Difference between revisions of "2006 AMC 12A Problems/Problem 14"

Revision as of 14:58, 1 November 2013

The following problem is from both the 2006 AMC 12A #14 and 2006 AMC 10A #22, so both problems redirect to this page.

Problem

Two farmers agree that pigs are worth $300$ dollars and that goats are worth $210$ dollars. When one farmer owes the other money, he pays the debt in pigs or goats, with "change" received in the form of goats or pigs as necessary. (For example, a $390$ dollar debt could be paid with two pigs, with one goat received in change.) What is the amount of the smallest positive debt that can be resolved in this way?

$\mathrm{(A) \ } 5\qquad \mathrm{(B) \ } 10\qquad \mathrm{(C) \ } 30\qquad \mathrm{(D) \ } 90\qquad \mathrm{(E) \ } 210$

Solutions

Solution 1

The problem can be restated as an equation of the form $300p + 210g = x$ , where $p$ is the number of pigs, $g$ is the number of goats, and $x$ is the positive debt. The problem asks us to find the lowest x possible. p and g must be integers, which makes the equation a Diophantine equation. The Euclidean algorithm tells us that there are integer solutions to the Diophantine equation $am + bn = c$ , where $c$ is the greatest common divisor of $a$ and $b$ , and no solutions for any smaller $c$ . Therefore, the answer is the greatest common divisor of 300 and 210, which is 30, $\mathrm{(C) \ }$

Solution 2

Alternatively, note that $300p + 210g = 30(10p + 7g)$ is divisible by 30 no matter what $p$ and $g$ are, so our answer must be divisible by 30. In addition, three goats minus two pigs give us $630 - 600 = 30$ exactly. Since our theoretical best can be achieved, it must really be the best, and the answer is $\mathrm{(C) \ }$ . debt that can be resolved.

=Solution 23

Let us think of it as if the farmers would like to pay off a certain debt, then the farmers take turns giving each other goats and pigs until it is resolved. Let us suppose they start at 0 debt, and they want to get to the smallest debt possible. Each time a pig or goat is exchanged, the remainder of the debt when divided by 30 stays the same. This is because a goat and pig are both worth multiples of 30. Since the debt is positive, the smallest possible achievable debt is 30 (C).

Solution 4

Let us simplify this problem. Dividing by $30$ , we get a pig to be: $\frac{300}{30} = 10$ , and a goat to be $\frac{210}{30}= 7$ . It becomes evident that if you exchange $5$ pigs for $4$ goats, we get the smallest positive difference - $5\cdot 10 - 4\cdot 7 = 50-49 = 1$ . Since we originally divided by $30$ , we need to multiply again, thus getting the answer: $1\cdot 30 = \mathrm{(C) 30}$

2006 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 13	Followed by Problem 15
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

2006 AMC 10A (Problems • Answer Key • Resources)
Preceded by Problem 21	Followed by Problem 23
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

@@ Line 7: / Line 7: @@
 <math> \mathrm{(A) \ } 5\qquad \mathrm{(B) \ } 10\qquad \mathrm{(C) \ } 30\qquad \mathrm{(D) \ } 90\qquad \mathrm{(E) \ }  210</math>
-== Solution ==
+== Solutions ==
-Solution 1:
+===Solution 1===
 The problem can be restated as an equation of the form <math>300p + 210g = x</math>, where <math>p</math> is the number of pigs, <math>g</math> is the number of goats, and <math>x</math> is the positive debt. The problem asks us to find the lowest ''x'' possible. ''p'' and ''g'' must be [[integer]]s, which makes the equation a [[Diophantine equation]].  The [[Euclidean algorithm]] tells us that there are integer solutions to the Diophantine equation <math>am + bn = c</math>, where <math>c</math> is the [[greatest common divisor]] of <math>a</math> and <math>b</math>, and no solutions for any smaller <math>c</math>. Therefore, the answer is the greatest common divisor of 300 and 210, which is 30, <math>\mathrm{(C) \ }</math>
+===Solution 2===
 Alternatively, note that <math>300p + 210g = 30(10p + 7g)</math> is divisible by 30 no matter what <math>p</math> and <math>g</math> are, so our answer must be divisible by 30.  In addition, three goats minus two pigs give us <math>630 - 600 = 30</math> exactly.  Since our theoretical best can be achieved, it must really be the best, and the answer is <math>\mathrm{(C) \ }</math>.
 debt that can be resolved.
-Solution 2:
+===Solution 23==
 Let us think of it as if the farmers would like to pay off a certain debt, then the farmers take turns giving each other goats and pigs until it is resolved. Let us suppose they start at 0 debt, and they want to get to the smallest debt possible. Each time a pig or goat is exchanged, the remainder of the debt when divided by 30 stays the same. This is because a goat and pig are both worth multiples of 30. Since the debt is positive, the smallest possible achievable debt is 30 (C).
-Solution 3:
+===Solution 4===
 Let us simplify this problem.  Dividing by <math>30</math>, we get a pig to be:  <math>\frac{300}{30} =  10</math>, and a goat to be <math>\frac{210}{30}= 7</math>.

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