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Difference between revisions of "2024 AMC 8 Problems/Problem 12"

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==Problem==
 
Rohan keeps a total of 90 guppies in 4 fish tanks.
 
Rohan keeps a total of 90 guppies in 4 fish tanks.
  
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==Solution 1==
 
==Solution 1==
−
Let <math>x</math> = 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:
  
−
*<math>x</math> + 1 = the number of guppies in the second tank
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*<math>x + 1 =</math> the number of guppies in the second tank
−
*<math>x</math> + 1 + 2 = the number of guppies in the third tank
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*<math>x + 1 + 2 =</math> the number of guppies in the third tank
−
*<math>x</math> + 1 + 2 + 3 = the number of guppies in the fourth tank
+
*<math>x + 1 + 2 + 3 =</math> the number of guppies in the fourth tank
  
 
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
  
−
<math>x</math> + <math>x</math> + 1 + <math>x</math> + 1 + 2 + <math>x</math> + 1 + 2 + 3 = 90.
+
<math>x + x + 1 + x + 1 + 2 + x + 1 + 2 + 3 = 90</math>.
  
 
Simplifying the equation gives
 
Simplifying the equation gives
  
−
4<math>x</math> + 10 = 90.
+
<math>4x + 10 = 90</math>.
  
−
Solving the resulting equation gives <math>x</math> = 20, so the number of guppies in the fourth tank is 20 + 1 + 2 + 3 = 26.
+
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>.
  
−
The correct answer is <math>\textbf{(E)}\ 26</math>.
+
Therefore, the correct answer is <math>\boxed{\textbf{(E)}\ 26}</math>.
  
−
- C. Ren, Thomas Grover Middle School
+
- C. Ren
 +
~andliu766 (Minor edits)
 +
 
 +
==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>
 +
 
 +
'''vladimir.shelomovskii@gmail.com, vvsss'''
 +
 
 +
==Video Solution by Math-X (First fully understand the problem!!!)==
 +
https://youtu.be/BaE00H2SHQM?si=XRr6hG8Mzai3OK-n&t=2749
 +
 
 +
~Math-X
 +
 
 +
==Video Solution (A Clever Explanation You’ll Get Instantly)==
 +
https://youtu.be/5ZIFnqymdDQ?si=tmv7XmwLdPVqHRxy&t=1271
 +
 
 +
~hsnacademy
 +
 
 +
==Video Solution 1 (easy to digest) by Power Solve==
 +
https://youtu.be/2UIVXOB4f0o?si=e1Q2EbdEfPB_Q5Ql&t=66
 +
hello
 +
 
 +
==Video Solution 2 by SpreadTheMathLove==
 +
https://www.youtube.com/watch?v=RRTxlduaDs8
 +
==Video Solution by NiuniuMaths (Easy to understand!)==
 +
https://www.youtube.com/watch?v=V-xN8Njd_Lc
 +
 
 +
~NiuniuMaths
 +
 
 +
== Video Solution by CosineMethod [🔥Fast and Easy🔥]==
 +
 
 +
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 WhyMath==
 +
https://youtu.be/Gub5HfwfI1A
 +
 
 +
==See Also==
 +
{{AMC8 box|year=2024|num-b=11|num-a=13}}
 +
{{MAA Notice}}

Latest revision as of 06:41, 15 November 2024

Problem

Rohan keeps a total of 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:

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

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 $20 + 1 + 2 + 3 = 26$.

Therefore, the correct answer is $\boxed{\textbf{(E)}\ 26}$.

- C. Ren ~andliu766 (Minor edits)

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 $20 + 6 = \boxed{\textbf{(E)}\ 26}.$

vladimir.shelomovskii@gmail.com, vvsss

Video Solution by Math-X (First fully understand the problem!!!)

https://youtu.be/BaE00H2SHQM?si=XRr6hG8Mzai3OK-n&t=2749

~Math-X

Video Solution (A Clever Explanation You’ll Get Instantly)

https://youtu.be/5ZIFnqymdDQ?si=tmv7XmwLdPVqHRxy&t=1271

~hsnacademy

Video Solution 1 (easy to digest) by Power Solve

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

Video Solution 2 by SpreadTheMathLove

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

Video Solution by NiuniuMaths (Easy to understand!)

https://www.youtube.com/watch?v=V-xN8Njd_Lc

~NiuniuMaths

Video Solution by CosineMethod [🔥Fast and Easy🔥]

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 WhyMath

https://youtu.be/Gub5HfwfI1A

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

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