Difference between revisions of "2020 AMC 10B Problems/Problem 3"

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==Problem 3==
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{{duplicate|[[2020 AMC 10B Problems|2020 AMC 10B #3]] and [[2020 AMC 12B Problems|2020 AMC 12B #3]]}}
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==Problem==
  
 
The ratio of <math>w</math> to <math>x</math> is <math>4:3</math>, the ratio of <math>y</math> to <math>z</math> is <math>3:2</math>, and the ratio of <math>z</math> to <math>x</math> is <math>1:6</math>. What is the ratio of <math>w</math> to <math>y?</math>
 
The ratio of <math>w</math> to <math>x</math> is <math>4:3</math>, the ratio of <math>y</math> to <math>z</math> is <math>3:2</math>, and the ratio of <math>z</math> to <math>x</math> is <math>1:6</math>. What is the ratio of <math>w</math> to <math>y?</math>
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<math>\textbf{(A)}\ 4:3 \qquad\textbf{(B)}\ 3:2 \qquad\textbf{(C)}\  8:3 \qquad\textbf{(D)}\ 4:1 \qquad\textbf{(E)}\ 16:3</math>
 
<math>\textbf{(A)}\ 4:3 \qquad\textbf{(B)}\ 3:2 \qquad\textbf{(C)}\  8:3 \qquad\textbf{(D)}\ 4:1 \qquad\textbf{(E)}\ 16:3</math>
  
==Solution 1==
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==Solution 1 (One Sentence)==
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We have <cmath>\frac wy = \frac wx \cdot \frac xz \cdot \frac zy = \frac43\cdot\frac61\cdot\frac23=\frac{16}{3},</cmath> from which <math>w:y=\boxed{\textbf{(E)}\ 16:3}.</math>
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~MRENTHUSIASM
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==Solution 2==
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We need to somehow link all three of the ratios together. We can start by connecting the last two ratios together by multiplying the last ratio by two.
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<math>z:x=1:6=2:12</math>, and since <math>y:z=3:2</math>, we can link them together to get <math>y:z:x=3:2:12</math>.
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Finally, since <math>x:w=3:4=12:16</math>, we can link this again to get: <math>y:z:x:w=3:2:12:16</math>, so <math>w:y = \boxed{\textbf{(E)}\ 16:3}</math>.
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~quacker88
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==Solution 3==
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We have the equations <math>\frac{w}{x}=\frac{4}{3}</math>, <math>\frac{y}{z}=\frac{3}{2}</math>, and <math>\frac{z}{x}=\frac{1}{6}</math>.
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Clearing denominators, we have <math>3w = 4x</math>, <math>2y = 3z</math>, and <math>6z = x</math>.
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Since we want <math>\frac{w}{y}</math>, we look to find <math>y</math> in terms of <math>x</math> since we know the relationship between <math>x</math> and <math>w</math>.
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We begin by multiplying both sides of <math>2y = 3z</math> by two, obtaining <math>4y = 6z</math>. We then substitute that into <math>6z = x</math>
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to get <math>4y = x</math> . Now, to be able to substitute this into out first equation, we need to have <math>4x</math> on the RHS.
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Multiplying both sides by <math>4</math>, we have <math>16y = 4x</math>.
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Substituting this into our first equation, we have <math>3w = 16y</math>, or <math>\frac{w}{y}=\frac{16}{3}</math>, so our answer is <math>\boxed{\textbf{(E)}\ 16:3}</math>.
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~Binderclips1
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==Solution 4==
  
 
WLOG, let <math>w=4</math> and <math>x=3</math>.  
 
WLOG, let <math>w=4</math> and <math>x=3</math>.  
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The ratio of <math>y</math> to <math>z</math> is <math>3:2</math>, so <math>\frac{y}{\frac{1}{2}}=\frac{3}{2} \implies y=\frac{3}{4}</math>.  
 
The ratio of <math>y</math> to <math>z</math> is <math>3:2</math>, so <math>\frac{y}{\frac{1}{2}}=\frac{3}{2} \implies y=\frac{3}{4}</math>.  
  
The ratio of <math>w</math> to <math>y</math> is then <math>\frac{4}{\frac{3}{4}}=\frac{16}{3}</math> so our answer is <math>\boxed{\textbf{(E)}\ 16:3}</math> ~quacker88
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The ratio of <math>w</math> to <math>y</math> is then <math>\frac{4}{\frac{3}{4}}=\frac{16}{3}</math> so our answer is <math>\boxed{\textbf{(E)}\ 16:3}</math>.
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~quacker88
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==Video Solution (HOW TO THINK CREATIVELY!!!)==
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https://www.youtube.com/watch?v=QqkNnsNEgiA (for AMC 10)
  
==Solution 2==
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https://youtu.be/Lz6XmHr8tJE (for AMC 12)
  
We need to somehow link all three of the ratios together. We can start by connecting the last two ratios together by multiplying the last ratio by two.
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~Education, the Study of Everything
  
<math>z:x=1:6=2:12</math>, and since <math>y:z=3:2</math>, we can link them together to get <math>y:z:x=3:2:12</math>.
 
  
Finally, since <math>x:w=3:4=12:16</math>, we can link this again to get: <math>y:z:x:w=3:2:12:16</math>, so <math>w:y = \boxed{\textbf{(E)}\ 16:3}</math> ~quacker88
 
  
 
==Video Solution==
 
==Video Solution==
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https://youtu.be/Gkm5rU5MlOU (for AMC 10)
 
https://youtu.be/Gkm5rU5MlOU (for AMC 10)
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https://youtu.be/WfTty8Fe5Fo (for AMC 12)
 
https://youtu.be/WfTty8Fe5Fo (for AMC 12)
  
~IceMatrix
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https://youtu.be/y4BbRapJepY
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~savannahsolver
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https://youtu.be/wH7xhYxwaFc
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~AlexExplains
  
 
==See Also==
 
==See Also==

Latest revision as of 13:32, 6 June 2023

The following problem is from both the 2020 AMC 10B #3 and 2020 AMC 12B #3, so both problems redirect to this page.

Problem

The ratio of $w$ to $x$ is $4:3$, the ratio of $y$ to $z$ is $3:2$, and the ratio of $z$ to $x$ is $1:6$. What is the ratio of $w$ to $y?$

$\textbf{(A)}\ 4:3 \qquad\textbf{(B)}\ 3:2 \qquad\textbf{(C)}\  8:3 \qquad\textbf{(D)}\ 4:1 \qquad\textbf{(E)}\ 16:3$

Solution 1 (One Sentence)

We have \[\frac wy = \frac wx \cdot \frac xz \cdot \frac zy = \frac43\cdot\frac61\cdot\frac23=\frac{16}{3},\] from which $w:y=\boxed{\textbf{(E)}\ 16:3}.$

~MRENTHUSIASM

Solution 2

We need to somehow link all three of the ratios together. We can start by connecting the last two ratios together by multiplying the last ratio by two.

$z:x=1:6=2:12$, and since $y:z=3:2$, we can link them together to get $y:z:x=3:2:12$.

Finally, since $x:w=3:4=12:16$, we can link this again to get: $y:z:x:w=3:2:12:16$, so $w:y = \boxed{\textbf{(E)}\ 16:3}$.

~quacker88

Solution 3

We have the equations $\frac{w}{x}=\frac{4}{3}$, $\frac{y}{z}=\frac{3}{2}$, and $\frac{z}{x}=\frac{1}{6}$. Clearing denominators, we have $3w = 4x$, $2y = 3z$, and $6z = x$. Since we want $\frac{w}{y}$, we look to find $y$ in terms of $x$ since we know the relationship between $x$ and $w$. We begin by multiplying both sides of $2y = 3z$ by two, obtaining $4y = 6z$. We then substitute that into $6z = x$ to get $4y = x$ . Now, to be able to substitute this into out first equation, we need to have $4x$ on the RHS. Multiplying both sides by $4$, we have $16y = 4x$. Substituting this into our first equation, we have $3w = 16y$, or $\frac{w}{y}=\frac{16}{3}$, so our answer is $\boxed{\textbf{(E)}\ 16:3}$.

~Binderclips1

Solution 4

WLOG, let $w=4$ and $x=3$.

Since the ratio of $z$ to $x$ is $1:6$, we can substitute in the value of $x$ to get $\frac{z}{3}=\frac{1}{6} \implies z=\frac{1}{2}$.

The ratio of $y$ to $z$ is $3:2$, so $\frac{y}{\frac{1}{2}}=\frac{3}{2} \implies y=\frac{3}{4}$.

The ratio of $w$ to $y$ is then $\frac{4}{\frac{3}{4}}=\frac{16}{3}$ so our answer is $\boxed{\textbf{(E)}\ 16:3}$.

~quacker88

Video Solution (HOW TO THINK CREATIVELY!!!)

https://www.youtube.com/watch?v=QqkNnsNEgiA (for AMC 10)

https://youtu.be/Lz6XmHr8tJE (for AMC 12)

~Education, the Study of Everything


Video Solution

https://youtu.be/Gkm5rU5MlOU (for AMC 10)

https://youtu.be/WfTty8Fe5Fo (for AMC 12)


https://youtu.be/y4BbRapJepY

~savannahsolver

https://youtu.be/wH7xhYxwaFc

~AlexExplains

See Also

2020 AMC 10B (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
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
2020 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 2
Followed by
Problem 4
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

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