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A ranch has both horses and ponies. Exactly 5/6 of the [#permalink]

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10 May 2012, 05:10

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A ranch has both horses and ponies. Exactly 5/6 of the ponies have horseshoes, and exactly 2/3 of the ponies with horseshoes are from Iceland. If there are 3 more horses than ponies, what is the minimum possible combined number of horses and ponies on the ranch?

A ranch has both horses and ponies. Exactly 5/6 of the ponies have horseshoes, and exactly 2/3 of the ponies with horseshoes are from Iceland. If there are 3 more horses than ponies, what is the minimum possible combined number of horses and ponies on the ranch?

A. 18 B. 21 C. 38 D. 39 E. 57

Say there are P ponies on the ranch.

Then # of ponies with horseshoes is P*5/6, which means that P must be a multiple of 6;

# of ponies from Iceland is (P*5/6)*2/3=P*5/9, which means that P must be a multiple of 9.

P to be multiple of both 6 and 9 it should be a multiple of 18. The least positive multiple of 18 is 18 itself, so P(min)=18 and H(min)=18+3=21, so the minimum possible combined number of horses and ponies on the ranch is 18+21=39.

Re: A ranch has both horses and ponies. Exactly 5/6 of the [#permalink]

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11 May 2012, 10:57

Bunuel wrote:

piyushksharma wrote:

A ranch has both horses and ponies. Exactly 5/6 of the ponies have horseshoes, and exactly 2/3 of the ponies with horseshoes are from Iceland. If there are 3 more horses than ponies, what is the minimum possible combined number of horses and ponies on the ranch?

A. 18 B. 21 C. 38 D. 39 E. 57

Say there are P ponies on the ranch. Then # of ponies with horseshoes, which are from Iceland is P*5/6*2/3=P*10/18. So, P must be a multiple of 18.

The least positive multiple of 18 is 18 itself, so P(min)=18 and H(min)=18+3=21, so the minimum possible combined number of horses and ponies on the ranch is 18+21=39.

Answer: C.

Hi Bunuel, I am confused with the question. it states, ranch has horses and ponies and horse= ponies+3. all the information regarding the ponies having horseshoes, isnt it extraneous? becasue the final question, is, what is the minimum # of horses and ponies? it doesnt ask ponies with horseshoes?

i came up with 21 based on my understanding.

considering your solution, you can reduce p*10/18 to p*5/9, hence p=9 to have mimimum ponies horse = 9+3 = 12 hence total is 12+9 = 21

A ranch has both horses and ponies. Exactly 5/6 of the ponies have horseshoes, and exactly 2/3 of the ponies with horseshoes are from Iceland. If there are 3 more horses than ponies, what is the minimum possible combined number of horses and ponies on the ranch?

A. 18 B. 21 C. 38 D. 39 E. 57

Say there are P ponies on the ranch. Then # of ponies with horseshoes, which are from Iceland is P*5/6*2/3=P*10/18. So, P must be a multiple of 18.

The least positive multiple of 18 is 18 itself, so P(min)=18 and H(min)=18+3=21, so the minimum possible combined number of horses and ponies on the ranch is 18+21=39.

Answer: C.

Hi Bunuel, I am confused with the question. it states, ranch has horses and ponies and horse= ponies+3. all the information regarding the ponies having horseshoes, isnt it extraneous? becasue the final question, is, what is the minimum # of horses and ponies? it doesnt ask ponies with horseshoes?

i came up with 21 based on my understanding.

considering your solution, you can reduce p*10/18 to p*5/9, hence p=9 to have mimimum ponies horse = 9+3 = 12 hence total is 12+9 = 21

can you please correct my mistake?

thanks in advance jay

If P=9 as you say then # of ponies with horseshoes would be P*5/6=15/2, which is not possible since it's not an integer.

AGAIN: Say there are P ponies on the ranch.

Then # of ponies with horseshoes is P*5/6, which means that P must be a multiple of 6;

# of ponies from Iceland is (P*5/6)*2/3=P*5/9, which means that P must be a multiple of 9.

P to be multiple of both 6 and 9 it should be a multiple of 18. The least positive multiple of 18 is 18 itself, so P(min)=18 and H(min)=18+3=21, so the minimum possible combined number of horses and ponies on the ranch is 18+21=39.

Re: A ranch has both horses and ponies. Exactly 5/6 of the [#permalink]

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16 Jul 2013, 00:16

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piyushksharma wrote:

A ranch has both horses and ponies. Exactly 5/6 of the ponies have horseshoes, and exactly 2/3 of the ponies with horseshoes are from Iceland. If there are 3 more horses than ponies, what is the minimum possible combined number of horses and ponies on the ranch?

A. 18 B. 21 C. 38 D. 39 E. 57

1. 5/6 of the ponies have horseshoes. We know that should be equal to some number of ponies. Since we are talking of ponies that number should be an integer. 2. Similarly 2/3 of 5/6 of ponies i.e., 5/9 of ponies should also be an integer. 3. Conditions (1) and (2) have to be satisfied by the minimum possible number of ponies 4. We have 5/6 of ponies and 5/9 of ponies to be integers. The minimum number of ponies that satisfy both is 18. 5. Therefore, the minimum possible combined number of horses and ponies is 21 + 18 =39
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Re: A ranch has both horses and ponies. Exactly 5/6 of the [#permalink]

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16 Jul 2013, 00:32

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Let H = Number of Horses and P= Number of Ponies, our target is H+P = minimum

Statement 1: - it is given (5/6)P = positive integer, i.e. exactly 5/6 of pony has horseshoe Statement 2: - it is also given (2/3)(5/6)P = positive integer, i.e. 2/3 of statement 1 is from iceland

From above two statements, if P has to be minimum, then it can be derived that P = 18, any number less than 18 will not satisfy both Statement-1 and Statement-2.

Statement 3: - informs that P+3 = H so if P=18, then H= 21 Hence the minimum combination of H & P is 18+21 = 39. Hence answer is D

Re: A ranch has both horses and ponies. Exactly 5/6 of the [#permalink]

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16 Jul 2013, 04:26

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piyushksharma wrote:

A ranch has both horses and ponies. Exactly 5/6 of the ponies have horseshoes, and exactly 2/3 of the ponies with horseshoes are from Iceland. If there are 3 more horses than ponies, what is the minimum possible combined number of horses and ponies on the ranch?

A. 18 B. 21 C. 38 D. 39 E. 57

Quite tricky.

If you are in a hurry and running against time here's what to do.

Step 1: Given that there are three more horses than ponies, subtract 3 from all the answer choices. Any one of them that has an even number is going to be the answer since the correct number of ponies multiplied by two go get the total number of numbers (less the 3 we initially subtracted) is even. Answer choices have been narrowed down to B, D or E. Guessing at this stage gives you a higher percentage of guessing right than picking any one of the five randomly.

Step 2: Divide the answer gotten in a by 2 to get the likely number of ponies and we have any of the following: B: (21-3) / 2 = 9 D: (39-3) / 2 = 18 E: (57-3) / 2 = 28

Step 3: We have 5/6 of ponies with horseshoes and then 2/3 of ponies with horseshoes from Iceland. That means we have to look for a number that is first divisible by 6 to get 5/6 of a number that is an integer. Only 18 is the possible answer since Options B and E are indivisible by 6. If any other one had been divisible by 6, we would have gone further to check which one is also divisible by 3 after dividing by 6 since we need to know which number will have 2/3 as an integer after dividing by 6. If we had more than one number that met these steps, then we would have gone for the lower number since we were looking for the "minimum possible combined number of horses and ponies".
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Re: A ranch has both horses and ponies. Exactly 5/6 of the [#permalink]

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17 Jul 2013, 10:17

We want to minimize the number of ponies in order to solve the question. This minimum number of ponies will be added by 3 to get the total # of horses.

I used a trial and error approach here to figure out the # of ponies because the fractions given give us clues as to which numbers to pick. For the ponies to have horseshoes, the number must be divisible by 6. If you use 6 you will end up with 5 poines which is not wholey divisible by 2/3 for the ponies that come from iceland. If you go to the next multiple of six, which is 12, you end up with the same problem ---> 10 is not wholey divisible by 2/3.

The next multiple of 6 which is 18 works well. You end up with 15 ponies w/horseshoes (5/6 of 18) of which 10 of those ponies (2/3 of 15) come from iceland.

Adding 3 to 18 gives you 21 horses. 21+18 = 39. Answer is D

Re: A ranch has both horses and ponies. Exactly 5/6 of the [#permalink]

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17 Dec 2013, 23:08

Let ponies = x, so horses should be x+3 Total no. of ponies & horses (minimum) should be any one of the five choices (18,21,38,39,57) So, 2x + 3 can be anything of (18,21,38,39,57)

Choice A (18), Choice C (38) ruled out as x returns a fraction value

Consider Choice B (21): x = 9; 5/6 of 9 comes to be a fraction, so this option ruled out

Consider Choice D (39): x = 18; 5/6 of 18 = 15 & 2/3 of 15 = 10

As we require to find minimum possible, Answer = D (39)
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Re: A ranch has both horses and ponies. Exactly 5/6 of the [#permalink]

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20 Dec 2013, 06:21

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Hi ,

Bunuel you calculated question correctly but provide answer C which should be D. 3 cheers for your solution.

I used back solving approach rather than attacking from from for easy and less time consuming. .Please find it.

As H=P+3. All answers are (H+P) i.e. total animal in ranch. SO we can check out P which are integers. H+P=P+3 + P=2P+3 A) 18 . P=15/2 .Not Integer. Eliminate B) 21 . P=18/2=9 .If 5/6 Ponies have horse shoe than P should be multiple of 6. It cannot be 9 . Eliminate C) 38 . P=35/2 .Not Integer. Eliminate D) 39 . P=36/2=18 . If 5/6 Ponies have horse shoe than P should be multiple of 6. 18 is multiple of 6 . Correct E) 57 . P=54/2 =27 . If 5/6 Ponies have horse shoe than P should be multiple of 6. It cannot be 27 . Eliminate

Bunuel you calculated question correctly but provide answer C which should be D. 3 cheers for your solution.

I used back solving approach rather than attacking from from for easy and less time consuming. .Please find it.

As H=P+3. All answers are (H+P) i.e. total animal in ranch. SO we can check out P which are integers. H+P=P+3 + P=2P+3 A) 18 . P=15/2 .Not Integer. Eliminate B) 21 . P=18/2=9 .If 5/6 Ponies have horse shoe than P should be multiple of 6. It cannot be 9 . Eliminate C) 38 . P=35/2 .Not Integer. Eliminate D) 39 . P=36/2=18 . If 5/6 Ponies have horse shoe than P should be multiple of 6. 18 is multiple of 6 . Correct E) 57 . P=54/2 =27 . If 5/6 Ponies have horse shoe than P should be multiple of 6. It cannot be 27 . Eliminate

Answer is D +1 for me.

Yes, 39 is answer D not C. Edited the typo. Thank you.
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