2021 AMC 10A Problems/Problem 2
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.