Difference between revisions of "2021 Fall AMC 12A Problems/Problem 3"
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~NH14 | ~NH14 | ||
| + | |||
| + | == Solution 3 == | ||
| + | The time spent on Route <math>A</math> is | ||
| + | <cmath> | ||
| + | \begin{align*} | ||
| + | \frac{6 \mbox{ miles}}{30 \mbox{ miles per hour}} | ||
| + | & = \frac{1}{5} \mbox{ hours} \\ | ||
| + | & = \frac{1}{5} \cdot 60 \mbox{ minutes} \\ | ||
| + | & = 12 \mbox{ minutes} . | ||
| + | \end{align*} | ||
| + | </cmath> | ||
| + | |||
| + | The time spent on Route <math>B</math> is | ||
| + | <cmath> | ||
| + | \begin{align*} | ||
| + | \frac{\left( 5 - \frac{1}{2} \right) \mbox{ miles}}{40 \mbox{ miles per hour}} | ||
| + | + \frac{\frac{1}{2} \mbox{ miles}}{20 \mbox{ miles per hour}} | ||
| + | & = \frac{9}{80} \mbox{ hours} | ||
| + | + \frac{1}{40} \mbox{ hours} \\ | ||
| + | & = \frac{11}{80} \mbox{ hours} \\ | ||
| + | & = \frac{11}{80} \cdot 60 \mbox{ minutes} \\ | ||
| + | & = \frac{33}{4} \mbox{ minutes} . | ||
| + | \end{align*} | ||
| + | </cmath> | ||
| + | |||
| + | Therefore, the number of minutes Route <math>B</math> is quicker than Route <math>A</math> is <math>12 - \frac{33}{4} = 3 \frac{3}{4}</math>. | ||
| + | |||
| + | Therefore, the answer is <math>\boxed{\textbf{(B) }3 \frac{3}{4}}</math>. | ||
| + | |||
| + | ~Steven Chen (www.professorchenedu.com) | ||
==See Also== | ==See Also== | ||
Revision as of 20:57, 25 November 2021
- The following problem is from both the 2021 Fall AMC 10A #4 and 2021 Fall AMC 12A #3, so both problems redirect to this page.
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
Solution 3
The time spent on Route is
The time spent on Route is
Therefore, the number of minutes Route is quicker than Route is 
.
Therefore, the answer is 
.
~Steven Chen (www.professorchenedu.com)
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. 
