al, pabo, and marsha

[ksc311]

al, pablo, and marsha shared the driving on a 1,500-mile trip. Which of the three drove the greatest distance on the trip?

1) al drove 1 hour longer than pablo but at an average rate of 5 mile per hour slower than pablo

2) marsha drove 9 hours and averaged 50 miles per hour

[RonPurewal]

attack the easy statement first. that's the second statement, which given information about only one of the players.

(2) alone: INSUFFICIENT
we know that marsha drove 450 miles, leaving 1050 miles of driving for the other two. however, we know nothing about the way those 1050 miles are split up. (we do at least know that marsha didn't drive the greatest distance, because, if two people drive a total of 1050 miles, then at least one of them drives 525 miles or more)

(1) alone: INSUFFICIENT
nowhere near enough information here, because we aren't told anything about (a) the speeds of al and pablo, and (b) anything involving marsha.

together: WARNING - this is probably nowhere close to the best way to solve this problem; it's just the first thing that jumped out of my mind
we know al & pablo together drove 1050 miles.
let t be al's time, and r be al's speed. then pablo's time is t - 1, and pablo's speed is r + 5. we have
rt + (t - 1)(r + 5) = 1050
2rt - r + 5t - 5 = 1050
2rt - r + 5t = 1055
r = (1055 - 5t) / (2t - 1)

we're trying to compare rt versus (r + 5)(t - 1)
--> try to pick one really short time (so that al's additional hour makes a huge difference)
let t = 3 hours --> r = 1040/5 = 208 mi/hr
so al drives 3 hrs @ 208 mph = 624 mi
pablo drives 2 hrs @ 213 mph = 426 mi
al wins

--> try to pick one really long time (so that pablo's bigger speed matters a lot)
let t = 100 hours --> r = 555/199 which is approximately 2.8 mph
al drives 100 hrs @ approx. 2.8 mph = about 280 mi
pablo drives 99 hrs @ approx. 7.8 mph = about 780 mi
pablo wins

INSUFFICIENT

answer = e

like i said, there is almost certanly a better way to solve this problem. perhaps there's a way to grind it down to one variable with both choices together? sigh ...

[vijaykumar.kondepudi]

Hi Ron,
For the case when we consider both the statements together, we see that there are 2 unknowns and a single equation.

we know al & pablo together drove 1050 miles.
let t be al's time, and r be al's speed. then pablo's time is t - 1, and pablo's speed is r + 5. we have
rt + (t - 1)(r + 5) = 1050

Shouldn't that be enough to conclude that "Both statements taken together are insufficient"?

Thanks.
Do we need to do the calculations that you showed?

[mschwrtz]

vijaykumar.kondepudi, that certainly means that we can't determine unique constant values for r and t, unless by some method we can eliminate one of the variables.

But the question doesn't ask us what information is sufficient to solve for r or t, it asks us, essentially, what information is sufficient to answer the question, "Is rt > (t - 1)(r + 5)?" or "Is rt > 525?"

Such questions can sometimes be answered even without solutions for the variables involved.

[i.ahmed111]

ron if instead we assign the variable "r" and "t" to denote Pablo's rate and time respectively and then correspondingly assign variables "r-5" and "t+1" to denote Al's rate and time, does the math (steps you go through to set their distance totals equal to 1050 and then isolation of r) work?

I flipped the assignment of variables, and then came up with more messy algebra!

If it doesn't work, how do you know which person gets the easier variable allocation?

thanks!

[tim]

that will work. sometimes the choice of which value to assign the variable is just a matter of having a feel for which one will work better - or just luck! :) with practice, you should get better at this process, but in most cases the assignment of variable won't affect the ultimate outcome of the problem (i.e. whether it can be solved and what the final answer is)..

[jp.jprasanna]

let t be al's time, and r be al's speed. then pablo's time is t - 1, and pablo's speed is r + 5. we have
rt + (t - 1)(r + 5) = 1050
2rt - r + 5t - 5 = 1050
2rt - r + 5t = 1055
r = (1055 - 5t) / (2t - 1)

let t = 3 hours --> r = 1040/5 = 208 mi/hr

Ron just one question here .... I might sound really stupid please bear with me...

Statement 2
Al drove 1 hour longer than Pablo but an average rate of 5 miles per hour slower than Pablo.

let t be al's time, and r be al's speed.
I translated the above equation as Al drove at his own speed took 1 hr longer but Pablo drove 5 mph faster than Al and took t hrs. - Is this paraphrase correct?

r * (t + 1) = d1
r+5 * (t) = d2
Adding above 2 equations...
So rt + r + rt + t5 = 1050
2rt + r + t5 = 1050
r(2t + 1) + t5 = 1050
r = 1050 -t5 / (2t + 1)

if I take t=3 i get some other value for r... So pls advise what's wrong with below eqs

r * (t + 1) = d1
r+5 * (t) = d2

Also when i saw the below equation i stopped working as we have we have 2 variables and 1 equation... Am I safe or should I go on to test nos too as you did in the real exam?

r(2t + 1) + t5 = 1050

Cheers
jp

[RonPurewal]

Also when i saw the below equation i stopped working as we have we have 2 variables and 1 equation... Am I safe or should I go on to test nos too as you did in the real exam?

the answer to "Should i test numbers anyway?" depends on your own degree of mastery/certainty.

if you have enough algebraic intuition to be absolutely 100.00000 percent sure that there are multiple solutions, then you don't have to test numbers. however, if you are anywhere from 0 to 99.99999 percent sure, you should go ahead and test numbers -- the test writers are very sneaky people, so making generalizations is risky.

--

besides, there are certain cases in which you can have 1 equation with 2 variables but still have a unique solution.
you will mostly see this happen in two kinds of cases:

1/ cases in which the signs of numbers restrict the possibilities. e.g., the equation x^2 + y^2 = 0 has only the solution (0, 0).

2/ cases in which your answers are restricted to positive integers (usually in word problems).
in these cases, you MUST test numbers -- there is absolutely no way to tell whether there will be a plurality of solutions just by looking.
for instance, if the variables x and y must be whole numbers, then 5x + 7y = 47 has two solutions, but 5x + 7y = 48 has only one solution. the only way to figure this out is to test numbers (try it yourself).