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by bblast » Mon May 16, 2011 7:47 am
A cyclist travels the length of a bike path that is 225 miles long, rounded to the nearest
mile. If the trip took him 5 hours, rounded to the nearest hour, then his average speed
must be between:
(A) 38 and 50 miles per hour
(B) 40 and 50 miles per hour
(C) 40 and 51 miles per hour
(D) 41 and 50 miles per hour
(E) 41 and 51 miles per hour

oa-C
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by Stuart@KaplanGMAT » Mon May 16, 2011 8:43 am
bblast wrote:A cyclist travels the length of a bike path that is 225 miles long, rounded to the nearest
mile. If the trip took him 5 hours, rounded to the nearest hour, then his average speed
must be between:
(A) 38 and 50 miles per hour
(B) 40 and 50 miles per hour
(C) 40 and 51 miles per hour
(D) 41 and 50 miles per hour
(E) 41 and 51 miles per hour

oa-C
I'll let someone else do the math on this one - I just want to discuss it from the viewpoint of strategic elimination.

The question is:
then his average speed must be between:
If we understand the question, we can actually eliminate 3 of the 5 choices without even looking at the math.

First, let's compare (A) and (B).

Well, if his average speed is between 40 and 50 miles an hour, then it's also between 38 and 50 miles per hour. So, if (B) were correct, (A) would also be correct. Since you can't have two correct choices, eliminate (B).

We can do an identical comparison between (D) and (A). Since the range in (D) is simply a subset of the range in (A), we can also eliminate (D).

Similarly, the range in (E) is a subset of the range in (C). Accordingly, (E) is out.

So, the only two possible answers to the question are (A) and (C). Since both (A) and (C) include the range of "between 40 and 50", we can ignore all values inside that range. If his average speed could be 40 or lower, then (A) is correct; if his average speed could be 50 or higher, then (C) is correct.
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by Whitney Garner » Thu May 19, 2011 11:14 am
bblast wrote:A cyclist travels the length of a bike path that is 225 miles long, rounded to the nearest
mile. If the trip took him 5 hours, rounded to the nearest hour, then his average speed
must be between:
(A) 38 and 50 miles per hour
(B) 40 and 50 miles per hour
(C) 40 and 51 miles per hour
(D) 41 and 50 miles per hour
(E) 41 and 51 miles per hour

oa-C
Love Stuart's strategy shortcut method but I'm here responding to a PM request for the algebra :)

If we know the miles are rounded to the nearest mile, then the cyclist could have traveled the following ranges of miles and hours:

224.5 <= Miles < 225.5

4.5 <= Hours < 5.5

Now we need to do some considering in order to determine the lower and upper bounds for the cyclists miles per hour. To do this it might be easiest to think of it as a fraction:

Miles
-----
Hours

To make a fraction as SMALL as possible, we make the numerator very small and the denominator very large. The smallest possible Miles would be 224.5, while the largest possible Hours would be essentially 5.5. That means the smallest miles per hour would be 224.5/5.5. I don't mix fractions and decimals, so converting to improper fractions we have (449/2)/(11/2) = 449/11 = approx 40.8.

TO make a fraction as LARGE as possible, we make the numerator very large and the denominator very small. The largest possible Miles would be basically 225.5, while the smallest possible Hours would be 4.5. That means the largest miles per hour would be 225.5/4.5. Again convert to improper fractions: (451/2)/(9/2) = 451/9 = approx 50.1.

So, the miles MUST be between 40.8 and 50.1 (be VERY careful rounding here). In order to be between 40.8 and 50.1, we must round OUT - meaning: we round away from the middle of the range (this is basically loosening the restriction on the range or making it a bit bigger to guarantee that all values we have listed will be included).

It must be between 40 and 51. The correct answer is C.
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by kannans3 » Fri May 20, 2011 1:20 am
Whitney Garner wrote:
224.5 <= Miles < 225.5

4.5 <= Hours < 5.5
Should it not be this way in the algebraic method:

224.5/5.4 (because the hours must be less than 5.5) & 225.4/4.5 ( Miles less than 225.5)?

Experts please COMMENT.
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by bblast » Fri May 20, 2011 10:43 am
Whitney Garner wrote:
bblast wrote:
So, the miles MUST be between 40.8 and 50.1 (be VERY careful rounding here). In order to be between 40.8 and 50.1, we must round OUT - meaning: we round away from the middle of the range (this is basically loosening the restriction on the range or making it a bit bigger to guarantee that all values we have listed will be included).

It must be between 40 and 51. The correct answer is C.
Whitney, i choose answer B as that is what the nearest rounding gives us. Whats the concept of rounding Out ? I get the basic of it what you have written.

This question is an alias of question 129 from OG-12 but thankfully GMAC has given fractions in the choices. Such numbers could trouble us in the real test.

Is it plain and simple that I convert
a lower limit say 30.1 to 30
an upper limit say 50.1 to 51
?
Cheers !!

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by gmatdriller » Fri May 20, 2011 1:32 pm
kannans3 wrote:
Whitney Garner wrote:
224.5 <= Miles < 225.5

4.5 <= Hours < 5.5
Should it not be this way in the algebraic method:

224.5/5.4 (because the hours must be less than 5.5) & 225.4/4.5 ( Miles less than 225.5)?

Experts please COMMENT.
I thought about this question as well, but look at these:
225.41/4.5
225.42/4.5
225.43/4.5
225.44/4.5

All the examples above are valid as they fall within the required range of speed.
In sum, we say speed < 225.5/4.5...the maximum average speed cannot be greater.

Hope that helps.
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by Whitney Garner » Mon Jun 06, 2011 9:17 am
kannans3 wrote:
Whitney Garner wrote:
224.5 <= Miles < 225.5

4.5 <= Hours < 5.5
Should it not be this way in the algebraic method:

224.5/5.4 (because the hours must be less than 5.5) & 225.4/4.5 ( Miles less than 225.5)?

Experts please COMMENT.
You are correct that the hours must be less than 5.5, but we don't want to forget that they could be 5.42, or 5.45925 or even 5.4999999999 out to infinity... or essentially 5.5. We know that this is a limit so we know that the resulting fraction 224.5/5.5 cannot be exactly achieved BUT we must assume that we can get infinitely close to it.

Your intuition would be absolutely correct if we were given items that could not take on fractional values. What if we had a question that was asking about cupcakes per student. If the number of students in the class had to be at least 10 but less than 15, and the number of cupcakes had to be more than 30 with a max of 40, then we would use your logic. The ranges would therefore be:

10 <= Students < 15
30 < Cupcakes <=40

But because students and cupcakes cannot have fractional values, the effective ranges would be:

10 <= Students <= 14 (because we cannot have 14.999 students, or any fraction of a student)
31 <= Cupcakes <= 40 (because we cannot bake 30.111 cupcakes, or any fraction of a cupcake)

Because both miles and hours can be expressed as infinitely smaller and smaller fractions, we need to use the endpoints (even if we cannot actually hit the endpoint itself).

:)
Whit
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by Whitney Garner » Mon Jun 06, 2011 9:35 am
bblast wrote:
Whitney Garner wrote:
bblast wrote:
So, the miles MUST be between 40.8 and 50.1 (be VERY careful rounding here). In order to be between 40.8 and 50.1, we must round OUT - meaning: we round away from the middle of the range (this is basically loosening the restriction on the range or making it a bit bigger to guarantee that all values we have listed will be included).

It must be between 40 and 51. The correct answer is C.
Whitney, i choose answer B as that is what the nearest rounding gives us. Whats the concept of rounding Out ? I get the basic of it what you have written.

This question is an alias of question 129 from OG-12 but thankfully GMAC has given fractions in the choices. Such numbers could trouble us in the real test.

Is it plain and simple that I convert
a lower limit say 30.1 to 30
an upper limit say 50.1 to 51
?
Hi bblast!

The concept of rounding "out" is my way of thinking about a given range. Let's look at the range we found in the problem:

40.8 and 50.1

Note that the answers are trying to capture EVERY value in that range. In order to accomplish that, we need to set a new range that either shares IDENTICAL endpoints with this one, OR we need to loosen the restriction (round to the OUTSIDE of the range). Notice what happens with answer choice B:

(B) 40 and 50 miles per hour

The 40 is a rounding OUT of 40.8: if something is greater than 40.8, it MUST BE greater than 40 - so we have achieved the lower bound.

The 50 is a rounding IN of 50.1 (we moved toward the inside of the range - but what did that do?) Well we must think of the fractional values between 50 and 50.1. For example, 50.05 is within the range we solved for (40.8 to 50.1), but it IS NOT inside of the compressed range 40 to 50.

I call this concept "rounding OUT" but it is probably better to think about the intuition behind it. If I am given a specific range of values, then the only way to guarantee that I catch ALL of those values is to give a bigger (or identical) range. Let's try some whole numbers:

If I know that the miles per hour had to be between 24 and 49, I could say any of the following:

- The mph must be between 24 and 49
- the mph must be between 20 and 50
- the mph must be between 0 and 1000
- the mph must be between (any range GREATER than 24-49 that includes 24-49 in its ranks).

The issue is when I try to compress the range or limit it further than what is given:

- mph between 25 and 48 (we have omitted anything between 24-25 and between 48-49, assuming fractional values are allowed)

I want to caution you against simply following some mechanical process without understanding the underlying intuition. The concept we have to attack for this question is two-fold:

(1) How do we conceptually make fractions as large or as small as possible
(2) How do I think about ranges of values, and can I be careful not to OMIT values due to poor rounding

:)
Whit
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by cans » Mon Jun 06, 2011 9:45 am
225 miles. as its rounded, so lies from 224.5 to 225.4
5 hours. 4.5 hours to 5.4 hours
speed=d/t
224.5/5.4=41
225.4/4.5=50.0
thus IMO D
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by bblast » Mon Jun 06, 2011 7:53 pm
OA- is C
Cheers !!

Quant 47-Striving for 50
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My gmat journey :
https://www.beatthegmat.com/710-bblast-s ... 90735.html
My take on the GMAT RC :
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https://www.youtube.com/watch?v=upz46D7 ... TWBZF14TKW_
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by gmatdriller » Sat Jun 11, 2011 6:50 pm
cans wrote:225 miles. as its rounded, so lies from 224.5 to 225.4
5 hours. 4.5 hours to 5.4 hours
speed=d/t
224.5/5.4=41
225.4/4.5=50.0
thus IMO D
Min Speed = Min Dist / Max Time
Min Dist = 224.5
MAX Time can be anything from 5.41, 5.42, to 5.499999 BUT LESS than 5.5
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