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Difference between revisions of "2024 AMC 8 Problems/Problem 12"
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==Problem==
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== Problem ==
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Rohan keeps a total of 90 guppies in 4 fish tanks.
 
  
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*There is 1 more guppy in the 2nd tank than in the 1st tank.
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Rohan keeps 90 guppies in 4 fish tanks.
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*There are 2 more guppies in the 3rd tank than in the 2nd tank.
+
 
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*There are 3 more guppies in the 4th tank than in the 3rd tank.
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* There is 1 more guppy in the 2nd tank than in the 1st tank.
 +
* There are 2 more guppies in the 3rd tank than in the 2nd tank.
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* There are 3 more guppies in the 4th tank than in the 3rd tank.
  
 
How many guppies are in the 4th tank?
 
How many guppies are in the 4th tank?
Line 10: Line 11:
 
<math>\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 26</math>
 
<math>\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 26</math>
  
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==Solution 1==
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== Solution 1 ==
 
Let <math>x</math> denote the number of guppies in the first tank.
 
Let <math>x</math> denote the number of guppies in the first tank.
  
 
Then, we have the following for the number of guppies in the rest of the tanks:
 
Then, we have the following for the number of guppies in the rest of the tanks:
  
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*<math>x + 1 =</math> the number of guppies in the second tank
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* The number of guppies in the second tank is <math>x+1</math>
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*<math>x + 1 + 2 =</math> the number of guppies in the third tank
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* The number of guppies in the third tank is <math>x+1+2</math>
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*<math>x + 1 + 2 + 3 =</math> the number of guppies in the fourth tank
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* The number of guppies in the fourth tank is <math>x+1+2+3</math>
  
 
The number of guppies in all of the tanks combined is 90, so we can write the equation
 
The number of guppies in all of the tanks combined is 90, so we can write the equation
  
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<math>x + x + 1 + x + 1 + 2 + x + 1 + 2 + 3 = 90</math>.
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<cmath>(x)+(x+1)+(x+1+2)+(x+1+2+3) = 90.</cmath>
  
 
Simplifying the equation gives
 
Simplifying the equation gives
  
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<math>4x + 10 = 90</math>.
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<cmath>4x + 10 = 90.</cmath>
 +
 
 +
Solving the resulting equation gives <math>x = 20</math>, so the number of guppies in the fourth tank is <math>20+1+2+3 = \boxed{\textbf{(E)}\ 26}</math>.
 +
 
 +
~ C. Ren, andliu766
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 +
== Solution 2 ==
 +
 
 +
Suppose there are no guppies in the first tank. Then, the number of guppies in the other tanks are <math>1,3,</math> and <math>6,</math> or <math>10</math> guppies in total. We need to add <math>90-10 = 80</math> guppies into <math>4</math> tanks or <math>20</math> guppies in each tank, so the number of guppies in the fourth tank is <math>20+6 = \boxed{\textbf{(E)}\ 26}.</math>
 +
 
 +
~vvsss
 +
 
 +
==Video by MathTalks 😉==
  
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Solving the resulting equation gives <math>x = 20</math>, so the number of guppies in the fourth tank is <math>20 + 1 + 2 + 3 = 26</math>.
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https://youtu.be/qAwRUj2N46c?si=QDUY8ZUVFP29Eg4c
  
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Therefore, the correct answer is <math>\boxed{\textbf{(E)}\ 26}</math>.
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  ~rc1219
  
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- C. Ren
 
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~andliu766 (Minor edits)
 
  
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==Solution 2==
 
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Suppose there are no guppies in the first tank.
 
  
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Then, the number of guppies in the other tanks are <math>1,3,</math> and <math>6,</math> or <math>10</math> guppies in total.
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== Video Solution 1 by Central Valley Math Circle ==
  
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We need to add <math>90 - 10 = 80</math> guppies into <math>4</math> tanks or <math>20</math> guppies in each tank.
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https://youtu.be/sldCPQRFgik
 +
~mr_mathman
  
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So the number of guppies in the fourth tank is <math>20 + 6 = \boxed{\textbf{(E)}\ 26}.</math>
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== Video Solution 2 ==
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+
 
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'''vladimir.shelomovskii@gmail.com, vvsss'''
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[//youtu.be/BaE00H2SHQM?si=XRr6hG8Mzai3OK-n&t=2749 ~Math-X]
 +
 
 +
==Video Solution 3 ==
  
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==Video Solution by Math-X (First fully understand the problem!!!)==
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[//youtu.be/5ZIFnqymdDQ?si=tmv7XmwLdPVqHRxy&t=1271 ~hsnacademy]
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https://youtu.be/BaE00H2SHQM?si=XRr6hG8Mzai3OK-n&t=2749
 
  
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~Math-X
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== Video Solution 4 by Power Solve ==
  
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==Video Solution 1 (easy to digest) by Power Solve==
 
 
https://youtu.be/2UIVXOB4f0o?si=e1Q2EbdEfPB_Q5Ql&t=66
 
https://youtu.be/2UIVXOB4f0o?si=e1Q2EbdEfPB_Q5Ql&t=66
  
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==Video Solution 2 by SpreadTheMathLove==
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== Video Solution 5 by SpreadTheMathLove ==
 +
 
 
https://www.youtube.com/watch?v=RRTxlduaDs8
 
https://www.youtube.com/watch?v=RRTxlduaDs8
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==Video Solution by NiuniuMaths (Easy to understand!)==
 
−
https://www.youtube.com/watch?v=V-xN8Njd_Lc
 
  
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~NiuniuMaths
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==Video Solution 6 ==
  
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== Video Solution by CosineMethod [🔥Fast and Easy🔥]==
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[//www.youtube.com/watch?v=V-xN8Njd_Lc ~NiuniuMaths]
 +
 
 +
== Video Solution 7 by CosineMethod ==
  
 
https://www.youtube.com/watch?v=u5sC6tftndU
 
https://www.youtube.com/watch?v=u5sC6tftndU
  
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==See Also==
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== Video Solution by Interstigation ==
 +
 
 +
https://youtu.be/ktzijuZtDas&t=1144
 +
 
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== Video Solution by Dr. David ==
 +
 
 +
https://youtu.be/GTSragg_268
 +
 
 +
==Video Solution by Daily Dose of Math ==
 +
 
 +
[https://youtu.be/x9CVizw-fz0 ~Thesmartgreekmathdude]
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 +
== See Also ==
 +
 
 
{{AMC8 box|year=2024|num-b=11|num-a=13}}
 
{{AMC8 box|year=2024|num-b=11|num-a=13}}
 
{{MAA Notice}}
 
{{MAA Notice}}
 +
[[Category:Introductory Algebra Problems]]

Latest revision as of 16:05, 2 June 2025

Problem

Rohan keeps 90 guppies in 4 fish tanks.

  • There is 1 more guppy in the 2nd tank than in the 1st tank.
  • There are 2 more guppies in the 3rd tank than in the 2nd tank.
  • There are 3 more guppies in the 4th tank than in the 3rd tank.

How many guppies are in the 4th tank?

$\textbf{(A)}\ 20 \qquad \textbf{(B)}\ 21 \qquad \textbf{(C)}\ 23 \qquad \textbf{(D)}\ 24 \qquad \textbf{(E)}\ 26$

Solution 1

Let $x$ denote the number of guppies in the first tank.

Then, we have the following for the number of guppies in the rest of the tanks:

  • The number of guppies in the second tank is $x+1$
  • The number of guppies in the third tank is $x+1+2$
  • The number of guppies in the fourth tank is $x+1+2+3$

The number of guppies in all of the tanks combined is 90, so we can write the equation

\[(x)+(x+1)+(x+1+2)+(x+1+2+3) = 90.\]

Simplifying the equation gives

\[4x + 10 = 90.\]

Solving the resulting equation gives $x = 20$, so the number of guppies in the fourth tank is $20+1+2+3 = \boxed{\textbf{(E)}\ 26}$.

~ C. Ren, andliu766

Solution 2

Suppose there are no guppies in the first tank. Then, the number of guppies in the other tanks are $1,3,$ and $6,$ or $10$ guppies in total. We need to add $90-10 = 80$ guppies into $4$ tanks or $20$ guppies in each tank, so the number of guppies in the fourth tank is $20+6 = \boxed{\textbf{(E)}\ 26}.$

~vvsss

Video by MathTalks 😉

https://youtu.be/qAwRUj2N46c?si=QDUY8ZUVFP29Eg4c 

 ~rc1219


Video Solution 1 by Central Valley Math Circle

https://youtu.be/sldCPQRFgik ~mr_mathman

Video Solution 2

~Math-X

Video Solution 3

~hsnacademy

Video Solution 4 by Power Solve

https://youtu.be/2UIVXOB4f0o?si=e1Q2EbdEfPB_Q5Ql&t=66

Video Solution 5 by SpreadTheMathLove

https://www.youtube.com/watch?v=RRTxlduaDs8

Video Solution 6

~NiuniuMaths

Video Solution 7 by CosineMethod

https://www.youtube.com/watch?v=u5sC6tftndU

Video Solution by Interstigation

https://youtu.be/ktzijuZtDas&t=1144

Video Solution by Dr. David

https://youtu.be/GTSragg_268

Video Solution by Daily Dose of Math

~Thesmartgreekmathdude

See Also

2024 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 11 		Followed by
Problem 13
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 AJHSME/AMC 8 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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