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Difference between revisions of "2020 AMC 8 Problems/Problem 1"
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==Problem==
 
Luka is making lemonade to sell at a school fundraiser. His recipe requires <math>4</math> times as much water as sugar and twice as much sugar as lemon juice. He uses <math>3</math> cups of lemon juice. How many cups of water does he need?
 
Luka is making lemonade to sell at a school fundraiser. His recipe requires <math>4</math> times as much water as sugar and twice as much sugar as lemon juice. He uses <math>3</math> cups of lemon juice. How many cups of water does he need?
  
 
<math>\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E) }24</math>
 
<math>\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E) }24</math>
  
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==Solution 1==
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==Solution 1 (Direct)==
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Luka will need <math>3\cdot 2=6</math> cups of sugar and <math>6\cdot 4=24</math> cups of water. The answer is <math>\boxed{\textbf{(E) } 24}</math>.
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We have <math>\text{water} : \text{sugar} : \text{lemon juice} = 4\cdot 2 : 2 : 1 = 8 : 2 : 1,</math> so Luka needs <math>3 \cdot 8 = \boxed{\textbf{(E) }24}</math> cups.
  
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==Solution 2==
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==Solution 2 (Stepwise)==
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Let <math>W, S,</math> and <math>L</math> represent the number of cups of water, sugar, and lemon juice that Luka needs for his recipe, respectively. We are given that <math>W=4S</math> and <math>S=2L</math>. Since <math>L=3</math>, it follows that <math>S=6</math>, which in turn implies that <math>W=24 \implies\boxed{\textbf{(E) }24}</math>.<br>
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Since the amount of sugar is twice the amount of lemon juice, Luka uses <math>3\cdot2=6</math> cups of sugar.
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~[http://artofproblemsolving.com/community/user/jmansuri junaidmansuri]
 
  
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==Solution 3==
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Since the amount of water is <math>4</math> times the amount of sugar, he uses <math>6\cdot4=\boxed{\textbf{(E) }24}</math> cups of water.
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We have that <math>\textsf{lemonade} : \textsf{water} : \textsf{lemon juice} = 8 : 2 : 1,</math> so we have <math>3 \cdot 8 = \boxed{\textbf{(E) }24}.</math>
 
  
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[pog]
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~MRENTHUSIASM
  
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==Solution 4==
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==Solution 3 (Combination of Solutions 1 and 2)==
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The ratio is <math>\text{Water}:\text{Sugar}:\text{Lemon Juice},</math> or <math>8:2:1.</math> Since we know that Luka used 3 cups of lemon juice, he needs <math>3\cdot2=6</math> cups of sugar. Because the amount of water is <math>4</math> times the amount of sugar Luka needs, he will need <math>6\cdot4=\boxed{\textbf{(E) }24}</math> cups of water.
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Thanks to MRENTHUSIASM for the inspiration!
  
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We are given that <math>4w:s</math> and <math>2s=l</math> which we combine to get <math>8w:2s:l</math>. Letting all the variables equal <math>3</math>, we find that the answer is <math>3\cdot 8=\textbf{(E)}\ 24</math>.
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EarthSaver 15:12, 11 June 2021 (EDT)
  
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-franzliszt
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==Video Solution by NiuniuMaths (Easy to understand!)==
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==Solution 5==
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https://www.youtube.com/watch?v=8hgK6rESdek&t=9s
  
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Put the numbers in ratios <math>4w:s</math> and <math>2s:lj</math> when w = water, s = sugar, and lj = lemon juice. then since we know there is <math>3</math> cups of lemon juice, do the math. <math>3\cdot2\cdot4=6\cdot4=24</math>
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~NiuniuMaths
  
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~ bsu1
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==Video Solution by Math-X (First understand the problem!!!)==
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https://youtu.be/UnVo6jZ3Wnk?si=KUWl6TL6PU9LtC7o&t=105
  
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==Video Solution==
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~Math-X
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==Video Solution (🚀Under 1 min🚀)==
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https://youtu.be/T1K8irAOxRk
 +
 
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~Education, the Study of Everything
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==Video Solution by WhyMath==
 
https://youtu.be/FPC792h-mGE
 
https://youtu.be/FPC792h-mGE
  
 
~savannahsolver
 
~savannahsolver
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==Video Solution by The Learning Royal==
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https://youtu.be/eSxzI8P9_h8
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 +
~The Learning Royal
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 +
==Video Solution by Interstigation==
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https://youtu.be/YnwkBZTv5Fw?t=34
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 +
~Interstigation
 +
 +
==Video Solution by North America Math Contest Go Go Go==
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https://www.youtube.com/watch?v=f428YRwoXO4
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 +
~North America Math Contest Go Go Go
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==Video Solution by STEMbreezy==
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https://youtu.be/L_vDc-i585o?list=PLFcinOE4FNL0TkI-_yKVEYyA_QCS9mBNS
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~STEMbreezy
  
 
==See also==
 
==See also==
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{{AMC8 box|year=2020|before=First problem|num-a=2}}
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{{AMC8 box|year=2020|before=First Problem|num-a=2}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 06:06, 24 January 2024

Problem

Luka is making lemonade to sell at a school fundraiser. His recipe requires $4$ times as much water as sugar and twice as much sugar as lemon juice. He uses $3$ cups of lemon juice. How many cups of water does he need?

$\textbf{(A) }6 \qquad \textbf{(B) }8 \qquad \textbf{(C) }12 \qquad \textbf{(D) }18 \qquad \textbf{(E) }24$

Solution 1 (Direct)

We have $\text{water} : \text{sugar} : \text{lemon juice} = 4\cdot 2 : 2 : 1 = 8 : 2 : 1,$ so Luka needs $3 \cdot 8 = \boxed{\textbf{(E) }24}$ cups.

Solution 2 (Stepwise)

Since the amount of sugar is twice the amount of lemon juice, Luka uses $3\cdot2=6$ cups of sugar.

Since the amount of water is $4$ times the amount of sugar, he uses $6\cdot4=\boxed{\textbf{(E) }24}$ cups of water.

~MRENTHUSIASM

Solution 3 (Combination of Solutions 1 and 2)

The ratio is $\text{Water}:\text{Sugar}:\text{Lemon Juice},$ or $8:2:1.$ Since we know that Luka used 3 cups of lemon juice, he needs $3\cdot2=6$ cups of sugar. Because the amount of water is $4$ times the amount of sugar Luka needs, he will need $6\cdot4=\boxed{\textbf{(E) }24}$ cups of water.

Thanks to MRENTHUSIASM for the inspiration!

EarthSaver 15:12, 11 June 2021 (EDT)

Video Solution by NiuniuMaths (Easy to understand!)

https://www.youtube.com/watch?v=8hgK6rESdek&t=9s

~NiuniuMaths

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

https://youtu.be/UnVo6jZ3Wnk?si=KUWl6TL6PU9LtC7o&t=105

~Math-X

Video Solution (🚀Under 1 min🚀)

https://youtu.be/T1K8irAOxRk

~Education, the Study of Everything

Video Solution by WhyMath

https://youtu.be/FPC792h-mGE

~savannahsolver

Video Solution by The Learning Royal

https://youtu.be/eSxzI8P9_h8

~The Learning Royal

Video Solution by Interstigation

https://youtu.be/YnwkBZTv5Fw?t=34

~Interstigation

Video Solution by North America Math Contest Go Go Go

https://www.youtube.com/watch?v=f428YRwoXO4

~North America Math Contest Go Go Go

Video Solution by STEMbreezy

https://youtu.be/L_vDc-i585o?list=PLFcinOE4FNL0TkI-_yKVEYyA_QCS9mBNS

~STEMbreezy

See also

2020 AMC 8 (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 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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