Problem Solving

Anonymous wrote: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

I don' understand this problem... how does one solve this.
If I may rephrase the question - What is this problem trying to test.?

Same here, I thought E was the answer but the answer seems to be D.

here the rounding off of numbers is tested...
distance rounded to nearest 10 miles = 290..so actual distance covered may be b/w 285 and 294.
gasoline used rounded to nearest gallon = 12...so actual gas used may be b/w 11.5 and 12.4. now to get range of of miles/gallon we'll use the formula
least value = the least of distance/the max of gas = 285/12.4
highest value = max of distance/least of gas = 294/11.5

Only D satisfies the above formula..........

Convincing solution. :D

anony wrote:

jangojess wrote:here the rounding off of numbers is tested...
distance rounded to nearest 10 miles = 290..so actual distance covered may be b/w 285 and 294.
gasoline used rounded to nearest gallon = 12...so actual gas used may be b/w 11.5 and 12.4. now to get range of of miles/gallon we'll use the formula
least value = the least of distance/the max of gas = 285/12.4
highest value = max of distance/least of gas = 294/11.5

Only D satisfies the above formula..........

I saw the soln in OG it says the range for miles is from 284 to 295
if you round 284 it will be 280
and similarly if you round 295 it will be 300

same for gasoline OG says the range is from 11.4 to 12.5
I don't understand how rounding works now

bump, can anybody answer how this rounding works

The Qs I saw in OG-11 and the Qs is -

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

Now according to me -
Cindy drove her car 290 miles, rounded to the nearest 10 miles - It means Cindy drove between 285 to 294.
used 12 gallons of gasoline, rounded to the nearest gallon. - It means the car used 11.5 to 12.4 gallons of gasoline to drive.
Now to find out miles/gasoline, we have to pick up the minimum value and maximum value to spread the range - Are u getting me? We know a denominator has to be smaller and numerator has to be larger. If we go with E, the range becomes narrower because it does not follow the rule mentioned above. But with D, we can get the desired answer. So D.
Guys am I clear to explain what I have thought of?

the key to this problem is the word "between" at the end of the question.

your rounding is correct, but if you specify 285/12.4 -- 294/11.5 as your solution, this range is not [b]between; instead, this range is [b]inclusive.

I know this is sounding like a post that is better suited for the verbal section, but hey that's gmac for you.

anyway, if you analyze each of the other options, you will see that only option d immediately "envelopes" the correct range cited above. Every other option has either the low end or the top end within the correct range.

in other words, only the range in option d includes all values between 285/12.4 and 294/11.5

Okay..I have seen multiple threads of this question but no clear explanation has yet been provided..so opening the post again..I would like the experts to give a detailed explanation.

Posting the question again(there was a typo previously):

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
OA:[spoiler=]D[/spoiler]

290 miles rounded to the nearest 10 miles: b/w 285 and 294
12 gallons rounded to the nearest gallon:b/w 11.5 and 12.4

Now,
worst possible mileage would happen with fewest miles traveled and the greatest gasoline consumption, and the best possible mileage would be the opposite.
whch should make it 285/12.4 and 294/11.5 which does not match with the OA.

The problem seems to arise from the word "between".

Please clarify!!

uptowngirl92 wrote:Okay..I have seen multiple threads of this question but no clear explanation has yet been provided..so opening the post again..I would like the experts to give a detailed explanation.

Posting the question again(there was a typo previously):

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
OA:[spoiler=]D[/spoiler]

290 miles rounded to the nearest 10 miles: b/w 285 and 294
12 gallons rounded to the nearest gallon:b/w 11.5 and 12.4

Now,
worst possible mileage would happen with fewest miles traveled and the greatest gasoline consumption, and the best possible mileage would be the opposite.
whch should make it 285/12.4 and 294/11.5 which does not match with the OA.

The problem seems to arise from the word "between".

Please clarify!!

You are correct.. here's the inequality we could set up:

285/12.4 <= x <= 294/11.5

However, the question doesn't ask "which of the following is the range of possible mileage for the car" - it asks what the mileage "must have been between".

For "must have been between", a range bigger than the minimum range certainly fits. Let's look at a different question:

If x=6, then x must be between which of the follwing:

A) 2 and 3
B) 4 and 5
C) 5 and 6
D) 1 and 1000000
E) 6 and 7

Now, if x=6, it certainly must be between 1 and 1000000 - even though that range is bigger than absolutely necessary, it meets the requirements of the question (while none of the other choices do so).

For this question, we need to recognize that:

285/12.5 is less than 285/12.4 (remember, when you increase the denominator, you decrease the fraction); and that

295/11.5 is greater than 294/11.5 (since increasing the numerator increases the fraction)

in order to see that the range in D is bigger than our inequality, and is therefore the correct answer to the question.

Awesome Stuart!Thanks a ton that really cleared up all the queries:)

Mr. Stuart Kovinsky to the rescue once again!!!

Sir, i can only dream of having your intelligence one day!!

Also, note that if, when rounded to the nearest multiple of 10, she drove 290 miles, the number of miles m she drove must satisfy the following inequality: 285 <= m < 295, as 294.7 would round to 290

ys2jax wrote:

anony wrote:

jangojess wrote:here the rounding off of numbers is tested...
distance rounded to nearest 10 miles = 290..so actual distance covered may be b/w 285 and 294.
gasoline used rounded to nearest gallon = 12...so actual gas used may be b/w 11.5 and 12.4. now to get range of of miles/gallon we'll use the formula
least value = the least of distance/the max of gas = 285/12.4
highest value = max of distance/least of gas = 294/11.5

Only D satisfies the above formula..........

I saw the soln in OG it says the range for miles is from 284 to 295
if you round 284 it will be 280
and similarly if you round 295 it will be 300

same for gasoline OG says the range is from 11.4 to 12.5
I don't understand how rounding works now

bump, can anybody answer how this rounding works

Thanks.

Stuart Kovinsky wrote:

uptowngirl92 wrote:Okay..I have seen multiple threads of this question but no clear explanation has yet been provided..so opening the post again..I would like the experts to give a detailed explanation.

Posting the question again(there was a typo previously):

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
OA:[spoiler=]D[/spoiler]

290 miles rounded to the nearest 10 miles: b/w 285 and 294
12 gallons rounded to the nearest gallon:b/w 11.5 and 12.4

Now,
worst possible mileage would happen with fewest miles traveled and the greatest gasoline consumption, and the best possible mileage would be the opposite.
whch should make it 285/12.4 and 294/11.5 which does not match with the OA.

The problem seems to arise from the word "between".

Please clarify!!

You are correct.. here's the inequality we could set up:

285/12.4 <= x <= 294/11.5

However, the question doesn't ask "which of the following is the range of possible mileage for the car" - it asks what the mileage "must have been between".

For "must have been between", a range bigger than the minimum range certainly fits. Let's look at a different question:

If x=6, then x must be between which of the follwing:

A) 2 and 3
B) 4 and 5
C) 5 and 6
D) 1 and 1000000
E) 6 and 7

Now, if x=6, it certainly must be between 1 and 1000000 - even though that range is bigger than absolutely necessary, it meets the requirements of the question (while none of the other choices do so).

For this question, we need to recognize that:

285/12.5 is less than 285/12.4 (remember, when you increase the denominator, you decrease the fraction); and that

295/11.5 is greater than 294/11.5 (since increasing the numerator increases the fraction)

in order to see that the range in D is bigger than our inequality, and is therefore the correct answer to the question.

Hello Stuart,

Thanks a lot for the detailed explanation. Choice D is clear now. I was stuck between choices D and E.

Choice D says that the number of miles per gallon must be between the best case and the worst case scenario.

Is this why we can eliminate E? Thanks a lot for your help.

Best Regards,
Sri

After thinking about this question for a while, here goes my approach. Just to solve it faster, read the answers before mine to understand the math!

First, translating GMAC language:
- "Must have been between" - only one range will apply, and for that, I need a number that is at least a bit less than 285/12.4 and another that is a bit more than 294/11.5

Using GMAC standandized shape in your favor:
- The numbers before the "and" in the answers are the lower extreme of the range (trust me, they could make your life even harder not using this obvious order)

Finally, use elimination:
1)Start with the first column (number before the "and" in the answers), they cannot be more than 285/12.4, otherwise this number would be left out of the range (remember, all the possibilities must be included!!!). Since, we only have numerators matching with this number (285/12.4), let's see what we can eliminate:
- To have a smaller fraction with 285 as the numerator, I need a bigger denominator (>12.4)
- Since C has the numerator 12, you can eliminate C
- No other answer has 285 as the numerator

2)Time for second column (number after the "and" in the answers), they cannot be less than 294/11.5, otherwise this number would be left out of the range (remember, all the possibilities must be included!!!). Since, we only have denominators matching this number, let's see what we can eliminate:
- To have a bigger fraction with 11.5 as the denominator, I need a bigger numerator (>294)
- Since A has the numerator 290, eliminate A
- Since B has the numerator 285, eliminate B
- Since E has the numerator 285, eliminate E
- Only D has a numerator bigger than 294 (and C is already out)

There's no way GMAC would require you to look further than this on those fractions. Remember to find a fast way to compare fractions and use elimination.
This is a great question to test rounding, and specially some fraction rules and patterns found both on the OG and MGMAT, like how changing only the numerator or denominator would impact the fraction.

Hope that can help,
niddy

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 to 290/11.5

B) 295/12 to 285/11.5

C) 285/12 to 295/12

D) 285/12.5 to 295/11.5

E) 205/12.5 to 285/11.5

We round UP when we reach the HALFWAY point between two values.
Any value BELOW the halfway point is rounded DOWN.

290 miles, rounded to the nearest 10 miles:
285 ≤ m < 295.
Please note that 295 is the UPPER LIMIT.
Any value below 295 -- even 294.999 -- is rounded DOWN to 290.

12 gallons of gasoline, rounded to the nearest gallon:
11.5 ≤ g < 12.5.
Please note that 12.5 is the UPPER LIMIT.
Any value below 12.5 -- even 12.499 -- is rounded DOWN to 12.

Worst miles per gallon = (lower limit for miles)/(upper limit for gas) = 285/12.5.
Best miles per gallon = (upper limit for miles)/(lower limit for gas) = 295/11.5.
Thus, miles per gallon is between 285/12.5 and 295/11.5.

The correct answer is D.