that the only way to get
cents using
coins is
. Thus, having three quarters, one nickel, and no dimes
Video Solution
Education, the Study of Everything
Video Solution by OmegaLearn
https://youtu.be/HISL2-N5NVg?t=1649
~ pi_is_3.14
See also
2004 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 10 |
Followed by Problem 12 |
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 12 Problems and Solutions |
2004 AMC 10A (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 AMC 10 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
Thank you for reading these solutions by the users of AoPS!