Difference between revisions of "2021 AMC 12A Problems/Problem 4"
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+ | {{duplicate|[[2021 AMC 10A Problems#Problem 7|2021 AMC 10A #7]] and [[2021 AMC 12A Problems#Problem 4|2021 AMC 12A #4]]}} | ||
+ | |||
==Problem== | ==Problem== | ||
− | Tom has a collection of <math>13</math> snakes, <math>4</math> of which are purple and <math>5</math> of which are happy. He observes that all of his happy snakes can add, none of his purple snakes can subtract, and all of his snakes that can't subtract also can't add. Which of these conclusions can be drawn about Tom's snakes? | + | Tom has a collection of <math>13</math> snakes, <math>4</math> of which are purple and <math>5</math> of which are happy. He observes that |
+ | |||
+ | * all of his happy snakes can add, | ||
+ | |||
+ | * none of his purple snakes can subtract, and | ||
+ | |||
+ | * all of his snakes that can't subtract also can't add. | ||
+ | |||
+ | Which of these conclusions can be drawn about Tom's snakes? | ||
<math>\textbf{(A) }</math> Purple snakes can add. | <math>\textbf{(A) }</math> Purple snakes can add. | ||
+ | |||
<math>\textbf{(B) }</math> Purple snakes are happy. | <math>\textbf{(B) }</math> Purple snakes are happy. | ||
+ | |||
<math>\textbf{(C) }</math> Snakes that can add are purple. | <math>\textbf{(C) }</math> Snakes that can add are purple. | ||
+ | |||
<math>\textbf{(D) }</math> Happy snakes are not purple. | <math>\textbf{(D) }</math> Happy snakes are not purple. | ||
+ | |||
<math>\textbf{(E) }</math> Happy snakes can't subtract. | <math>\textbf{(E) }</math> Happy snakes can't subtract. | ||
+ | |||
+ | ==Solution 1 (Comprehensive Explanation of Logic)== | ||
+ | We are given that | ||
+ | <cmath>\begin{align*} | ||
+ | \text{happy}&\Longrightarrow\text{can add}, &(1) \\ | ||
+ | \text{purple}&\Longrightarrow\text{cannot subtract}, \hspace{15mm} &(2) \\ | ||
+ | \text{cannot subtract}&\Longrightarrow\text{cannot add}. &(3) | ||
+ | \end{align*}</cmath> | ||
+ | Two solutions follow from here: | ||
+ | |||
+ | ===Solution 1.1 (Intuitive)=== | ||
+ | Combining <math>(2)</math> and <math>(3)</math> gives | ||
+ | <cmath>\begin{align*} | ||
+ | \text{happy}&\Longrightarrow\text{can add}, &(1) \\ | ||
+ | \lefteqn{\underbrace{\phantom{\text{purple}\Longrightarrow\text{cannot subtract}}}_{(2)}}\text{purple}&\Longrightarrow\overbrace{\text{cannot subtract}\Longrightarrow\text{cannot add}}^{(3)}. \hspace{2.5mm} &(*) | ||
+ | \end{align*}</cmath> | ||
+ | Clearly, the answer is <math>\boxed{\textbf{(D)}}.</math> | ||
+ | |||
+ | ~MRENTHUSIASM (credit given to abhinavg0627) | ||
+ | |||
+ | ===Solution 1.2 (Rigorous)=== | ||
+ | <i><b>Recall that every conditional statement <math>\boldsymbol{p\Longrightarrow q}</math> is always logically equivalent to its contrapositive <math>\boldsymbol{\lnot q\Longrightarrow\lnot p.}</math></b></i> | ||
+ | |||
+ | Combining <math>(1),(2)</math> and <math>(3)</math> gives <cmath>\lefteqn{\underbrace{\phantom{\text{purple}\Longrightarrow\text{cannot subtract}}}_{(2)}}\text{purple}\Longrightarrow\overbrace{\text{cannot subtract}\Longrightarrow\lefteqn{\underbrace{\phantom{\text{cannot add}\Longrightarrow\text{not happy}}}_{\text{Contrapositive of }(1)}}\text{cannot add}}^{(3)}\Longrightarrow\text{not happy}. \hspace{15mm}(**)</cmath> Applying the hypothetical syllogism to <math>(**),</math> we conclude that <cmath>\text{purple}\Longrightarrow\text{not happy},</cmath> whose contrapositive is <cmath>\text{happy}\Longrightarrow\text{not purple}.</cmath> Therefore, the answer is <math>\boxed{\textbf{(D)}}.</math> | ||
+ | |||
+ | <u><b>Remark</b></u> | ||
+ | |||
+ | The conclusions in the other choices do not follow from <math>(**):</math> | ||
+ | |||
+ | <math>\textbf{(A) }\text{purple}\Longrightarrow\text{can add}</math> | ||
+ | |||
+ | <math>\textbf{(B) }\text{purple}\Longrightarrow\text{happy}</math> | ||
+ | |||
+ | <math>\textbf{(C) }\text{can add}\Longrightarrow\text{purple}</math> | ||
+ | |||
+ | <math>\textbf{(E) }\text{happy}\Longrightarrow\text{cannot subtract}</math> | ||
+ | |||
+ | ~MRENTHUSIASM | ||
+ | |||
+ | ==Solution 2 (Process of Elimination)== | ||
+ | From Solution 1.1, we can also see this through the process of elimination. | ||
+ | Statement <math>A</math> is false because purple snakes cannot add. <math>B</math> is false as well because since happy snakes can add and purple snakes can not add, purple snakes are not happy snakes. <math>E</math> is false using the same reasoning, purple snakes are not happy snakes so happy snakes can subtract since purple snakes cannot subtract. <math>C</math> is false since snakes that can add are happy, not purple. That leaves statement D. <math>\boxed{\textbf{(D)}}</math> is the only correct statement. | ||
+ | |||
+ | ~Bakedpotato66 | ||
+ | |||
+ | ==Solution 3 (Rigorous)== | ||
+ | We first convert each statement to "If X, then Y" form: | ||
+ | |||
+ | * If a snake is happy, then it can add. | ||
+ | |||
+ | * If a snake is purple, then it can't subtract. | ||
+ | |||
+ | * If a snake can't subtract, then it can't add. | ||
+ | |||
+ | Now, we simply check the truth value for each statement: | ||
+ | <ol style="margin-left: 1.5em;" type="A"> | ||
+ | <li>Combining the last two propositions, we have | ||
+ | * If a snake is purple, then it can't add. <p> | ||
+ | Thus, <math>\textbf{(A)}</math> is never true.</li><p> | ||
+ | <li>From the last part, we found that | ||
+ | * If a snake is purple, then it can't add. <p> | ||
+ | Also, since the contrapositive of a proposition has the same truth value as the proposition itself, we know, from the first statement, that | ||
+ | * If a snake can't add, then it isn't happy. <p> | ||
+ | Combining these two propositions, we find that | ||
+ | * If a snake is purple, then it isn't happy. Purple snakes are not happy.</li><p> | ||
+ | Thus, <math>\textbf{(B)}</math> is never true.</li><p> | ||
+ | <li>From part <math>\textbf{(A)},</math> we found that "If a snake is purple, then it can't add." This implies its contrapositive, "If a snake can add, then it is not purple." is true, meaning <math>\textbf{(C)}</math> is NEVER true. [Thanks again to MRENTHUSIASM for pointing this out!]</li><p> | ||
+ | <li>From the first statement, we have | ||
+ | * If a snake is happy, then it can add.<p> | ||
+ | From the contrapositive of the third statement, we have | ||
+ | * If a snake can add, then it can subtract. <p> | ||
+ | Then, from the contrapositive of the second statement, we have | ||
+ | * If a snake can subtract, then it is not purple. <p> | ||
+ | Combining all of these yields | ||
+ | * If a snake is happy, then it is not purple.<p> | ||
+ | Thus, <math>\textbf{(D)}</math> is always true.</li><p> | ||
+ | <li>From the first proposition, we have | ||
+ | * If a snake is happy, then it can add. <p> | ||
+ | From the contrapositive of the third proposition, we have | ||
+ | * If a snake can add, then it can subtract. <p> | ||
+ | Combining these two propositions gives | ||
+ | * If a snake is happy, then it can subtract. <p> | ||
+ | Thus, <math>\textbf{(E)}</math> is never true.</li><p> | ||
+ | </ol> | ||
+ | Therefore, <math>\boxed{\textbf{(D)}}</math> is our answer. | ||
+ | |||
+ | ~ Peace09 (My First Wiki Solution!) | ||
+ | |||
+ | ~ MRENTHUSIASM (Revision Suggestions and Code Adjustments) | ||
+ | |||
+ | ==Video Solution (Simple & Quick)== | ||
+ | https://youtu.be/hJKHaIcyIxA | ||
+ | |||
+ | ~ Education the Study of Everything | ||
+ | |||
+ | ==Video Solution by Aaron He (Sets)== | ||
+ | https://www.youtube.com/watch?v=xTGDKBthWsw&t=164 | ||
==Video Solution by Punxsutawney Phil== | ==Video Solution by Punxsutawney Phil== | ||
https://youtube.com/watch?v=MUHja8TpKGw&t=259s (Note that there's a slight error in the video I corrected in the description) | https://youtube.com/watch?v=MUHja8TpKGw&t=259s (Note that there's a slight error in the video I corrected in the description) | ||
+ | ==Video Solution by Hawk Math== | ||
+ | https://www.youtube.com/watch?v=P5al76DxyHY | ||
+ | |||
+ | == Video Solution (Using Logic to Eliminate Choices) == | ||
+ | https://youtu.be/Mofw3VXHPyg | ||
+ | |||
+ | ~ pi_is_3.14 | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/uDJv06-cNrI | ||
+ | ~savannahsolver | ||
− | == | + | ==Video Solution by TheBeautyofMath== |
+ | https://youtu.be/s6E4E06XhPU?t=202 (AMC10A) | ||
− | + | https://youtu.be/rEWS75W0Q54?t=353 (AMC12A) | |
+ | |||
+ | ~IceMatrix | ||
+ | |||
+ | ==Video Solution by The Learning Royal== | ||
+ | https://youtu.be/AWjOeBFyeb4 | ||
==See also== | ==See also== | ||
+ | {{AMC10 box|year=2021|ab=A|num-b=6|num-a=8}} | ||
{{AMC12 box|year=2021|ab=A|num-b=3|num-a=5}} | {{AMC12 box|year=2021|ab=A|num-b=3|num-a=5}} | ||
− | |||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 07:57, 4 September 2021
- The following problem is from both the 2021 AMC 10A #7 and 2021 AMC 12A #4, so both problems redirect to this page.
Contents
- 1 Problem
- 2 Solution 1 (Comprehensive Explanation of Logic)
- 3 Solution 2 (Process of Elimination)
- 4 Solution 3 (Rigorous)
- 5 Video Solution (Simple & Quick)
- 6 Video Solution by Aaron He (Sets)
- 7 Video Solution by Punxsutawney Phil
- 8 Video Solution by Hawk Math
- 9 Video Solution (Using Logic to Eliminate Choices)
- 10 Video Solution
- 11 Video Solution by TheBeautyofMath
- 12 Video Solution by The Learning Royal
- 13 See also
Problem
Tom has a collection of snakes, of which are purple and of which are happy. He observes that
- all of his happy snakes can add,
- none of his purple snakes can subtract, and
- all of his snakes that can't subtract also can't add.
Which of these conclusions can be drawn about Tom's snakes?
Purple snakes can add.
Purple snakes are happy.
Snakes that can add are purple.
Happy snakes are not purple.
Happy snakes can't subtract.
Solution 1 (Comprehensive Explanation of Logic)
We are given that Two solutions follow from here:
Solution 1.1 (Intuitive)
Combining and gives Clearly, the answer is
~MRENTHUSIASM (credit given to abhinavg0627)
Solution 1.2 (Rigorous)
Recall that every conditional statement is always logically equivalent to its contrapositive
Combining and gives Applying the hypothetical syllogism to we conclude that whose contrapositive is Therefore, the answer is
Remark
The conclusions in the other choices do not follow from
~MRENTHUSIASM
Solution 2 (Process of Elimination)
From Solution 1.1, we can also see this through the process of elimination. Statement is false because purple snakes cannot add. is false as well because since happy snakes can add and purple snakes can not add, purple snakes are not happy snakes. is false using the same reasoning, purple snakes are not happy snakes so happy snakes can subtract since purple snakes cannot subtract. is false since snakes that can add are happy, not purple. That leaves statement D. is the only correct statement.
~Bakedpotato66
Solution 3 (Rigorous)
We first convert each statement to "If X, then Y" form:
- If a snake is happy, then it can add.
- If a snake is purple, then it can't subtract.
- If a snake can't subtract, then it can't add.
Now, we simply check the truth value for each statement:
- Combining the last two propositions, we have
- If a snake is purple, then it can't add.
- From the last part, we found that
- If a snake is purple, then it can't add.
- If a snake can't add, then it isn't happy.
- If a snake is purple, then it isn't happy. Purple snakes are not happy.
- From part we found that "If a snake is purple, then it can't add." This implies its contrapositive, "If a snake can add, then it is not purple." is true, meaning is NEVER true. [Thanks again to MRENTHUSIASM for pointing this out!]
- From the first statement, we have
- If a snake is happy, then it can add.
- If a snake can add, then it can subtract.
- If a snake can subtract, then it is not purple.
- If a snake is happy, then it is not purple.
- From the first proposition, we have
- If a snake is happy, then it can add.
- If a snake can add, then it can subtract.
- If a snake is happy, then it can subtract.
Therefore, is our answer.
~ Peace09 (My First Wiki Solution!)
~ MRENTHUSIASM (Revision Suggestions and Code Adjustments)
Video Solution (Simple & Quick)
~ Education the Study of Everything
Video Solution by Aaron He (Sets)
https://www.youtube.com/watch?v=xTGDKBthWsw&t=164
Video Solution by Punxsutawney Phil
https://youtube.com/watch?v=MUHja8TpKGw&t=259s (Note that there's a slight error in the video I corrected in the description)
Video Solution by Hawk Math
https://www.youtube.com/watch?v=P5al76DxyHY
Video Solution (Using Logic to Eliminate Choices)
~ pi_is_3.14
Video Solution
~savannahsolver
Video Solution by TheBeautyofMath
https://youtu.be/s6E4E06XhPU?t=202 (AMC10A)
https://youtu.be/rEWS75W0Q54?t=353 (AMC12A)
~IceMatrix
Video Solution by The Learning Royal
See also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 6 |
Followed by Problem 8 | |
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 |
2021 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 3 |
Followed by Problem 5 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.