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2000 AMC 12 Problems/Problem 11: Difference between revisions

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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>.
<math>\frac{a}{b}+\frac{b}{a}-ab=(1+a)+(1-b)-(a-b)=2</math>. The answer is <math>\boxed{E}</math>.
==Solution 6==
We are told <math>ab=a-b</math>.
Dividing by <math>a</math>, we get <math>b = 1 - \frac{b}{a} \Rightarrow \boxed{ \frac{b}{a} = 1 - b}</math>.
We can also solve for <math>\frac{a}{b}</math> by dividing by <math>b</math>. Dividing, we get <math>a = \frac{a}{b} - 1 \Rightarrow \boxed{ \frac{a}{b} = a + 1}</math>.
Plugging these values into <math>\frac{a}{b} + \frac{b}{a} - ab</math>, we get:
<cmath>(a + 1) + (1 - b) - (a - b) = 2</cmath>
Therefore, the final answer is <math>\boxed{\text{E}}</math>
-Dustin_Links


== Video Solution ==
== Video Solution ==

Revision as of 15:39, 15 June 2025

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 $\boxed{E}$.

Solution 6

We are told $ab=a-b$.

Dividing by $a$, we get $b = 1 - \frac{b}{a} \Rightarrow \boxed{ \frac{b}{a} = 1 - b}$.

We can also solve for $\frac{a}{b}$ by dividing by $b$. Dividing, we get $a = \frac{a}{b} - 1 \Rightarrow \boxed{ \frac{a}{b} = a + 1}$.

Plugging these values into $\frac{a}{b} + \frac{b}{a} - ab$, we get:

\[(a + 1) + (1 - b) - (a - b) = 2\]

Therefore, the final answer is $\boxed{\text{E}}$

-Dustin_Links

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

These problems are copyrighted © by the Mathematical Association of America, as part of the American Mathematics Competitions. Error creating thumbnail: File missing

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