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Difference between revisions of "2008 AMC 10B Problems/Problem 8"

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==Solution 1==
 
==Solution 1==
  
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The cost of a rose is odd, hence we need an even number of roses. Let there be <math>2r</math> roses for some <math>r\geq 0</math>. Then we have <math>50-3\cdot 2r = 50-6r</math> dollars left. We can always reach the sum exactly <math>50</math> by buying <math>(50-6r)/2 = 25-3r</math> carnations. Of course, the number of roses must be such that the number of carnations is non-negative. We get the inequality <math>25-3r \geq 0</math> which solves to <math>r\leq \frac{25}{3}</math>. <math>r</math> must be an integer, so there are <math>\boxed{9 \text{ (C)}}</math> possible values of <math>r</math>, and each gives us one solution.
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The cost of a rose is odd, hence we need an even number of roses. Let there be <math>2r</math> roses for some <math>r\geq 0</math>. Then we have <math>50-3\cdot 2r = 50-6r</math> dollars left. We can always reach the sum exactly <math>50</math> by buying <math>(50-6r)/2 = 25-3r</math> carnations. Of course, the number of roses must be such that the number of carnations is non-negative. We get the inequality <math>25-3r \geq 0</math> which solves to <math>r\leq 8 \frac13</math>. <math>r</math> must be an integer, so there are <math>\boxed{9 \text{ (C)}}</math> possible values of <math>r</math>, and each gives us one solution.
  
 
==Solution 2==
 
==Solution 2==
  
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Let <math>x</math> and <math>y</math> be the number of roses and carnations bought. The equation should be <math>3x+2y = 50</math>. Since <math>50</math> is an odd number, the product of <math>3x</math> must be even and smaller than <math>50</math>. You can try all the even substitutes for <math>x</math> and you will end up with the numbers <math>0</math>, <math>2</math>, <math>4</math>, <math>6</math>, <math>8</math>, <math>10</math>, <math>12</math>, <math>14</math>, and <math>16</math>. There are <math>9</math> numbers in total, so the answer is  <math>\boxed{9 \text{ (C)}}</math>.
+
Let <math>x</math> and <math>y</math> be the number of roses and carnations bought. The equation should be <math>3x+2y = 50</math>. Since <math>50</math> is an even number, the product of <math>3x</math> must be even and smaller than <math>50</math>. You can try nonnegative even integers for <math>x</math> and you will end up with the numbers <math>0</math>, <math>2</math>, <math>4</math>, <math>6</math>, <math>8</math>, <math>10</math>, <math>12</math>, <math>14</math>, and <math>16</math>. There are <math>9</math> numbers in total, so the answer is  <math>\boxed{9 \text{ (C)}}</math>.
  
 
==Solution 3==
 
==Solution 3==
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Using the Euclidean algorithm, we get the initial solutions to be <math>r_0=50</math> and <math>c_0=-50</math>, meaning the complete solution will be,
 
Using the Euclidean algorithm, we get the initial solutions to be <math>r_0=50</math> and <math>c_0=-50</math>, meaning the complete solution will be,
 
<math>r=50+\frac{2}{\gcd(2,3)}</math> <math>k=50+2k</math>, <math>c=-50-\frac{3}{\gcd(2,3)}k=-50-3k</math>
 
<math>r=50+\frac{2}{\gcd(2,3)}</math> <math>k=50+2k</math>, <math>c=-50-\frac{3}{\gcd(2,3)}k=-50-3k</math>
 +
 
The solution range for which both <math>r</math> and <math>c</math> are positive is <math>17</math> <math>\leq k</math> <math>\leq</math> <math>25</math>. There are <math>\boxed{9 \text{ (C)}}</math> possible values for <math>k</math>.
 
The solution range for which both <math>r</math> and <math>c</math> are positive is <math>17</math> <math>\leq k</math> <math>\leq</math> <math>25</math>. There are <math>\boxed{9 \text{ (C)}}</math> possible values for <math>k</math>.
 
{{AMC10 box|year=2008|ab=B|num-b=7|num-a=9}}
 
{{AMC10 box|year=2008|ab=B|num-b=7|num-a=9}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Latest revision as of 19:32, 26 July 2022

Problem

A class collects 50 dollars to buy flowers for a classmate who is in the hospital. Roses cost 3 dollars each, and carnations cost 2 dollars each. No other flowers are to be used. How many different bouquets could be purchased for exactly 50 dollars?

$\mathrm{(A)}\ 1 \qquad \mathrm{(B)}\ 7 \qquad \mathrm{(C)}\ 9 \qquad \mathrm{(D)}\ 16 \qquad \mathrm{(E)}\ 17$

Solution 1

The cost of a rose is odd, hence we need an even number of roses. Let there be $2r$ roses for some $r\geq 0$. Then we have $50-3\cdot 2r = 50-6r$ dollars left. We can always reach the sum exactly $50$ by buying $(50-6r)/2 = 25-3r$ carnations. Of course, the number of roses must be such that the number of carnations is non-negative. We get the inequality $25-3r \geq 0$ which solves to $r\leq 8 \frac13$. $r$ must be an integer, so there are $\boxed{9 \text{ (C)}}$ possible values of $r$, and each gives us one solution.

Solution 2

Let $x$ and $y$ be the number of roses and carnations bought. The equation should be $3x+2y = 50$. Since $50$ is an even number, the product of $3x$ must be even and smaller than $50$. You can try nonnegative even integers for $x$ and you will end up with the numbers $0$, $2$, $4$, $6$, $8$, $10$, $12$, $14$, and $16$. There are $9$ numbers in total, so the answer is $\boxed{9 \text{ (C)}}$.

Solution 3

Let $r$ represent the number of roses, and let $c$ represent the number of carnations. Then, we get the linear Diophantine equation, $3r+2c=50$. Using the Euclidean algorithm, we get the initial solutions to be $r_0=50$ and $c_0=-50$, meaning the complete solution will be, $r=50+\frac{2}{\gcd(2,3)}$ $k=50+2k$, $c=-50-\frac{3}{\gcd(2,3)}k=-50-3k$

The solution range for which both $r$ and $c$ are positive is $17$ $\leq k$ $\leq$ $25$. There are $\boxed{9 \text{ (C)}}$ possible values for $k$.

2008 AMC 10B (Problems • Answer Key • Resources)
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Problem 7 		Followed by
Problem 9
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