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Difference between revisions of "2024 AMC 10B Problems/Problem 1"

(→‎Solution 3)
(→‎Solution for Certain China Testpapers)
 
(19 intermediate revisions by 14 users not shown)
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==Problem==
 
==Problem==
−
In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in line?
+
In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?
  
 
<math>\textbf{(A) } 2021 \qquad\textbf{(B) } 2022 \qquad\textbf{(C) } 2023 \qquad\textbf{(D) } 2024 \qquad\textbf{(E) } 2025</math>
 
<math>\textbf{(A) } 2021 \qquad\textbf{(B) } 2022 \qquad\textbf{(C) } 2023 \qquad\textbf{(D) } 2024 \qquad\textbf{(E) } 2025</math>
  
−
 
+
===Modified Problem in Certain China Testpapers===
−
 
 
−
Certain China Testpapers:
 
  
 
In a long line of people arranged left to right, the 1015th person from the left is also the 1010th person from the right. How many people are in line?
 
In a long line of people arranged left to right, the 1015th person from the left is also the 1010th person from the right. How many people are in line?
Line 14: Line 12:
 
<math>\textbf{(A) } 2021 \qquad\textbf{(B) } 2022 \qquad\textbf{(C) } 2023 \qquad\textbf{(D) } 2024 \qquad\textbf{(E) } 2025</math>
 
<math>\textbf{(A) } 2021 \qquad\textbf{(B) } 2022 \qquad\textbf{(C) } 2023 \qquad\textbf{(D) } 2024 \qquad\textbf{(E) } 2025</math>
  
−
==Solution 1==
+
===Solution for Certain China Testpapers===
−
If the person is the 1013th from the left, that means there is 1012 people to their left.
 
−
If the person is the 1010th from the right, that means there is 1009 people to their right.
 
−
Therefore, there are <math>1012 + 1 + 1009 = \boxed{\textbf{(B)} 2022}</math> people in line.
 
−
 
 
−
Solution for certain China test papers:
 
  
 
If the person is the 1015th from the left, that means there is 1014 people to their left.
 
If the person is the 1015th from the left, that means there is 1014 people to their left.
 
If the person is the 1010th from the right, that means there is 1009 people to their right.
 
If the person is the 1010th from the right, that means there is 1009 people to their right.
−
Therefore, there are <math>1014 + 1 + 1009 = \boxed{\textbf{(D)} 2024}</math> people in line.
+
Therefore, there are <math>1014 + 1 + 1009 = \boxed{\textbf{(D) } 2024}</math> people in line.
  
 
~Aray10 (Main Solution) and RULE101 (Modifications for certain China test papers)
 
~Aray10 (Main Solution) and RULE101 (Modifications for certain China test papers)
 +
 +
==Solution 1==
 +
If the person is the 1013th from the left, that means there is 1012 people to their left.
 +
If the person is the 1010th from the right, that means there is 1009 people to their right.
 +
Therefore, there are <math>1012 + 1 + 1009 = \boxed{\textbf{(B) } 2022}</math> people in line.
  
 
==Solution 2==
 
==Solution 2==
  
−
The person is 1013th person from the left is also the 1010th person from the right, so the same person is counted twice.
+
The person is <math>1013</math>rd person from the left is also the <math>1010</math>th person from the right, so the same person is counted twice.
  
−
Therefore, there are 1013 + 1010 - 1 = <math>\boxed{\textbf{(B)} 2022}</math> people in line.
+
Therefore, there are <math>1013+1010-1=\boxed{\textbf{(B) } 2022}</math> people in line.
  
 
~Kathan_17
 
~Kathan_17
−
[[Image: 2024_AMC_12B_P1.png|thumb|center|600px|]]
+
~megaboy6679
 +
~Pomfun (fixed + vs. -)
  
 
==Solution 3==
 
==Solution 3==
−
We can look at a smaller case where it is the <math>4</math>th person from the left and the <math>2</math>nd person from the right. Listing out the people as numbers gives us a list of <math>1,2,3,4,5.</math> We see that the total number of people is the position from the left + the position from the right - 1, which in the case above is <math>4+2-1=5.</math> Plugging in <math>1013</math> and <math>1010</math> gives us <math>1013+1010-1=\boxed{\textbf{(B)} 2022}</math>
+
We can look at a smaller case where it is the <math>4</math>th person from the left and the <math>2</math>nd person from the right. Listing out the people as numbers gives us a list of <math>1,2,3,4,5.</math> We see that the total number of people is the position from the left + the position from the right - 1, which in the case above is <math>4+2-1=5.</math> Plugging in <math>1013</math> and <math>1010</math> gives us <math>1013+1010-1=\boxed{\textbf{(B) } 2022}</math>
  
 
~cownav
 
~cownav
Line 44: Line 43:
 
Since the person is the 1013th person from the left and the 1010th person from the right, there are 1009 people after the 1013th person.  
 
Since the person is the 1013th person from the left and the 1010th person from the right, there are 1009 people after the 1013th person.  
  
−
Therefore, there are 1013 + 1009 = <math>\boxed{\textbf{(B)} 2022}</math> people in line.
+
Therefore, there are 1013 + 1009 = <math>\boxed{\textbf{(B) } 2022}</math> people in line.
  
 
(Note: Similarly, you can perceive this information as 1012 people to the left of 1010 people and proceed.)
 
(Note: Similarly, you can perceive this information as 1012 people to the left of 1010 people and proceed.)
Line 50: Line 49:
 
~ElaineGu
 
~ElaineGu
  
−
==Video Solution 2 by SpreadTheMathLove==
+
==Solution 5==
 +
 
 +
Think of a Pattern
 +
 
 +
If 4 people: p1, p2, p3, p4 and you are 2nd from right and third from left
 +
 
 +
So just add 1013+1010 and subtract 1 since you are overcounting yourself twice.
 +
 
 +
We then get, 1013 + 1009 = <math>\boxed{\textbf{(B) } 2022}</math>
 +
 
 +
~ Aarav22
 +
 
 +
==Video Solution 1 by SpreadTheMathLove==
 
https://www.youtube.com/watch?v=24EZaeAThuE
 
https://www.youtube.com/watch?v=24EZaeAThuE
  
−
== Video Solution by Daily Dose of Math ==
+
== Video Solution 2 by Daily Dose of Math ==
  
 
https://youtu.be/XTUUEgYW_Fc
 
https://youtu.be/XTUUEgYW_Fc
Line 59: Line 70:
 
~Thesmartgreekmathdude
 
~Thesmartgreekmathdude
  
−
== Video Solution (TheBeautyofMath) ==
+
== Video Solution 3 (TheBeautyofMath) ==
  
 
https://www.youtube.com/watch?v=mpnm_r6D3-w
 
https://www.youtube.com/watch?v=mpnm_r6D3-w

Latest revision as of 14:50, 4 July 2025

The following problem is from both the 2024 AMC 10B #1 and 2024 AMC 12B #1, so both problems redirect to this page.

Problem

In a long line of people arranged left to right, the 1013th person from the left is also the 1010th person from the right. How many people are in the line?

$\textbf{(A) } 2021 \qquad\textbf{(B) } 2022 \qquad\textbf{(C) } 2023 \qquad\textbf{(D) } 2024 \qquad\textbf{(E) } 2025$

Modified Problem in Certain China Testpapers

In a long line of people arranged left to right, the 1015th person from the left is also the 1010th person from the right. How many people are in line?

$\textbf{(A) } 2021 \qquad\textbf{(B) } 2022 \qquad\textbf{(C) } 2023 \qquad\textbf{(D) } 2024 \qquad\textbf{(E) } 2025$

Solution for Certain China Testpapers

If the person is the 1015th from the left, that means there is 1014 people to their left. If the person is the 1010th from the right, that means there is 1009 people to their right. Therefore, there are $1014 + 1 + 1009 = \boxed{\textbf{(D) } 2024}$ people in line.

~Aray10 (Main Solution) and RULE101 (Modifications for certain China test papers)

Solution 1

If the person is the 1013th from the left, that means there is 1012 people to their left. If the person is the 1010th from the right, that means there is 1009 people to their right. Therefore, there are $1012 + 1 + 1009 = \boxed{\textbf{(B) } 2022}$ people in line.

Solution 2

The person is $1013$rd person from the left is also the $1010$th person from the right, so the same person is counted twice.

Therefore, there are $1013+1010-1=\boxed{\textbf{(B) } 2022}$ people in line.

~Kathan_17 ~megaboy6679 ~Pomfun (fixed + vs. -)

Solution 3

We can look at a smaller case where it is the $4$th person from the left and the $2$nd person from the right. Listing out the people as numbers gives us a list of $1,2,3,4,5.$ We see that the total number of people is the position from the left + the position from the right - 1, which in the case above is $4+2-1=5.$ Plugging in $1013$ and $1010$ gives us $1013+1010-1=\boxed{\textbf{(B) } 2022}$

~cownav

Solution 4

Since the person is the 1013th person from the left and the 1010th person from the right, there are 1009 people after the 1013th person.

Therefore, there are 1013 + 1009 = $\boxed{\textbf{(B) } 2022}$ people in line.

(Note: Similarly, you can perceive this information as 1012 people to the left of 1010 people and proceed.)

~ElaineGu

Solution 5

Think of a Pattern

If 4 people: p1, p2, p3, p4 and you are 2nd from right and third from left

So just add 1013+1010 and subtract 1 since you are overcounting yourself twice.

We then get, 1013 + 1009 = $\boxed{\textbf{(B) } 2022}$

~ Aarav22

Video Solution 1 by SpreadTheMathLove

https://www.youtube.com/watch?v=24EZaeAThuE

Video Solution 2 by Daily Dose of Math

https://youtu.be/XTUUEgYW_Fc

~Thesmartgreekmathdude

Video Solution 3 (TheBeautyofMath)

https://www.youtube.com/watch?v=mpnm_r6D3-w

See also

2024 AMC 10B (Problems • Answer Key • Resources)
Preceded by
First Problem 		Followed by
Problem 2
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
2024 AMC 12B (Problems • Answer Key • Resources)
Preceded by
First Problem 		Followed by
Problem 2
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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. AMC logo.png

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