At a certain instant in time, the number of cars, N , traveling on a portion of a certain highway can be estimated by the formula N=( 20*L*d)/(600+s^2)
where L is the number of lanes in the same direction, d is the length of the portion of the highway, in feet, and s is the average speed of the cars, in miles per hour. Based on the formula, what is the estimated number of cars traveling on a 21 mile portion of the highway if the highway has 2 lanes in the same direction and the average speed of the cars is 40 miles per hour? (5,280 feet = 1 mile)
(A) 155
(B) 96
(C) 80
(D) 48
(E) 24
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- RonPurewal
- Students
- Posts: 19744
- Joined: Tue Aug 14, 2007 8:23 am
well, ok, all you have to do is plug into the formula. but the key here is the word "estimated": you don't want to work with EXACT numbers such as 5280. too much pain there, unless you're ridiculously fast at arithmetic (like "rain man" fast).
the formula says: N = (20 x 2 x 21 x 5280) / (600 + 1600) -- note that 'd' is 21 x 5280, not just 21, because it has to be converted into feet.
estimate as (20 x 2 x 20 x 5000) / (2000) -- note that we're rounding down BOTH the numerator and the denominator, so that the roundings will offset each other a bit. you don't want to, say, round down the denominator and round up the numerator, because those changes would amplify each other. if we round things, we want to make changes that will offset each other ('cancel each other out').
you can take out the '000's from both sides, leaving 20x2x20x5 / 2, or 20x20x5, or 2000.
my calculator says the exact value is 2016, so this is pretty good.
but, um ... the answer choices?
are you sure the problem statement didn't say PER MILE of the stretch of highway? that would make a lot more sense, as that would be approx. 2000/21 ~ 2000/20 ~ 100 cars per mile, so, (b).
in fact, i'm pretty sure the problem did say PER MILE, because my calculator says that the exact value of 2016/21 is EXACTLY 96 cars per mile.
not to mention that this makes a lot more sense - all of the answer choices are preposterously low for the number of cars traveling on twenty-one miles of double-lane highway, but they're perfectly reasonable per mile of highway.
the formula says: N = (20 x 2 x 21 x 5280) / (600 + 1600) -- note that 'd' is 21 x 5280, not just 21, because it has to be converted into feet.
estimate as (20 x 2 x 20 x 5000) / (2000) -- note that we're rounding down BOTH the numerator and the denominator, so that the roundings will offset each other a bit. you don't want to, say, round down the denominator and round up the numerator, because those changes would amplify each other. if we round things, we want to make changes that will offset each other ('cancel each other out').
you can take out the '000's from both sides, leaving 20x2x20x5 / 2, or 20x20x5, or 2000.
my calculator says the exact value is 2016, so this is pretty good.
but, um ... the answer choices?
are you sure the problem statement didn't say PER MILE of the stretch of highway? that would make a lot more sense, as that would be approx. 2000/21 ~ 2000/20 ~ 100 cars per mile, so, (b).
in fact, i'm pretty sure the problem did say PER MILE, because my calculator says that the exact value of 2016/21 is EXACTLY 96 cars per mile.
not to mention that this makes a lot more sense - all of the answer choices are preposterously low for the number of cars traveling on twenty-one miles of double-lane highway, but they're perfectly reasonable per mile of highway.
- Guest
RPurewal Wrote:well, ok, all you have to do is plug into the formula. but the key here is the word "estimated": you don't want to work with EXACT numbers such as 5280. too much pain there, unless you're ridiculously fast at arithmetic (like "rain man" fast).
the formula says: N = (20 x 2 x 21 x 5280) / (600 + 1600) -- note that 'd' is 21 x 5280, not just 21, because it has to be converted into feet.
estimate as (20 x 2 x 20 x 5000) / (2000) -- note that we're rounding down BOTH the numerator and the denominator, so that the roundings will offset each other a bit. you don't want to, say, round down the denominator and round up the numerator, because those changes would amplify each other. if we round things, we want to make changes that will offset each other ('cancel each other out').
you can take out the '000's from both sides, leaving 20x2x20x5 / 2, or 20x20x5, or 2000.
my calculator says the exact value is 2016, so this is pretty good.
but, um ... the answer choices?
are you sure the problem statement didn't say PER MILE of the stretch of highway? that would make a lot more sense, as that would be approx. 2000/21 ~ 2000/20 ~ 100 cars per mile, so, (b).
in fact, i'm pretty sure the problem did say PER MILE, because my calculator says that the exact value of 2016/21 is EXACTLY 96 cars per mile.
not to mention that this makes a lot more sense - all of the answer choices are preposterously low for the number of cars traveling on twenty-one miles of double-lane highway, but they're perfectly reasonable per mile of highway.
Ron,
Can you plz. help me understand this.
First of all I am not clear why we are converting d from miles to feet, when s is also in miles per hour.
Also I am not understanding this Formula.
How can you get Number of cars when units donot cancel each other.
Leaving apart the constants, we are left with d which is in miles and s which is in miles per hour.
How are we getting number of cars from this.
Plz. help me understand where I am going wrong.
- klm
Re: At a certain instant in time, the number of cars
guest2 Wrote:At a certain instant in time, the number of cars, N , traveling on a portion of a certain highway can be estimated by the formula N=( 20*L*d)/(600+s^2)
where L is the number of lanes in the same direction, d is the length of the portion of the highway, in feet, and s is the average speed of the cars, in miles per hour. Based on the formula, what is the estimated number of cars traveling on a 21 mile portion of the highway if the highway has 2 lanes in the same direction and the average speed of the cars is 40 miles per hour? (5,280 feet = 1 mile)
(A) 155
(B) 96
(C) 80
(D) 48
(E) 24
http://gmatclub.com/forum/7-p527110?t=71520#p527110
- RonPurewal
- Students
- Posts: 19744
- Joined: Tue Aug 14, 2007 8:23 am
hi.
you're converting d into feet because the problem statement requires that d be expressed in feet. go back and look at it again.
admittedly, this is an unusual problem, because the units don't combine in any intuitively accessible way. however, the paramount theme here is that you must FOLLOW THE DIRECTIONS. if they tell you that d is in feet, then d is in feet. period. end of story.
well, again, just follow the directions. shoot first, ask questions later...
the way to understand formulas like this is as follows: the constants in these formulas contain hidden units that do cancel out the remaining units. they just aren't expressed in the formula, because they would add needless bulk without aiding in the computations in any way.
for instance, the "600" must be in mi^2/hr^2, because it must have the same units as does s^2 (otherwise the addition in the denominator would be impossible). but it's not as though we're going to go around writing that as "600 mi^2/hr^2". no way jose.
same goes for the "20", which can be understood to have whatever units are necessary to produce the required cancellation of units.
hth.
Anonymous Wrote:First of all I am not clear why we are converting d from miles to feet, when s is also in miles per hour.
you're converting d into feet because the problem statement requires that d be expressed in feet. go back and look at it again.
admittedly, this is an unusual problem, because the units don't combine in any intuitively accessible way. however, the paramount theme here is that you must FOLLOW THE DIRECTIONS. if they tell you that d is in feet, then d is in feet. period. end of story.
How can you get Number of cars when units donot cancel each other.
well, again, just follow the directions. shoot first, ask questions later...
the way to understand formulas like this is as follows: the constants in these formulas contain hidden units that do cancel out the remaining units. they just aren't expressed in the formula, because they would add needless bulk without aiding in the computations in any way.
for instance, the "600" must be in mi^2/hr^2, because it must have the same units as does s^2 (otherwise the addition in the denominator would be impossible). but it's not as though we're going to go around writing that as "600 mi^2/hr^2". no way jose.
same goes for the "20", which can be understood to have whatever units are necessary to produce the required cancellation of units.
hth.
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