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Difference between revisions of "2020 AMC 8 Problems/Problem 11"

(→‎Solution 3 -SweetMango77)
(→‎Video Solution by Math-X (First understand the problem!!!))
 
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==Solution 1==
 
==Solution 1==
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Naomi travels <math>6</math> miles in a time of <math>10</math> minutes, which is equivalent to <math>\dfrac{1}{6}</math> of an hour. Since <math>\text{speed} = \frac{\text{distance}}{\text{time}}</math>, her speed is <math>\frac{6}{\left(\frac{1}{6}\right)} = 36</math> mph. By a similar calculation, Maya's speed is <math>12</math> mph, so the answer is <math>36-12 = \boxed{\textbf{(E) }24}</math>.
  
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We use the formula <math>\text{speed}=\dfrac{\text{distance}}{\text{time}}</math>. Naomi's distance is <math>6</math> miles, and her time is <math>10</math> minutes, which is equivalent to <math>\dfrac{1}{6}</math> of an hour.
+
==Solution 2 (Variant of Solution 1)==
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Since speed is distance over time, Naomi's speed is <math>36</math> mph.
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Naomi's speed of <math>6</math> miles in <math>10</math> minutes is equivalent to <math>6 \cdot 6 = 36</math> miles per hour, while Maya's speed of <math>6</math> miles in <math>30</math> minutes (i.e. half an hour) is equivalent to <math>6 \cdot 2 = 12</math> miles per hour. The difference is consequently <math>36-12=\boxed{\textbf{(E) }24}</math>.
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Using the same process, Maya's speed is <math>12</math> mph. Subtracting those, we get an answer of <math>\boxed{(\text{E}) 24}</math>.
 
  
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==Solution 2==
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==Video Solution by NiuniuMaths (Easy to understand!)==
 +
https://www.youtube.com/watch?v=bHNrBwwUCMI
  
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Notice that Naomi travels at a rate of <math>6</math> miles every <math>10</math> minutes or <math>36</math> miles an hour. Maya travels at a rate of <math>6</math> miles every <math>30</math> minutes or <math>12</math> miles an hour. Hence, the answer is <math>36-12=\textbf{(E) }24</math>.
+
~NiuniuMaths
  
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-franzliszt
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==Video Solution by Math-X (First understand the problem!!!)==
 +
https://youtu.be/UnVo6jZ3Wnk?si=MfI4I15mLBXNvGMm&t=1375
  
−
==Solution 3 -SweetMango77==
+
~Math-X
  
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Notice that the difference between Maya's and Naomi's arrival time is 4 units on the graph, or twice as slow as Naomi. Since Naomi's time to go is <math>12</math> minutes, their difference in speed is <math>2\cdot12=\boxed{\textbf{(E) }24}</math>
+
==Video Solution (CLEVER MANIPULATIONS!!!)==
 +
https://youtu.be/gQrRdpw_tu8
  
−
==See also== {{AMC8 box|year=2020|num-b=10|num-a=12}} {{MAA Notice}}
+
~Education, the Study of Everything
 +
 
 +
==Video Solution by North America Math Contest Go Go Go==
 +
 
 +
https://www.youtube.com/watch?v=ND0y051eYm0
 +
 
 +
~North America Math Contest Go Go Go
 +
 
 +
==Video Solution by WhyMath==
 +
https://youtu.be/y__IHWpXprY
 +
 
 +
~savannahsolver
 +
 
 +
==Video Solution==
 +
https://youtu.be/xjwDsaRE_Wo
 +
 
 +
==Video Solution by Interstigation==
 +
https://youtu.be/YnwkBZTv5Fw?t=456
 +
 
 +
~Interstigation
 +
 
 +
 
 +
 
 +
 
 +
==See also== {{AMC8 box|year=2020|num-b=10|num-a=12}}  
 +
{{MAA Notice}}

Latest revision as of 15:34, 26 January 2024

Problem 11

After school, Maya and Naomi headed to the beach, $6$ miles away. Maya decided to bike while Naomi took a bus. The graph below shows their journeys, indicating the time and distance traveled. What was the difference, in miles per hour, between Naomi's and Maya's average speeds?

[asy] unitsize(1.25cm); dotfactor = 10; pen shortdashed=linetype(new real[] {2.7,2.7}); for (int i = 0; i < 6; ++i) { for (int j = 0; j < 6; ++j) { draw((i,0)--(i,6), grey); draw((0,j)--(6,j), grey); } } for (int i = 1; i <= 6; ++i) { draw((-0.1,i)--(0.1,i),linewidth(1.25)); draw((i,-0.1)--(i,0.1),linewidth(1.25)); label(string(5*i), (i,0), 2*S); label(string(i), (0, i), 2*W); } draw((0,0)--(0,6)--(6,6)--(6,0)--(0,0)--cycle,linewidth(1.25)); label(rotate(90) * "Distance (miles)", (-0.5,3), W); label("Time (minutes)", (3,-0.5), S); dot("Naomi", (2,6), 3*dir(305)); dot((6,6)); label("Maya", (4.45,3.5)); draw((0,0)--(1.15,1.3)--(1.55,1.3)--(3.15,3.2)--(3.65,3.2)--(5.2,5.2)--(5.4,5.2)--(6,6),linewidth(1.35)); draw((0,0)--(0.4,0.1)--(1.15,3.7)--(1.6,3.7)--(2,6),linewidth(1.35)+shortdashed); [/asy]

$\textbf{(A) }6 \qquad \textbf{(B) }12 \qquad \textbf{(C) }18 \qquad \textbf{(D) }20 \qquad \textbf{(E) }24$

Solution 1

Naomi travels $6$ miles in a time of $10$ minutes, which is equivalent to $\dfrac{1}{6}$ of an hour. Since $\text{speed} = \frac{\text{distance}}{\text{time}}$, her speed is $\frac{6}{\left(\frac{1}{6}\right)} = 36$ mph. By a similar calculation, Maya's speed is $12$ mph, so the answer is $36-12 = \boxed{\textbf{(E) }24}$.

Solution 2 (Variant of Solution 1)

Naomi's speed of $6$ miles in $10$ minutes is equivalent to $6 \cdot 6 = 36$ miles per hour, while Maya's speed of $6$ miles in $30$ minutes (i.e. half an hour) is equivalent to $6 \cdot 2 = 12$ miles per hour. The difference is consequently $36-12=\boxed{\textbf{(E) }24}$.

Video Solution by NiuniuMaths (Easy to understand!)

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

~NiuniuMaths

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

https://youtu.be/UnVo6jZ3Wnk?si=MfI4I15mLBXNvGMm&t=1375

~Math-X

Video Solution (CLEVER MANIPULATIONS!!!)

https://youtu.be/gQrRdpw_tu8

~Education, the Study of Everything

Video Solution by North America Math Contest Go Go Go

https://www.youtube.com/watch?v=ND0y051eYm0

~North America Math Contest Go Go Go

Video Solution by WhyMath

https://youtu.be/y__IHWpXprY

~savannahsolver

Video Solution

https://youtu.be/xjwDsaRE_Wo

Video Solution by Interstigation

https://youtu.be/YnwkBZTv5Fw?t=456

~Interstigation



See also

2020 AMC 8 (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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

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