Difference between revisions of "2021 Fall AMC 12A Problems/Problem 3"
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+ | ==Problem== | ||
Mr. Lopez has a choice of two routes to get to work. Route A is <math>6</math> miles long, and his average speed along this route is <math>30</math> miles per hour. Route B is <math>5</math> miles long, and his average speed along this route is <math>40</math> miles per hour, except for a <math>\frac{1}{2}</math>-mile stretch in a school zone where his average speed is <math>20</math> miles per hour. By how many minutes is Route B quicker than Route A? | Mr. Lopez has a choice of two routes to get to work. Route A is <math>6</math> miles long, and his average speed along this route is <math>30</math> miles per hour. Route B is <math>5</math> miles long, and his average speed along this route is <math>40</math> miles per hour, except for a <math>\frac{1}{2}</math>-mile stretch in a school zone where his average speed is <math>20</math> miles per hour. By how many minutes is Route B quicker than Route A? | ||
<math>\textbf{(A)}\ 2 \frac{3}{4} \qquad\textbf{(B)}\ 3 \frac{3}{4} \qquad\textbf{(C)}\ 4 \frac{1}{2} \qquad\textbf{(D)}\ | <math>\textbf{(A)}\ 2 \frac{3}{4} \qquad\textbf{(B)}\ 3 \frac{3}{4} \qquad\textbf{(C)}\ 4 \frac{1}{2} \qquad\textbf{(D)}\ | ||
5 \frac{1}{2} \qquad\textbf{(E)}\ 6 \frac{3}{4}</math> | 5 \frac{1}{2} \qquad\textbf{(E)}\ 6 \frac{3}{4}</math> | ||
+ | |||
+ | ==Solution 1== | ||
+ | If Mr. Lopez chooses Route A, then he will spend <math>\frac{6}{30}=\frac{1}{5}</math> hour, or <math>12</math> minutes. | ||
+ | |||
+ | If Mr. Lopez chooses Route B, then he will spend <math>\frac{9/2}{40}+\frac{1/2}{20}=\frac{11}{80}</math> hour, or <math>8\frac14</math> minutes. | ||
+ | |||
+ | Therefore, Route B is quicker than Route A by <math>12-8\frac14=\boxed{\textbf{(B)}\ 3 \frac{3}{4}}</math> minutes. | ||
+ | |||
+ | ~MRENTHUSIASM | ||
+ | |||
+ | == Solution 2 == | ||
+ | We use the equation <math>d=st</math> to solve this problem. Recall that <math>1\text{ mile per hour}=\frac{1}{60}\text{ mile per minute}.</math> | ||
+ | |||
+ | On route <math>A,</math> the distance is <math>6</math> miles and the speed to travel this distance is <math>\frac{1}{2}</math> mile per minute. Thus, the time it takes on route <math>A</math> is <math>12</math> minutes. For route <math>B</math> we have to use the equation twice, once for the distance of <math>5- \frac{1}{2} = \frac{9}{2}</math> miles with a speed of <math>\frac{2}{3}</math> mile per minute and a distance of <math>\frac{1}{2}</math> miles at a speed of <math>\frac{1}{3}</math> mile per minute. Thus, the time it takes to go on Route <math>B</math> is <math>\frac{9}{2} \cdot \frac{3}{2} + \frac{1}{2} \cdot 3 = \frac{27}{4} + \frac{3}{2} = \frac{33}{4}</math> minutes. Thus, Route B is <math>12 - \frac{33}{4} = \frac{15}{4} = 3\frac{3}{4}</math> faster than Route <math>A.</math> Thus, the answer is <math>\boxed{\textbf{(B)}\ 3 \frac{3}{4}}.</math> | ||
+ | |||
+ | ~NH14 | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC12 box|year=2021 Fall|ab=A|num-b=2|num-a=4}} | ||
+ | {{AMC10 box|year=2021 Fall|ab=A|num-b=3|num-a=5}} | ||
+ | {{MAA Notice}} |
Revision as of 18:44, 23 November 2021
Contents
Problem
Mr. Lopez has a choice of two routes to get to work. Route A is miles long, and his average speed along this route is
miles per hour. Route B is
miles long, and his average speed along this route is
miles per hour, except for a
-mile stretch in a school zone where his average speed is
miles per hour. By how many minutes is Route B quicker than Route A?
Solution 1
If Mr. Lopez chooses Route A, then he will spend hour, or
minutes.
If Mr. Lopez chooses Route B, then he will spend hour, or
minutes.
Therefore, Route B is quicker than Route A by minutes.
~MRENTHUSIASM
Solution 2
We use the equation to solve this problem. Recall that
On route the distance is
miles and the speed to travel this distance is
mile per minute. Thus, the time it takes on route
is
minutes. For route
we have to use the equation twice, once for the distance of
miles with a speed of
mile per minute and a distance of
miles at a speed of
mile per minute. Thus, the time it takes to go on Route
is
minutes. Thus, Route B is
faster than Route
Thus, the answer is
~NH14
See Also
2021 Fall AMC 12A (Problems • Answer Key • Resources) | |
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 |
2021 Fall AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.