Difference between revisions of "2021 AMC 10A Problems/Problem 2"
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<math>\textbf{(A)} ~600 \qquad\textbf{(B)} ~650 \qquad\textbf{(C)} ~1950 \qquad\textbf{(D)} ~2000\qquad\textbf{(E)} ~2050</math> | <math>\textbf{(A)} ~600 \qquad\textbf{(B)} ~650 \qquad\textbf{(C)} ~1950 \qquad\textbf{(D)} ~2000\qquad\textbf{(E)} ~2050</math> | ||
− | ==Solution 1== | + | ==Solution 1 (Two Variables)== |
The following system of equations can be formed with <math>p</math> representing the number of students in Portia's high school and <math>l</math> representing the number of students in Lara's high school. | The following system of equations can be formed with <math>p</math> representing the number of students in Portia's high school and <math>l</math> representing the number of students in Lara's high school. | ||
<cmath>p=3l</cmath> | <cmath>p=3l</cmath> | ||
Line 18: | Line 18: | ||
==Solution 3 (Arithmetic)== | ==Solution 3 (Arithmetic)== | ||
− | Clearly, <math>2600</math> | + | Clearly, <math>2600</math> is <math>4</math> times the number of students in Lara's high school. Therefore, Lara's high school has <math>2600\div4=650</math> students, and Portia's high school has <math>650\cdot3=\boxed{\textbf{(C)} ~1950}</math> students. |
~MRENTHUSIASM | ~MRENTHUSIASM | ||
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~MRENTHUSIASM | ~MRENTHUSIASM | ||
− | ==Video Solution (Very | + | ==Video Solution 1 (Very Fast & Simple)== |
https://youtu.be/DOtysU-a1B4 | https://youtu.be/DOtysU-a1B4 | ||
~ Education, the Study of Everything | ~ Education, the Study of Everything | ||
− | ==Video Solution | + | ==Video Solution 2 (Setting Variables) == |
https://youtu.be/qNf6SiIpIsk?t=119 | https://youtu.be/qNf6SiIpIsk?t=119 | ||
~ThePuzzlr | ~ThePuzzlr | ||
− | ==Video Solution | + | ==Video Solution 3 (Solving by Equation) == |
https://www.youtube.com/watch?v=aOpgeMfvUpE&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=1 | https://www.youtube.com/watch?v=aOpgeMfvUpE&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=1 | ||
~North America Math Contest Go Go Go | ~North America Math Contest Go Go Go | ||
− | ==Video Solution== | + | ==Video Solution 4== |
https://youtu.be/xXx0iP1tn8k | https://youtu.be/xXx0iP1tn8k | ||
Line 73: | Line 73: | ||
~IceMatrix | ~IceMatrix | ||
− | ==Video Solution (Problems 1-3)== | + | ==Video Solution 7 (Problems 1-3)== |
https://youtu.be/CupJpUzKPB0 | https://youtu.be/CupJpUzKPB0 | ||
Latest revision as of 19:58, 26 May 2021
Contents
- 1 Problem
- 2 Solution 1 (Two Variables)
- 3 Solution 2 (One Variable)
- 4 Solution 3 (Arithmetic)
- 5 Solution 4 (Answer Choices)
- 6 Video Solution 1 (Very Fast & Simple)
- 7 Video Solution 2 (Setting Variables)
- 8 Video Solution 3 (Solving by Equation)
- 9 Video Solution 4
- 10 Video Solution 5
- 11 Video Solution 6
- 12 Video Solution 7 (Problems 1-3)
- 13 See Also
Problem
Portia's high school has times as many students as Lara's high school. The two high schools have a total of students. How many students does Portia's high school have?
Solution 1 (Two Variables)
The following system of equations can be formed with representing the number of students in Portia's high school and representing the number of students in Lara's high school. Substituting with we get . Solving for , we get . Since we need to find we multiply by 3 to get , which is
-happykeeper
Solution 2 (One Variable)
Suppose Lara's high school has students, so Portia's high school has students. We have or The answer is
~MRENTHUSIASM
Solution 3 (Arithmetic)
Clearly, is times the number of students in Lara's high school. Therefore, Lara's high school has students, and Portia's high school has students.
~MRENTHUSIASM
Solution 4 (Answer Choices)
Solution 4.1 (Quick Inspection)
The number of students in Portia's high school must be a multiple of This eliminates and Since is too small (as it is clear that ), we are left with
~MRENTHUSIASM
Solution 4.2 (Plug in the Answer Choices)
For we have So, is incorrect.
For we have So, is incorrect.
For we have So, is correct.
For completeness, we will check answer choices and
For we have So, is incorrect.
For we have So, is incorrect.
~MRENTHUSIASM
Video Solution 1 (Very Fast & Simple)
~ Education, the Study of Everything
Video Solution 2 (Setting Variables)
https://youtu.be/qNf6SiIpIsk?t=119 ~ThePuzzlr
Video Solution 3 (Solving by Equation)
https://www.youtube.com/watch?v=aOpgeMfvUpE&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=1 ~North America Math Contest Go Go Go
Video Solution 4
- pi_is_3.14
Video Solution 5
~savannahsolver
Video Solution 6
https://youtu.be/50CThrk3RcM?t=66
~IceMatrix
Video Solution 7 (Problems 1-3)
~MathWithPi
See Also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 1 |
Followed by Problem 3 | |
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.