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Difference between revisions of "2000 AMC 12 Problems/Problem 11"

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{{duplicate|[[2000 AMC 12 Problems|2000 AMC 12 #11]] and [[2000 AMC 10 Problems|2000 AMC 10 #15]]}}
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==Problem==
 
==Problem==
 
Two non-zero [[real number]]s, <math>a</math> and <math>b,</math> satisfy <math>ab = a - b</math>. Which of the following is a possible value of <math>\frac {a}{b} + \frac {b}{a} - ab</math>?
 
Two non-zero [[real number]]s, <math>a</math> and <math>b,</math> satisfy <math>ab = a - b</math>. Which of the following is a possible value of <math>\frac {a}{b} + \frac {b}{a} - ab</math>?
  
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<math>\text{(A)} \ - 2 \qquad \text{(B)} \ \frac { - 1}{2} \qquad \text{(C)} \ \frac {1}{3} \qquad \text{(D)} \ \frac {1}{2} \qquad \text{(E)} \ 2</math>
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<math>\textbf{(A)} \ - 2 \qquad \textbf{(B)} \ \frac { -1 }{2} \qquad \textbf{(C)} \ \frac {1}{3} \qquad \textbf{(D)} \ \frac {1}{2} \qquad \textbf{(E)} \ 2</math>
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==Solution 1==
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<math>\frac {a}{b} + \frac {b}{a} - ab = \frac{a^2 + b^2}{ab} - (a - b) = \frac{a^2 + b^2}{a-b} - \frac{(a-b)^2}{(a-b)} = \frac{2ab}{a-b} = \frac{2(a-b)}{a-b} =2 \Rightarrow \boxed{\text{E}}</math>.
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Another way is to solve the equation for <math>b,</math> giving <math>b = \frac{a}{a+1};</math> then substituting this into the expression and simplifying gives the answer of <math>2.</math>
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==Solution 2==
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This simplifies to <math>ab+b-a=0 \Rightarrow (a+1)(b-1) = -1</math>. The two integer solutions to this are <math>(-2,2)</math> and <math>(0,0)</math>. The problem states than <math>a</math> and <math>b</math> are non-zero, so we consider the case of <math>(-2,2)</math>. So, we end up with <math>\frac{-2}{2} + \frac{2}{-2} - 2 \cdot -2 = 4 - 1 - 1 = 2 \Rightarrow \boxed{\text{E}}</math>
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==Solution 3==
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Just realize that two such numbers are <math>a = 1</math> and <math>b = \frac{1}{2}</math>. You can see this by plugging in <math>a = 1</math> and then solving for b. With this, you can solve and get <math>2 \Rightarrow\boxed{\text{E}}</math>
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==Solution 4 ==
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Set <math>a</math> to some nonzero number. In this case, I'll set it to <math>4</math>.
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Then solve for <math>b</math>. In this case, <math>b=0.8</math>.
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Now just simply evaluate. In this case it's 2. So since 2 is a possible value of the original expression, select <math>\boxed{E}</math>.
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~hastapasta
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==Solution 5==
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Notice that <math>a=\frac{a}{b}-1</math> and <math>b=1-\frac{b}{a}</math>. Then, <math>\frac{a}{b}=1+a</math> and <math>\frac{b}{a}=1-b</math>.
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<math>\frac{a}{b}+\frac{b}{a}-ab=(1+a)+(1-b)-(a-b)=2</math>. The answer is <math>\boxed{E}</math>.
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== Video Solution ==
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https://www.youtube.com/watch?v=7-RloNHTnXM
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== Video Solution ==
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https://youtu.be/ZWqHxc0i7ro?t=6
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 +
~ pi_is_3.14
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==Video Solution==
 +
 
 +
https://www.youtube.com/watch?v=8nxvuv5oZ7A&t=3s
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==Video Solution by Daily Dose of Math==
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https://youtu.be/Q_th4G-xGLo?si=4VwtJirZjREyyQuO
  
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==Solution==
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~Thesmartgreekmathdude
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<math>\frac {a}{b} + \frac {b}{a} - ab = \frac{a^2 + b^2}{ab} - (a - b) = \frac{a^2 + b^2}{a-b} - \frac{(a-b)^2}{(a-b)} = \frac{2ab}{a-b} = 2 \Rightarrow \text{(E)}</math>.
 
  
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Alternatively, we could test simple values, like <math>(a,b)=\left(1, \frac{1}{2}\right)</math>, which would yield <math>\frac {a}{b} + \frac {b}{a} - ab=2</math>.
 
  
 
==See also==
 
==See also==
 
{{AMC12 box|year=2000|num-b=10|num-a=12}}
 
{{AMC12 box|year=2000|num-b=10|num-a=12}}
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{{AMC10 box|year=2000|num-b=14|num-a=16}}
  
 
[[Category:Introductory Algebra Problems]]
 
[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 23:54, 14 July 2024

The following problem is from both the 2000 AMC 12 #11 and 2000 AMC 10 #15, so both problems redirect to this page.

Problem

Two non-zero real numbers, $a$ and $b,$ satisfy $ab = a - b$. Which of the following is a possible value of $\frac {a}{b} + \frac {b}{a} - ab$?

$\textbf{(A)} \ - 2 \qquad \textbf{(B)} \ \frac { -1 }{2} \qquad \textbf{(C)} \ \frac {1}{3} \qquad \textbf{(D)} \ \frac {1}{2} \qquad \textbf{(E)} \ 2$

Solution 1

$\frac {a}{b} + \frac {b}{a} - ab = \frac{a^2 + b^2}{ab} - (a - b) = \frac{a^2 + b^2}{a-b} - \frac{(a-b)^2}{(a-b)} = \frac{2ab}{a-b} = \frac{2(a-b)}{a-b} =2 \Rightarrow \boxed{\text{E}}$.

Another way is to solve the equation for $b,$ giving $b = \frac{a}{a+1};$ then substituting this into the expression and simplifying gives the answer of $2.$

Solution 2

This simplifies to $ab+b-a=0 \Rightarrow (a+1)(b-1) = -1$. The two integer solutions to this are $(-2,2)$ and $(0,0)$. The problem states than $a$ and $b$ are non-zero, so we consider the case of $(-2,2)$. So, we end up with $\frac{-2}{2} + \frac{2}{-2} - 2 \cdot -2 = 4 - 1 - 1 = 2 \Rightarrow \boxed{\text{E}}$

Solution 3

Just realize that two such numbers are $a = 1$ and $b = \frac{1}{2}$. You can see this by plugging in $a = 1$ and then solving for b. With this, you can solve and get $2 \Rightarrow\boxed{\text{E}}$

Solution 4

Set $a$ to some nonzero number. In this case, I'll set it to $4$.

Then solve for $b$. In this case, $b=0.8$.

Now just simply evaluate. In this case it's 2. So since 2 is a possible value of the original expression, select $\boxed{E}$.

~hastapasta

Solution 5

Notice that $a=\frac{a}{b}-1$ and $b=1-\frac{b}{a}$. Then, $\frac{a}{b}=1+a$ and $\frac{b}{a}=1-b$.

$\frac{a}{b}+\frac{b}{a}-ab=(1+a)+(1-b)-(a-b)=2$. The answer is $\boxed{E}$.

Video Solution

https://www.youtube.com/watch?v=7-RloNHTnXM

Video Solution

https://youtu.be/ZWqHxc0i7ro?t=6

~ pi_is_3.14

Video Solution

https://www.youtube.com/watch?v=8nxvuv5oZ7A&t=3s

Video Solution by Daily Dose of Math

https://youtu.be/Q_th4G-xGLo?si=4VwtJirZjREyyQuO

~Thesmartgreekmathdude


See also

2000 AMC 12 (Problems • Answer Key • Resources)
Preceded by
Problem 10 		Followed by
Problem 12
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
2000 AMC 10 (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
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

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