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

(D) The trick to this question is that the number of Icelandic ponies cannot be fractional. It must be an integer.
The number of Icelandic ponies is:
(2/3) × (5/6) × total number of ponies, or
10/18 × total number of ponies.

9 is the smallest positive integer that yields another integer when multiplied by 5/9, and all multiples of 9 will yield a whole number of Icelandic ponies.

However, the number of ponies must also be a multiple of 6 so that the number of ponies with horseshoes is an integer. If there were only 9 ponies, the number of ponies with horseshoes would be:
(5/6) × 9 = 15/2. This is impossible.

The least common multiple of 6 and 9 is 18, so there are 18 ponies at minimum.

If there are 18 ponies, then there must be 18 + 3 = 21 horses. Since 18 + 21 = 39, the minimum number of horses and ponies on the ranch is 39.

The correct answer is choice (D).
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The answer can also be 21 (12+9). i.e. 12 horses and 9 ponies. if there are 9 ponies,the total no.of iceland ponies will be 10/18 x 9 = 5 ponies. So combined no. of animals shall be 21(minimum).

If there were 9 ponies, then the number of the ponies with horseshoes would be 9 × 5/6 = 15/5 = 7.5. That is the fractional value and can NOT be true.

The key to solve this problem is to keep in mind that all the values must NOT be fractional: the number of the ponies with horseshoes and the number of the ponies with horseshoes from Iceland.

When you are confused with an explanation, think which logical step you are exactly confused with.

For example, you can be confused with any of these logical steps:
- "The trick to this question is that the number of Icelandic ponies cannot be fractional. It must be an integer."

- "The number of Icelandic ponies is: (2/3) × (5/6) × total number of ponies"

- "9 is the smallest positive integer that yields another integer when multiplied by 5/9"

- "all multiples of 9 will yield a whole number of Icelandic ponies"

- " the number of ponies must also be a multiple of 6 so that the number of ponies with horseshoes is an integer."

- " If there were only 9 ponies, the number of ponies with horseshoes would be: (5/6) × 9 = 15/2. This is impossible."

- "The least common multiple of 6 and 9 is 18, so there are 18 ponies at minimum."

- "If there are 18 ponies, then there must be 18 + 3 = 21 horses. Since 18 + 21 = 39, the minimum number of horses and ponies on the ranch is 39."


When you know which logical step is confusing, you'll be able to look at it more precisely. In many cases you'll find the answer on your own. If not, I would be able to help you. But I can't help you if I don't know what it is that you don't understand. There are many options: fractions are not clear to you, you don't know how to calculate the LCM, you do not understand why we calculate the LCM, you do not understand why the numbers must be integers, etc.

So please, specify, what exactly is NOT clear to you and I'll be able to help. For now, I can only suggest that you plug in some values and see what happens. For example:

If there are 4 ponies, then there are 3 1/3 ponies with horseshoes, and 2 2/9 ponies with horseshoes are from Iceland.

If there are 6 ponies, then there are 5 ponies with horseshoes, and 3 1/3 ponies with horseshoes are from Iceland.

If there are 18 ponies, then there are 15 ponies with horseshoes, and 10 ponies with horseshoes are from Iceland.

If there are 36 ponies, then there are 30 ponies with horseshoes, and 20 ponies with horseshoes are from Iceland.

Suppose the answer is B. Then there are 12 horses and 9 ponies on the ranch (because "there are 3 more horses than ponies").

If there are 9 ponies, then there must be 9 × (5/6) ponies with horseshoes. But 9 × (5/6) = 7.5, which is fractional, can not be the number of ponies with horseshoes! It is the contradiction with the question statement, thus 21 does NOT fit as an answer.

The key to solve this problem is to keep in mind that all the values must NOT be fractional: the number of ponies with horseshoes and the number of ponies with horseshoes from Iceland.

It is very important here.

If the total number of ponies and horses was 21, then there would be 12 horses and 9 ponies. But let's go further. There would be

(5/6) × 12 = 10 ponies with horseshoes
and
(2/3) × 10 = 6 2/3 ponies with horseshoes from Iceland.

The number of ponies with horseshoes from Iceland cannot be fractional! Thus B is not the correct answer.

That is why the given information (fractions) about the groups of ponies is important. Above you may find the explanation of how we make sure those groups are not fractional numbers.