Difference between revisions of "2021 Fall AMC 10A Problems/Problem 4"
MRENTHUSIASM (talk | contribs) |
(→See Also) |
||
| Line 5: | Line 5: | ||
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== | + | ==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 A, then he will spend <math>\frac{6}{30}=\frac{1}{5}</math> hour, or <math>12</math> minutes. | ||
| Line 13: | Line 13: | ||
~MRENTHUSIASM | ~MRENTHUSIASM | ||
| + | |||
| + | == Solution 2 == | ||
| + | |||
| + | We use the equation <math>d=st</math> to solve this problem. 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> mph. 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> mph and a distance of <math>\frac{1}{2}</math> miles at a speed of <math>\frac{1}{3}</math> mph. 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)}.}</math> | ||
| + | |||
| + | ~NH14 | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2021 Fall|ab=A|num-b=3|num-a=5}} | {{AMC10 box|year=2021 Fall|ab=A|num-b=3|num-a=5}} | ||
{{MAA Notice}} | {{MAA Notice}} | ||
Revision as of 19:44, 22 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. On route 
the distance is miles and the speed to travel this distance is mph. 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 
mph and a distance of miles at a speed of 
mph. 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 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. 
