Difference between revisions of "2008 iTest Problems/Problem 33"
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One night, over dinner Jerry poses a challenge to his younger children: "Suppose we travel <math>50</math> miles per hour while heading to our final vacation destination..." | One night, over dinner Jerry poses a challenge to his younger children: "Suppose we travel <math>50</math> miles per hour while heading to our final vacation destination..." | ||
| − | Hannah teases her husband, "You | + | Hannah teases her husband, "You would drive that slowly" |
Jerry smirks at Hannah, then starts over, "So that we get a good view of all the beautiful landscape your mother likes to photograph from the passenger's seat, we | Jerry smirks at Hannah, then starts over, "So that we get a good view of all the beautiful landscape your mother likes to photograph from the passenger's seat, we | ||
travel at a constant rate of <math>50</math> miles per hour on the way to the beach. However, on the way back we travel at a faster constant rate along the exact same route. | travel at a constant rate of <math>50</math> miles per hour on the way to the beach. However, on the way back we travel at a faster constant rate along the exact same route. | ||
| − | If our faster return rate is an integer number of miles per hour, and our average speed for the | + | If our faster return rate is an integer number of miles per hour, and our average speed for the whole round trip is also an integer number of |
miles per hour, what must be our speed during the return trip?" | miles per hour, what must be our speed during the return trip?" | ||
Michael pipes up, "How about <math>4950</math> miles per hour?!" | Michael pipes up, "How about <math>4950</math> miles per hour?!" | ||
| − | Wendy smiles, "For the sake of your | + | Wendy smiles, "For the sake of your other children, please don't let Michael drive." |
| − | Jerry adds, "How about we assume that we never | + | Jerry adds, "How about we assume that we never ever drive more than <math>100</math> miles per hour. Michael and Wendy, let Josh and Alexis try this one." |
Joshua ignores the problem in favor of the huge pile of mashed potatoes on his plate. | Joshua ignores the problem in favor of the huge pile of mashed potatoes on his plate. | ||
| − | But Alexis scribbles some work on her napkin and declares the correct answer. What answer did Alexis find? | + | But Alexis scribbles some work on her napkin and declares the correct answer. What answer did Alexis find? |
==Solution== | ==Solution== | ||
Latest revision as of 19:19, 6 August 2023
Problem
One night, over dinner Jerry poses a challenge to his younger children: "Suppose we travel 
miles per hour while heading to our final vacation destination..."
Hannah teases her husband, "You would drive that slowly"
Jerry smirks at Hannah, then starts over, "So that we get a good view of all the beautiful landscape your mother likes to photograph from the passenger's seat, we
travel at a constant rate of miles per hour on the way to the beach. However, on the way back we travel at a faster constant rate along the exact same route.
If our faster return rate is an integer number of miles per hour, and our average speed for the whole round trip is also an integer number of
miles per hour, what must be our speed during the return trip?"
Michael pipes up, "How about 
miles per hour?!"
Wendy smiles, "For the sake of your other children, please don't let Michael drive."
Jerry adds, "How about we assume that we never ever drive more than 
miles per hour. Michael and Wendy, let Josh and Alexis try this one."
Joshua ignores the problem in favor of the huge pile of mashed potatoes on his plate.
But Alexis scribbles some work on her napkin and declares the correct answer. What answer did Alexis find?
Solution
Let 
be the time (in hours) it takes to drive on the way to the final vacation destination. That means the distance to the destination is 
miles, and the time the family takes on the way back is 
hours, where 
is the return trip rate in mph. Also, let the average rate of the trip be 
mph, where is an integer. Using the distance-rate-time formula,
![\[\frac{50t+50t}{t+\frac{50t}{r}} = n\]](http://latex.artofproblemsolving.com/2/6/9/269a4ee5bb55e38d022681cc0912fe2a1a91016c.png)
![\[\frac{100}{1+\frac{50}{r}} = n\]](http://latex.artofproblemsolving.com/a/7/7/a77977bcce0fabdc505dea923d1a99c26c83246d.png)
![\[100 = n + \frac{50n}{r}\]](http://latex.artofproblemsolving.com/d/8/8/d88d1dc974f051cd451016a0b1a3626324f7b9e7.png)
![\[100r = nr + 50n\]](http://latex.artofproblemsolving.com/2/d/f/2df826289775c03092268b91ebf232891846b9d1.png)
![\[0 = nr - 100r + 50n\]](http://latex.artofproblemsolving.com/0/0/6/00600d598a74debd09c3e0db6b813c0dd944c2dc.png)
Using Simon's Favorite Factoring Trick,
![\[0 = (n-100)(r+50) + 5000\]](http://latex.artofproblemsolving.com/3/d/4/3d430d9d75a3859d73308b6b56d14d09f6507e41.png)
![\[5000 = (100-n)(r+50)\]](http://latex.artofproblemsolving.com/2/8/6/286893c61defd50a70d2503f853badd31d773ba4.png)
Going through each factor of 
and solving for and , the only value less than that makes an integer is 
. The speed of the return trip is 
mph.
See Also
| 2008 iTest (Problems) | ||
| Preceded by: Problem 32 |
Followed by: Problem 34 | |
| 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 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40 • 41 • 42 • 43 • 44 • 45 • 46 • 47 • 48 • 49 • 50 • 51 • 52 • 53 • 54 • 55 • 56 • 57 • 58 • 59 • 60 • 61 • 62 • 63 • 64 • 65 • 66 • 67 • 68 • 69 • 70 • 71 • 72 • 73 • 74 • 75 • 76 • 77 • 78 • 79 • 80 • 81 • 82 • 83 • 84 • 85 • 86 • 87 • 88 • 89 • 90 • 91 • 92 • 93 • 94 • 95 • 96 • 97 • 98 • 99 • 100 | ||
