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Difference between revisions of "2021 AMC 10A Problems/Problem 2"

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==Problem==
 
==Problem==
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Lorem ipsum dolor sit amet, consectetur adipiscing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Nam libero justo laoreet sit amet. Eget arcu dictum varius duis at consectetur. Amet venenatis urna cursus eget nunc. Feugiat nisl pretium fusce id velit. Quam elementum pulvinar etiam non quam lacus. Feugiat nisl pretium fusce id velit. Felis eget nunc lobortis mattis aliquam faucibus purus. Iaculis urna id volutpat lacus laoreet non curabitur gravida arcu. Et ultrices neque ornare aenean euismod elementum nisi quis eleifend. Tellus id interdum velit laoreet id donec ultrices. Nascetur ridiculus mus mauris vitae ultricies. Ut placerat orci nulla pellentesque dignissim enim sit. Varius vel pharetra vel turpis. Aliquam id diam maecenas ultricies mi eget mauris. At tempor commodo ullamcorper a lacus vestibulum sed arcu non. Nulla malesuada pellentesque elit eget gravida cum sociis. Auctor augue mauris augue neque. Risus ultricies tristique nulla aliquet enim.
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Portia's high school has <math>3</math> times as many students as Lara's high school. The two high schools have a total of <math>2600</math> students. How many students does Portia's high school have?
  
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==Solution==
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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>
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lorem ipsum is annoying, so therefore the answer is F.
 
  
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==See also==
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==Solution 1 (Two Variables)==
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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:
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<cmath>\begin{align*}
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P&=3L, \\
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P+L&=2600.
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\end{align*}</cmath>
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Substituting <math>P=3L</math> gives <math>4L=2600.</math> Solving for <math>L</math> gives <math>L=650.</math> Since we need to find <math>P,</math> we multiply <math>650</math> by <math>3</math> to get <math>P=\boxed{\textbf{(C)} ~1950}.</math>
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~happykeeper (Solution)
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~MRENTHUSIASM (Reformatting)
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==Solution 2 (One Variable)==
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Suppose Lara's high school has <math>x</math> students, so Portia's high school has <math>3x</math> students. We have <math>x+3x=2600,</math> or <math>4x=2600.</math> The answer is <cmath>3x=2600\cdot\frac 34=650\cdot3=\boxed{\textbf{(C)} ~1950}.</cmath>
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~MRENTHUSIASM
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==Solution 3 (Arithmetic)==
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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.
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~MRENTHUSIASM
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==Solution 4 (Observations)==
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The number of students in Portia's high school must be a multiple of <math>3.</math> This eliminates <math>\textbf{(B)},\textbf{(D)},</math> and <math>\textbf{(E)}.</math> Since <math>\textbf{(A)}</math> is too small (as it is clear that <math>600+\frac{600}{3}<2600</math>), we are left with <math>\boxed{\textbf{(C)} ~1950}.</math>
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 +
~MRENTHUSIASM
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 +
==Video Solutions==
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===Video Solution 1 (Very Fast & Simple)===
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https://youtu.be/DOtysU-a1B4
 +
 
 +
~Education, the Study of Everything
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===Video Solution 2 (Setting Variables)===
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https://youtu.be/qNf6SiIpIsk?t=119
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~ThePuzzlr
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===Video Solution 3 (Solving by Equation)===
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https://www.youtube.com/watch?v=aOpgeMfvUpE&list=PLexHyfQ8DMuKqltG3cHT7Di4jhVl6L4YJ&index=1
 +
 
 +
~North America Math Contest Go Go Go
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===Video Solution 4 by OmegaLearn===
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https://youtu.be/xXx0iP1tn8k
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~pi_is_3.14
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===Video Solution 5===
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https://youtu.be/GwwDQYqptlQ
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~savannahsolver
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===Video Solution 6===
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https://youtu.be/50CThrk3RcM?t=66
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~IceMatrix
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===Video Solution 7 (Problems 1-3)===
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https://youtu.be/CupJpUzKPB0
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~MathWithPi
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===Video Solution 8===
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https://youtu.be/slVBYmcDMOI
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~The Learning Royal
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==See Also==
 
{{AMC10 box|year=2021|ab=A|num-b=1|num-a=3}}
 
{{AMC10 box|year=2021|ab=A|num-b=1|num-a=3}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 09:06, 11 July 2024

Problem

Portia's high school has $3$ times as many students as Lara's high school. The two high schools have a total of $2600$ students. How many students does Portia's high school have?

$\textbf{(A)} ~600 \qquad\textbf{(B)} ~650 \qquad\textbf{(C)} ~1950 \qquad\textbf{(D)} ~2000\qquad\textbf{(E)} ~2050$

Solution 1 (Two Variables)

The following system of equations can be formed with $P$ representing the number of students in Portia's high school and $L$ representing the number of students in Lara's high school: \begin{align*} P&=3L, \\ P+L&=2600. \end{align*} Substituting $P=3L$ gives $4L=2600.$ Solving for $L$ gives $L=650.$ Since we need to find $P,$ we multiply $650$ by $3$ to get $P=\boxed{\textbf{(C)} ~1950}.$

~happykeeper (Solution)

~MRENTHUSIASM (Reformatting)

Solution 2 (One Variable)

Suppose Lara's high school has $x$ students, so Portia's high school has $3x$ students. We have $x+3x=2600,$ or $4x=2600.$ The answer is \[3x=2600\cdot\frac 34=650\cdot3=\boxed{\textbf{(C)} ~1950}.\]

~MRENTHUSIASM

Solution 3 (Arithmetic)

Clearly, $2600$ is $4$ times the number of students in Lara's high school. Therefore, Lara's high school has $2600\div4=650$ students, and Portia's high school has $650\cdot3=\boxed{\textbf{(C)} ~1950}$ students.

~MRENTHUSIASM

Solution 4 (Observations)

The number of students in Portia's high school must be a multiple of $3.$ This eliminates $\textbf{(B)},\textbf{(D)},$ and $\textbf{(E)}.$ Since $\textbf{(A)}$ is too small (as it is clear that $600+\frac{600}{3}<2600$), we are left with $\boxed{\textbf{(C)} ~1950}.$

~MRENTHUSIASM

Video Solutions

Video Solution 1 (Very Fast & Simple)

https://youtu.be/DOtysU-a1B4

~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 by OmegaLearn

https://youtu.be/xXx0iP1tn8k

~pi_is_3.14

Video Solution 5

https://youtu.be/GwwDQYqptlQ

~savannahsolver

Video Solution 6

https://youtu.be/50CThrk3RcM?t=66

~IceMatrix

Video Solution 7 (Problems 1-3)

https://youtu.be/CupJpUzKPB0

~MathWithPi

Video Solution 8

https://youtu.be/slVBYmcDMOI

~The Learning Royal

See Also

2021 AMC 10A (Problems • Answer Key • Resources)
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Problem 1 		Followed by
Problem 3
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All AMC 10 Problems and Solutions

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