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Difference between revisions of "2000 AIME I Problems/Problem 5"

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== Problem ==
 
== Problem ==
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Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is <math>25.</math> One marble is taken out of each box randomly. The probability that both marbles are black is <math>27/50,</math> and the probability that both marbles are white is <math>m/n,</math> where <math>m</math> and <math>n</math> are relatively prime positive integers. What is <math>m + n</math>?
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Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is <math>25.</math> One marble is taken out of each box randomly. The [[probability]] that both marbles are black is <math>27/50,</math> and the probability that both marbles are white is <math>m/n,</math> where <math>m</math> and <math>n</math> are [[relatively prime]] positive integers. What is <math>m + n</math>?
  
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== Solution ==
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== Solution 1==
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{{solution}}
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If we work with the problem for a little bit, we quickly see that there is no direct combinatorics way to calculate <math>m/n</math>. The [[Principle of Inclusion-Exclusion]] still requires us to find the individual probability of each box.
 +
 
 +
Let <math>a, b</math> represent the number of marbles in each box, and [[without loss of generality]] let <math>a>b</math>. Then, <math>a + b = 25</math>, and since the <math>ab</math> may be reduced to form <math>50</math> on the denominator of <math>\frac{27}{50}</math>, <math>50|ab</math>. It follows that <math>5|a,b</math>, so there are 2 pairs of <math>a</math> and <math>b: (20,5),(15,10)</math>.
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*'''Case 1''': Then the product of the number of black marbles in each box is <math>54</math>, so the only combination that works is <math>18</math> black in first box, and <math>3</math> black in second. Then, <math>P(\text{both white}) = \frac{2}{20} \cdot \frac{2}{5} = \frac{1}{25},</math> so <math>m + n = 26</math>.
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*'''Case 2''': The only combination that works is 9 black in both. Thus, <math>P(\text{both white}) = \frac{1}{10}\cdot \frac{6}{15} = \frac{1}{25}</math>. <math>m + n = 26</math>.
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Thus, <math>m + n = \boxed{026}</math>.
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== Solution 2==
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Let <math>w_1, w_2, b_1,</math> and <math>b_2</math> represent the white and black marbles in boxes 1 and 2.
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Since there are <math>25</math> marbles in the box:
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<math>w_1 + w_2 + b_1 + b_2 = 25</math>
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From the fact that there is a <math>\frac{27}{50}</math> chance of drawing one black marble from each box:
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<math>\frac{b_1 \cdot b_2}{(b_1 + w_1)(b_2 + w_2)} = \frac{27}{50} = \frac{54}{100} = \frac{81}{150}</math>
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 +
Thinking of the numerator and denominator separately, if <math>\frac{27}{50}</math> was not a reduced fraction when calculating out the probability, then <math>b_1 \cdot b_2 = 27</math>.  Since <math>b_1 < 25</math>, this forces the variables to be <math>3</math> and <math>9</math> in some permutation.  Without loss of generality, let <math>b_1 = 3</math> and <math>b_2 = 9</math>.
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 +
The denominator becomes:
 +
<math>(3 + w_1)(9 + w_2) = 50</math>
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 +
Since there have been <math>12</math> black marbles used, there must be <math>13</math> white marbles.  Substituting that in:
 +
 
 +
<math>(3 + w_1)(9 + (13 - w_1)) = 50</math>
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 +
<math>(3 + w_1)(22 - w_1) = 50</math>
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Since the factors of <math>50</math> that are greater than <math>3</math> are <math>5, 10, 25,</math> and <math>50</math>, the quantity <math>3 + w_1</math> must equal one of those.  However, since <math>w_1 < 13</math>, testing <math>2</math> and <math>7</math> for <math>w_1</math> does not give a correct product.  Thus, <math>\frac{27}{50}</math> must be a reduced form of the actual fraction.
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First assume that the fraction was reduced from <math>\frac{54}{100}</math>, yielding the equations <math>b_1\cdot b_2 = 54</math> and <math>(b_1 + w_1)(b_2 + w_2) = 100</math>.
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Factoring <math>b_1 \cdot b_2 = 54</math> and saying WLOG that <math>b_1 <  b_2 < 25</math> gives <math>(b_1, b_2) = (3, 18)</math> or <math> (6, 9)</math>.  Trying the first pair and setting the denominator equal to 100 gives:
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<math>(3 + w_1)(18 + w_2) = 100</math>
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Since <math>w_1 + w_2 = 4</math>, the pairs <math>(w_1, w_2) = (1, 3), (2,2),</math> and <math>(3,1)</math> can be tried, since each box must contain at least one white marble.  Plugging in <math>w_1 = w_2 = 2</math> gives the true equation <math>(3 + 2)(18 + 2) =100</math>, so the number of marbles are <math>(w_1, w_2, b_1, b_2) = (2, 2, 3, 18)</math>
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Thus, the chance of drawing 2 white marbles is <math>\frac{w_1 \cdot w_2 }{(w_1+ b_1)(w_2 + b_2)} = \frac{4}{100} = \frac{1}{25}</math> in lowest terms, and the answer to the problem is  <math>1 + 25 = \boxed{026}.</math>
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For completeness, the fraction <math>\frac{81}{150}</math> may be tested.  <math>150</math> is the highest necessary denominator that needs to be tested, since the maximum the denominator <math>(w_1+ b_1)(w_2 + b_2)</math> can be when the sum of all integer variables is <math>25</math> is when the variables are <math>6, 6, 6, </math> and <math>7</math>, in some permutation, which gives <math>154</math>.  If <math>b_1 \cdot b_2 = 81</math>, this forces <math>b_1 = b_2 = 9</math>, since all variables must be integers under <math>25</math>.  The denominator becomes <math>(9 + w_1)(9 + w_2) = 150</math>, and since there are now <math>25 - 18 = 7</math> white marbles total, the denominator becomes <math>(9 + w_1)(16 - w_1) = 150</math>.  Testing <math>w_1 = 1</math> gives a solution, and thus <math>w_2 = 6</math>.  The complete solution for this case is <math>(w_1, w_2, b_1, b_2) = (1, 6, 9, 9)</math>.  Although the distribution and colors of the marbles is different from the last case, the probability of drawing two white marbles is <math>\frac{6 \cdot 1}{ 150}</math>, which still simplifies to <math>\frac {1}{25}</math>.
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==Solution 3==
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We know that <math>\frac{27}{50} = \frac{b_1}{t_1} \cdot \frac{b_2}{t_2}</math>, where <math>b_1</math> and <math>b_2</math> are the number of black marbles in the first and the second box respectively, and <math>t_1</math> and <math>t_2</math> is the total number of marbles in the first and the second boxes respectively. So, <math>t_1 + t_2 = 25</math>. Then, we can realize that <math>\frac{27}{50} = \frac{9}{10} \cdot \frac{3}{5} = \frac{9}{10} \cdot \frac{9}{15}</math>, which means that having 9 black marbles out of 10 total in the first box and 9 marbles out of 15 total the second box is valid. Then there is 1 white marble in the first box and 6 in the second box. So, the probability of drawing two white marbles becomes <math>\frac{1}{10} \cdot \frac{6}{15} = \frac{1}{25}</math>. The answer is <math>1 + 25 = \boxed{026}</math>
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Note: Note that if <math>t_1=5, t_2=20</math>, and <math>b_1=3, b_2=18</math>, it also works since <math>\frac{b_1}{t_1} \cdot \frac{b_2}{t_2} = \frac{3}{5} \cdot \frac{18}{20} = \frac{27}{50}</math>, so the probability of drawing a white marble is <math>\frac{2}{5} \cdot \frac{2}{20} = \frac{1}{25}</math>. Therefore, our answer is <math>1+25=\boxed{026}.</math>
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 +
~Yiyj1
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2000|n=I|num-b=4|num-a=6}}
 
{{AIME box|year=2000|n=I|num-b=4|num-a=6}}
 +
 +
[[Category:Intermediate Number Theory Problems]]
 +
{{MAA Notice}}

Latest revision as of 19:09, 4 January 2024

Problem

Each of two boxes contains both black and white marbles, and the total number of marbles in the two boxes is $25.$ One marble is taken out of each box randomly. The probability that both marbles are black is $27/50,$ and the probability that both marbles are white is $m/n,$ where $m$ and $n$ are relatively prime positive integers. What is $m + n$?

Solution 1

If we work with the problem for a little bit, we quickly see that there is no direct combinatorics way to calculate $m/n$. The Principle of Inclusion-Exclusion still requires us to find the individual probability of each box.

Let $a, b$ represent the number of marbles in each box, and without loss of generality let $a>b$. Then, $a + b = 25$, and since the $ab$ may be reduced to form $50$ on the denominator of $\frac{27}{50}$, $50|ab$. It follows that $5|a,b$, so there are 2 pairs of $a$ and $b: (20,5),(15,10)$.

  • Case 1: Then the product of the number of black marbles in each box is $54$, so the only combination that works is $18$ black in first box, and $3$ black in second. Then, $P(\text{both white}) = \frac{2}{20} \cdot \frac{2}{5} = \frac{1}{25},$ so $m + n = 26$.
  • Case 2: The only combination that works is 9 black in both. Thus, $P(\text{both white}) = \frac{1}{10}\cdot \frac{6}{15} = \frac{1}{25}$. $m + n = 26$.

Thus, $m + n = \boxed{026}$.

Solution 2

Let $w_1, w_2, b_1,$ and $b_2$ represent the white and black marbles in boxes 1 and 2.

Since there are $25$ marbles in the box:

$w_1 + w_2 + b_1 + b_2 = 25$

From the fact that there is a $\frac{27}{50}$ chance of drawing one black marble from each box:

$\frac{b_1 \cdot b_2}{(b_1 + w_1)(b_2 + w_2)} = \frac{27}{50} = \frac{54}{100} = \frac{81}{150}$

Thinking of the numerator and denominator separately, if $\frac{27}{50}$ was not a reduced fraction when calculating out the probability, then $b_1 \cdot b_2 = 27$. Since $b_1 < 25$, this forces the variables to be $3$ and $9$ in some permutation. Without loss of generality, let $b_1 = 3$ and $b_2 = 9$.

The denominator becomes: $(3 + w_1)(9 + w_2) = 50$

Since there have been $12$ black marbles used, there must be $13$ white marbles. Substituting that in:

$(3 + w_1)(9 + (13 - w_1)) = 50$

$(3 + w_1)(22 - w_1) = 50$

Since the factors of $50$ that are greater than $3$ are $5, 10, 25,$ and $50$, the quantity $3 + w_1$ must equal one of those. However, since $w_1 < 13$, testing $2$ and $7$ for $w_1$ does not give a correct product. Thus, $\frac{27}{50}$ must be a reduced form of the actual fraction.

First assume that the fraction was reduced from $\frac{54}{100}$, yielding the equations $b_1\cdot b_2 = 54$ and $(b_1 + w_1)(b_2 + w_2) = 100$. Factoring $b_1 \cdot b_2 = 54$ and saying WLOG that $b_1 < b_2 < 25$ gives $(b_1, b_2) = (3, 18)$ or $(6, 9)$. Trying the first pair and setting the denominator equal to 100 gives: $(3 + w_1)(18 + w_2) = 100$


Since $w_1 + w_2 = 4$, the pairs $(w_1, w_2) = (1, 3), (2,2),$ and $(3,1)$ can be tried, since each box must contain at least one white marble. Plugging in $w_1 = w_2 = 2$ gives the true equation $(3 + 2)(18 + 2) =100$, so the number of marbles are $(w_1, w_2, b_1, b_2) = (2, 2, 3, 18)$

Thus, the chance of drawing 2 white marbles is $\frac{w_1 \cdot w_2 }{(w_1+ b_1)(w_2 + b_2)} = \frac{4}{100} = \frac{1}{25}$ in lowest terms, and the answer to the problem is $1 + 25 = \boxed{026}.$


For completeness, the fraction $\frac{81}{150}$ may be tested. $150$ is the highest necessary denominator that needs to be tested, since the maximum the denominator $(w_1+ b_1)(w_2 + b_2)$ can be when the sum of all integer variables is $25$ is when the variables are $6, 6, 6,$ and $7$, in some permutation, which gives $154$. If $b_1 \cdot b_2 = 81$, this forces $b_1 = b_2 = 9$, since all variables must be integers under $25$. The denominator becomes $(9 + w_1)(9 + w_2) = 150$, and since there are now $25 - 18 = 7$ white marbles total, the denominator becomes $(9 + w_1)(16 - w_1) = 150$. Testing $w_1 = 1$ gives a solution, and thus $w_2 = 6$. The complete solution for this case is $(w_1, w_2, b_1, b_2) = (1, 6, 9, 9)$. Although the distribution and colors of the marbles is different from the last case, the probability of drawing two white marbles is $\frac{6 \cdot 1}{ 150}$, which still simplifies to $\frac {1}{25}$.

Solution 3

We know that $\frac{27}{50} = \frac{b_1}{t_1} \cdot \frac{b_2}{t_2}$, where $b_1$ and $b_2$ are the number of black marbles in the first and the second box respectively, and $t_1$ and $t_2$ is the total number of marbles in the first and the second boxes respectively. So, $t_1 + t_2 = 25$. Then, we can realize that $\frac{27}{50} = \frac{9}{10} \cdot \frac{3}{5} = \frac{9}{10} \cdot \frac{9}{15}$, which means that having 9 black marbles out of 10 total in the first box and 9 marbles out of 15 total the second box is valid. Then there is 1 white marble in the first box and 6 in the second box. So, the probability of drawing two white marbles becomes $\frac{1}{10} \cdot \frac{6}{15} = \frac{1}{25}$. The answer is $1 + 25 = \boxed{026}$

Note: Note that if $t_1=5, t_2=20$, and $b_1=3, b_2=18$, it also works since $\frac{b_1}{t_1} \cdot \frac{b_2}{t_2} = \frac{3}{5} \cdot \frac{18}{20} = \frac{27}{50}$, so the probability of drawing a white marble is $\frac{2}{5} \cdot \frac{2}{20} = \frac{1}{25}$. Therefore, our answer is $1+25=\boxed{026}.$

~Yiyj1

See also

2000 AIME I (Problems • Answer Key • Resources)
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