So You Think You're Smart? Take 3
Time for another problem.
Anne and Betty set out on a trip. On their return, Betty's mother asked her, "Who drove?"
"Well," replied Betty, "on the way there, Ann drove the first 40 miles and I drove the rest."
"And you reversed that on the way home?" asked Betty's mom.
"Well, we drove the same route home, but Anne drove the first part of the trip, then I drove the last 50 miles."
"Why, that's not fair," Betty's mom said. "You too should have shared the driving equally. Instead, you drove farther."
Now, Betty's mom didn't know where the two girls went, so how could she have known they didn't drive the same amount? And (according to Betty's mom) she even knew how much farther Betty had driven.
How's that possible?
on the way there, ann drove 40 miles and betty drove 60
On the return trip, they both drove 50 miles.
Using the "first part" reference in the description of the trip home, I take that to mean that the trip was two equal parts of 50 miles.
Betty drove 10 more miles than Ann.
and "too" should be "two" in Betty's mom's reply about sharing equally ;)
We know the trip is at LEAST 51 miles because the most than any one person drove was 50, and it can be reasonably assumed that the other person drove at least SOME mileage, minimum of 1 mile.
This would yield (incorrect) math as follows:
Departure:
Ann drove 40, Betty drove 11 = 51
Return:
Ann drove 11, Betty drove 50 = 61 (!)
So the actual trip is DEFINITELY 61 miles, and the correct math is:
Departure:
Ann drove 40, Betty drove 21
Return:
Ann drove 11, betty drove 50
Total Mileage per person, round trip:
Ann: 51
Betty: 61
Expressed in some sort of crappy algebra (at which I suck):
y = total trip, one way
40 + x = y
x + 50 = y
I don't know how to find the solution formula other than the logic I presented above, but I'm sure there's an equation that properly finishes it.
y is how far Ann drove
x+40 = y+50, since the distance there and back is the same.
x = y+10, subtract forty from both sides.
How far Betty drove is always ten miles more than how far Ann drove.
Then, Ann dove <strong>Z</strong>, while Beth <strong>50</strong> miles and the total was same <strong>Y</strong>. So, <strong>Z + 50 = Y</strong>.<br>
So we know Ann drove: <strong>40 + Z</strong>, Beth drove <strong>X + 50</strong>.<br>
<br>
From the first 2 equations we have: <strong>40 + X = Z + 50</strong>. This way, <strong>X = Z + 10</strong>.<br>
As result we have:<br>
Ann: <strong>40 + Z</strong><br>
Beth <strong>X + 50 = Z + 10 + 50 = 60 + Z</strong><br>
<br>
This way we know Beth drove more by <strong>20</strong> miles.<br>
First way Ann drove 40 miles, Beth X miles and te total was Y. So, 40 + X = Y.
Then, Ann dove Z, while Beth 50 miles and the total was same Y. So, Z + 50 = Y.
So we know Ann drove: 40 + Z, Beth drove X + 50.
From the first 2 equations we have: 40 + X = Z + 50. This way, X = Z + 10.
As result we have:
Ann: 40 + Z
Beth X + 50 = Z + 10 + 50 = 60 + Z
This way we know Beth drove more by 20 miles.
Mom: Who Drove (assumes the entire trip).
Ann drove 40. at which point betty drove the remainder of the trip.
so she drove 50.
40 + 50 = 90
45 miles each way.
Sorry to be so short with my answer but I did not want to get beat to the punch. :)
Duh.
In which case, I still think my claim that the distance from point a to point b is 61 miles... it could be more, but that would have to be the minimum, right?
40 + Betty = Anne + 50
Betty = Anne + 10
Done.
I still state Betty drove 20 miles more than Ann. Am i the only one here thinking so? hehe
here is an example. The distance between points A and B is 100 miles. Then one way Ann made 40 miles, Betty - 60 miles. When coming back, because the 100 miles distance, they both drove by 50 miles. This way, we have Ann making 40+50=90 miles. Betty made 60+50=110 miles.
110 miles - 90 miles = 20 miles.
Betty drove 20 miles more than Ann.
Questions? :D
Therefore, Betty drove 20 miles more than Ann.
All it takes is a couple proof narratives:
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If total distance round trip = 61, then:
A to B, Anne drives 40, Betty drives 21.
B to A, Anne drives 11, and Betty drives 50.
Total distance driven by Anne = 40 + 11 = 51
Total distance driven by Betty = 21 + 51 = 71
A difference of 20 miles.
===================
If total distance round trip = 584, then:
A to B, Anne drives 40, Betty drives 544.
B to A, Anne drives 534 and Betty drives 50.
Total distance driven by Anne = 40 + 534 = 574
Total distance driven by Betty = 544 + 50 = 594
Again, a difference of 20 miles
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You can plug in any number and the answer is still 20 miles difference. Betty's anal-retentive mom is correct -- Betty drove more, by a mere 20 miles, no matter if they went to the next city or across the country.
wait, that doesn't make me smart, that makes me a smartypants. Never mind.