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Intern  Joined: 24 Feb 2011
Posts: 27
A cyclist travels the length of a bike path that is 225  [#permalink]

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Difficulty:   95% (hard)

Question Stats: 28% (02:17) correct 72% (02:32) wrong based on 427 sessions

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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 hrs, rounded to the nearest hour, then his average speed must be between:

A. 38 and 50 mph
B. 40 and 50 mph
C. 40 and 51 mph
D. 41 and 50 mph
E. 41 and 51 mph
Math Expert V
Joined: 02 Sep 2009
Posts: 58991

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12
6
ENAFEX wrote:
Isn't the answer supposed to be D?

224.5/5.4 = 41.57
AND
225.5/4.5 = 50.1

41 and 50 mph

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 hrs, rounded to the nearest hour, then his average speed must be between:
A. 38 and 50 mph
B. 40 and 50 mph
C. 40 and 51 mph
D. 41 and 50 mph
E. 41 and 51 mph

Length of a path is 225 miles long, rounded to the nearest mile --> $$224.5\leq{distance}<225.5$$;
The trip took him 5 hrs, rounded to the nearest hour --> $$4.5\leq{time}<5.5$$;

Lowest average rate is $$\frac{224.5}{5.5}\approx{40.8}$$ (take the lowest value of nominator and highest value of denominator);
Highest average rate is $$\frac{225.5}{4.5}\approx{50.1}$$ (take the highest value of nominator and lowest value of denominator);

$$40.8<rate<50.1$$.

Now, the question is: "the average speed must be between..." hence the range from correct answer choice MUST cover all possible values of rate, so must cover all the range: $$40.8<rate<50.1$$. Only C does that: $$(40)<40.8<rate<50.1<(51)$$. D can not be the answer as if $$rate=40.9$$ or if $$rate=50.01$$ then these possible values of the average rate are not covered by the range from this answer choice, which is (41-50).

Alternative approach.
It's based on observing the answer choices. On the PS section always look at the answer choices before you start to solve a problem. They might often give you a clue on how to approach the question.
A. 38 and 50 mph
B. 40 and 50 mph
C. 40 and 51 mph
D. 41 and 50 mph
E. 41 and 51 mph

Notice that since the range from A covers entire range from B and D, then B and D are out (if B or D is correct so is A and we cannot have two correct answer, leave the bigger range). Similarly since the range from C covers entire range from E, then E is out too (if E is correct so is C and we cannot have two correct answer, leave the bigger range).

Thus we are left only with two answer choices A (38, 50) and C (40, 51). From here it's much easier to get the correct answer.

Hope it's clear.
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1
A bit lenghty method though as below.

225 -> 224.5 to 225.4
5 -> 4.5 to 5.4

224.5/5.4 apprx 40.09
225.4/4.5 apprx 50.1

Hence C
Intern  Joined: 24 Feb 2011
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1
IF
225.4/4.5 apprx 50.1
why would you round it up to 51? Isn't rounding from 50.5 and up?
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[Hi Surabhhi

Does 50.1 equal to 51? I disagree with ur solution. R u from Univ of Central Missouri? ]A bit lenghty method though as below.

225 -> 224.5 to 225.4
5 -> 4.5 to 5.4

224.5/5.4 apprx 40.09
225.4/4.5 apprx 50.1

Hence C[/quote]
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Intern  Joined: 14 Mar 2010
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224.5/5.4 = 41.57 . how did u approximate it to 40.09?
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Isn't the answer supposed to be D?

224.5/5.4 = 41.57
AND
225.5/4.5 = 50.1

41 and 50 mph
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Re: A cyclist travels the length of a bike path that is 225  [#permalink]

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Bunuel,

Thanks for the explanations and the alternative approach. Great as always. I am having a problem in selecting the denominator range. In some problems, I noticed that the lowest denominator value is usually taken as <0.5, i.e. 0.4, e.g. in the above problem I thought it should be 4.4. How do you choose between .4 and .5?

Hope my question makes sense. :D
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Re: A cyclist travels the length of a bike path that is 225  [#permalink]

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ENAFEX wrote:
Bunuel,

Thanks for the explanations and the alternative approach. Great as always. I am having a problem in selecting the denominator range. In some problems, I noticed that the lowest denominator value is usually taken as <0.5, i.e. 0.4, e.g. in the above problem I thought it should be 4.4. How do you choose between .4 and .5?

Hope my question makes sense. :D

It's because of the rounding rules: math-number-theory-88376.html (check chapter for Rounding). 4.5 rounded equals to 5, while 4.4 rounded equals to 4.

So, "the trip took him 5 hrs, rounded to the nearest hour" means $$4.5\leq{time}<5.5$$.

Hope it's clear.
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Re: A cyclist travels the length of a bike path that is 225  [#permalink]

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Bunuel, I understand the rounding concept but sometimes it is just confusing to apply it in problems.

e.g.
On a recent trip Cindy drove her car 290 miles, rounded to the nearest 10 miles and used 12 gallons of gasoline rounded to the nearest gallon.
The actual number of miles per gallon that Cindy's car got on this trip must have been between:

A) 290/12.5 and 290/11.4
B) 294/12 and 284/11.4
C) 284/12 and 295/12
D) 284/12.5 and 295/11.4
E) 295/12.5 and 284/11.4

In this is the correct choice is D
12 rounded to the nearest gasoline should be 11.5<=Value<12.5. But why is it 11.4 in the correct choice?
Also why is it 284? Is it not supposed to be 285?
I thought the answer will be 285/12.5 to 295/11.5?
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Re: A cyclist travels the length of a bike path that is 225  [#permalink]

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Have to start working on the basics, the concepts are good just taking me too long to divide the numbers and get an accurate answer.
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Re: A cyclist travels the length of a bike path that is 225  [#permalink]

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ENAFEX wrote:
Bunuel, I understand the rounding concept but sometimes it is just confusing to apply it in problems.

e.g.
On a recent trip Cindy drove her car 290 miles, rounded to the nearest 10 miles and used 12 gallons of gasoline rounded to the nearest gallon.
The actual number of miles per gallon that Cindy's car got on this trip must have been between:

A) 290/12.5 and 290/11.4
B) 294/12 and 284/11.4
C) 284/12 and 295/12
D) 284/12.5 and 295/11.4
E) 295/12.5 and 284/11.4

In this is the correct choice is D
12 rounded to the nearest gasoline should be 11.5<=Value<12.5. But why is it 11.4 in the correct choice?
Also why is it 284? Is it not supposed to be 285?
I thought the answer will be 285/12.5 to 295/11.5?

On a recent trip, Cindy drove her car 290 miles, rounded to the nearest 10 miles, and used 12 gallons of gasoline, rounded to the nearest gallon. The actual number of miles per gallon that Cindy's car got on this trip must have been between
A. 290/12.5 and 290/11.5
B. 295/12 and 285/11.5
C. 285/12 and 295/12
D. 285/12.5 and 295/11.5
E. 295/12.5 and 285/11.5

Cindy drove her car 290 miles, rounded to the nearest 10 miles --> $$285\leq{m}<295$$;
Used 12 gallons of gasoline, rounded to the nearest gallon --> $$11.5\leq{g}<12.5$$;

Minimum Miles per gallon, m/g --> $$\frac{285}{12.5}<\frac{m}{g}<\frac{295}{11.5}$$ (to get lower limit take min possible for nominator and max possible for denominator, and for upper limit take max possible for nominator and min possible for denominator).

See more here: on-a-recent-trip-cindy-drove-her-car-290-miles-rounded-to-99378.html

Hope it helps.
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Re: A cyclist travels the length of a bike path that is 225  [#permalink]

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Bumping for review and further discussion*. Get a kudos point for an alternative solution!

*New project from GMAT Club!!! Check HERE

All DS Min/Max Problems to practice: search.php?search_id=tag&tag_id=42
All PS Min/Max Problems to practice: search.php?search_id=tag&tag_id=63

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Bunuel wrote:
ENAFEX wrote:
Isn't the answer supposed to be D?

224.5/5.4 = 41.57
AND
225.5/4.5 = 50.1

41 and 50 mph

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 hrs, rounded to the nearest hour, then his average speed must be between:
A. 38 and 50 mph
B. 40 and 50 mph
C. 40 and 51 mph
D. 41 and 50 mph
E. 41 and 51 mph

Length of a path is 225 miles long, rounded to the nearest mile --> $$224.5\leq{distance}<225.5$$;
The trip took him 5 hrs, rounded to the nearest hour --> $$4.5\leq{time}<5.5$$;

Lowest average rate is $$\frac{224.5}{5.5}\approx{40.8}$$ (take the lowest value of nominator and highest value of denominator);
Highest average rate is $$\frac{225.5}{4.5}\approx{50.1}$$ (take the highest value of nominator and lowest value of denominator);

$$40.8<rate<50.1$$.

Now, the question is: "the average speed must be between..." hence the range from correct answer choice MUST cover all possible values of rate, so must cover all the range: $$40.8<rate<50.1$$. Only C does that: $$(40)<40.8<rate<50.1<(51)$$. D can not be the answer as if $$rate=40.9$$ or if $$rate=50.01$$ then these possible values of the average rate are not covered by the range from this answer choice, which is (41-50).

Hi Bunuel,

Refering to the rounding chapter in Number Theory examples

Rounding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep.

Example:
5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.
5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.
5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.

Shouldn't the range for distance be 224.5 < D< 225.4 ------> If we take the higher value of distance i.e 225.4 and drop the last digit (4) then it gets rounded off to 225 but if we keep 225.5 then after dropping (5) it should be rounded off to 226.

Similarly for Time it should 4.5 <T< 5.4

Calculating Lowest Avg rate : 224.5/5.4 = 41.57
Highest : 225.4/4.5 = 50.08

Thanks
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mridulparashar1 wrote:
Bunuel wrote:
ENAFEX wrote:
Isn't the answer supposed to be D?

224.5/5.4 = 41.57
AND
225.5/4.5 = 50.1

41 and 50 mph

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 hrs, rounded to the nearest hour, then his average speed must be between:
A. 38 and 50 mph
B. 40 and 50 mph
C. 40 and 51 mph
D. 41 and 50 mph
E. 41 and 51 mph

Length of a path is 225 miles long, rounded to the nearest mile --> $$224.5\leq{distance}<225.5$$;
The trip took him 5 hrs, rounded to the nearest hour --> $$4.5\leq{time}<5.5$$;

Lowest average rate is $$\frac{224.5}{5.5}\approx{40.8}$$ (take the lowest value of nominator and highest value of denominator);
Highest average rate is $$\frac{225.5}{4.5}\approx{50.1}$$ (take the highest value of nominator and lowest value of denominator);

$$40.8<rate<50.1$$.

Now, the question is: "the average speed must be between..." hence the range from correct answer choice MUST cover all possible values of rate, so must cover all the range: $$40.8<rate<50.1$$. Only C does that: $$(40)<40.8<rate<50.1<(51)$$. D can not be the answer as if $$rate=40.9$$ or if $$rate=50.01$$ then these possible values of the average rate are not covered by the range from this answer choice, which is (41-50).

Hi Bunuel,

Refering to the rounding chapter in Number Theory examples

Rounding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep.

Example:
5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.
5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.
5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.

Shouldn't the range for distance be 224.5 < D< 225.4 ------> If we take the higher value of distance i.e 225.4 and drop the last digit (4) then it gets rounded off to 225 but if we keep 225.5 then after dropping (5) it should be rounded off to 226.

Similarly for Time it should 4.5 <T< 5.4

Calculating Lowest Avg rate : 224.5/5.4 = 41.57
Highest : 225.4/4.5 = 50.08

Thanks

Should be as written. What about 225.499999? So, take 225.5 but exclude the endpoints, by writing < and > instead of <= and >=.

Hope it's clear.
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Bunuel wrote:
mridulparashar1 wrote:
Bunuel 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 hrs, rounded to the nearest hour, then his average speed must be between:
A. 38 and 50 mph
B. 40 and 50 mph
C. 40 and 51 mph
D. 41 and 50 mph
E. 41 and 51 mph

Length of a path is 225 miles long, rounded to the nearest mile --> $$224.5\leq{distance}<225.5$$;
The trip took him 5 hrs, rounded to the nearest hour --> $$4.5\leq{time}<5.5$$;

Lowest average rate is $$\frac{224.5}{5.5}\approx{40.8}$$ (take the lowest value of nominator and highest value of denominator);
Highest average rate is $$\frac{225.5}{4.5}\approx{50.1}$$ (take the highest value of nominator and lowest value of denominator);

$$40.8<rate<50.1$$.

Now, the question is: "the average speed must be between..." hence the range from correct answer choice MUST cover all possible values of rate, so must cover all the range: $$40.8<rate<50.1$$. Only C does that: $$(40)<40.8<rate<50.1<(51)$$. D can not be the answer as if $$rate=40.9$$ or if $$rate=50.01$$ then these possible values of the average rate are not covered by the range from this answer choice, which is (41-50).

Hi Bunuel,

Refering to the rounding chapter in Number Theory examples

Rounding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep.

Example:
5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.
5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.
5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.

Shouldn't the range for distance be 224.5 < D< 225.4 ------> If we take the higher value of distance i.e 225.4 and drop the last digit (4) then it gets rounded off to 225 but if we keep 225.5 then after dropping (5) it should be rounded off to 226.

Similarly for Time it should 4.5 <T< 5.4

Calculating Lowest Avg rate : 224.5/5.4 = 41.57
Highest : 225.4/4.5 = 50.08

Thanks

Should be as written. What about 225.499999? So, take 225.5 but exclude the endpoints, by writing < and > instead of <= and >=.

Hope it's clear.

Hi Bunuel,

Can you elaborate a bit more. I am not able to get the message. Thanks
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mridulparashar1 wrote:
Bunuel wrote:
mridulparashar1 wrote:
Hi Bunuel,

Refering to the rounding chapter in Number Theory examples

Rounding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep.

Example:
5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.
5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.
5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.

Shouldn't the range for distance be 224.5 < D< 225.4 ------> If we take the higher value of distance i.e 225.4 and drop the last digit (4) then it gets rounded off to 225 but if we keep 225.5 then after dropping (5) it should be rounded off to 226.

Similarly for Time it should 4.5 <T< 5.4

Calculating Lowest Avg rate : 224.5/5.4 = 41.57
Highest : 225.4/4.5 = 50.08

Thanks

Should be as written. What about 225.499999? So, take 225.5 but exclude the endpoints, by writing < and > instead of <= and >=.

Hope it's clear.

Hi Bunuel,

Can you elaborate a bit more. I am not able to get the message. Thanks

When you take maximum distance as 225.4 you exclude distance from 225.4 to 225.5. For example d could be 225.47, 225.43, 225.4111111. So, take 225.5 as max distance but write rate<225.5/4.5 instead of rate<=225.5/4.5.
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WoundedTiger wrote:
Bunuel wrote:
ENAFEX wrote:
Isn't the answer supposed to be D?

224.5/5.4 = 41.57
AND
225.5/4.5 = 50.1

41 and 50 mph

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 hrs, rounded to the nearest hour, then his average speed must be between:
A. 38 and 50 mph
B. 40 and 50 mph
C. 40 and 51 mph
D. 41 and 50 mph
E. 41 and 51 mph

Length of a path is 225 miles long, rounded to the nearest mile --> $$224.5\leq{distance}<225.5$$;
The trip took him 5 hrs, rounded to the nearest hour --> $$4.5\leq{time}<5.5$$;

Lowest average rate is $$\frac{224.5}{5.5}\approx{40.8}$$ (take the lowest value of nominator and highest value of denominator);
Highest average rate is $$\frac{225.5}{4.5}\approx{50.1}$$ (take the highest value of nominator and lowest value of denominator);

$$40.8<rate<50.1$$.

Now, the question is: "the average speed must be between..." hence the range from correct answer choice MUST cover all possible values of rate, so must cover all the range: $$40.8<rate<50.1$$. Only C does that: $$(40)<40.8<rate<50.1<(51)$$. D can not be the answer as if $$rate=40.9$$ or if $$rate=50.01$$ then these possible values of the average rate are not covered by the range from this answer choice, which is (41-50).

Hi Bunuel,

Refering to the rounding chapter in Number Theory examples

Rounding is simplifying a number to a certain place value. To round the decimal drop the extra decimal places, and if the first dropped digit is 5 or greater, round up the last digit that you keep. If the first dropped digit is 4 or smaller, round down (keep the same) the last digit that you keep.

Example:
5.3485 rounded to the nearest tenth = 5.3, since the dropped 4 is less than 5.
5.3485 rounded to the nearest hundredth = 5.35, since the dropped 8 is greater than 5.
5.3485 rounded to the nearest thousandth = 5.349, since the dropped 5 is equal to 5.

Shouldn't the range for distance be 224.5 < D< 225.4 ------> If we take the higher value of distance i.e 225.4 and drop the last digit (4) then it gets rounded off to 225 but if we keep 225.5 then after dropping (5) it should be rounded off to 226.

Similarly for Time it should 4.5 <T< 5.4

Calculating Lowest Avg rate : 224.5/5.4 = 41.57
Highest : 225.4/4.5 = 50.08

Thanks

Also, your method would actually change the answer to E (between 41 and 51 miles) since 41.57 would not need the 40 min set by answer C
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Should be as written. What about 225.499999? So, take 225.5 but exclude the endpoints, by writing < and > instead of <= and >=.

Hi Bunuel,

Found your explanations so helpful, thanks! I just came across this problem and given your answer above, I understand why we would use 225.5 and 5.5 as the maximum distance and time for the calculation, but still, the inequality range for distance and time excludes these values as the maximum endpoints, so isn't our answer going to be inaccurate? After all, 225.5 rounds up to 226 mi and 5.5 to 6 hr...

I calculated with max's 225.4 and 5.4 but yes, that would exclude the .4-.499's... so does this mean that for similar questions we have to use the max endpoint even if it is NOT inclusive in the range?

This was a problem in a MGMAT book, but are there similar examples of actual GMAT questions where we would have to use the maximum endpoint like we had to for this question?

Thanks much!

Fabs
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fcy wrote:
This was a problem in a MGMAT book, but are there similar examples of actual GMAT questions where we would have to use the maximum endpoint like we had to for this question?

Thanks much!

Fabs

Similar questions:
on-a-recent-trip-cindy-drove-her-car-290-miles-rounded-to-99378.html
andrew-drove-his-truck-100-miles-rounded-to-the-nearest-145227.html
did-it-take-pei-more-than-2-hours-to-walk-a-distance-of-63638.html
if-an-automobile-averaged-22-5-miles-per-gallon-of-gasoline-109083.html

Hope this helps/
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