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

Revision as of 07:43, 20 November 2020

Problem

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 of $36-12 = \boxed{\textbf{(E) }24}$ .

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 $36-12=\boxed{\textbf{(E) }24}$ .

Video Solution

https://youtu.be/y__IHWpXprY

@@ Line 1: / Line 1: @@
-==Problem 11==
+==Problem==
 After school, Maya and Naomi headed to the beach, <math>6</math> 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?
@@ Line 38: / Line 38: @@
 ==Solution 1==
+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 of <math>36-12 = \boxed{\textbf{(E) }24}</math>.
-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)==
-Since speed is distance over time, Naomi's speed is <math>36</math> mph.
+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 <math>36-12=\boxed{\textbf{(E) }24}</math>.
-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>.
--wuwang2002
-==Solution 2==
-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>.
--franzliszt
-==Solution 3==
-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>~SweetMango77
-==Solution 4==
-We see that Naomi travels <math>6</math> miles in <math>\dfrac{1}{6}</math> of an hour. Thus, his speed is <math>36</math> mph. Maya's speed is <math>12</math> mph. The difference is <math>36 - 12 = \boxed{(\text{E}) 24}.</math>
--[[User:A_mathemagician|A_MatheMagician]]
 ==Video Solution==
 https://youtu.be/y__IHWpXprY
-~savannahsolver
+==See also== {{AMC8 box|year=2020|num-b=10|num-a=12}}
+{{MAA Notice}}
-==See also== {{AMC8 box|year=2020|num-b=10|num-a=12}} {{MAA Notice}}

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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