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Difference between revisions of "1980 AHSME Problems/Problem 8"

(Created page with "==Problem== How many pairs <math>(a,b)</math> of non-zero real numbers satisfy the equation <cmath> \frac{1}{a} + \frac{1}{b} = \frac{1}{a+b} </cmath> <math>\text{(A)} \ \text...")
 
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<math>\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{one pair for each} ~b \neq 0</math>
 
<math>\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{one pair for each} ~b \neq 0</math>
 
<math>\text{(E)} \ \text{two pairs for each} ~b \neq 0</math>
 
<math>\text{(E)} \ \text{two pairs for each} ~b \neq 0</math>
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== Solution ==
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Adding the two fractions on the left side yields <math>\frac{a+b}{ab}=\frac{1}{a+b}</math>. 
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Cross-multiplying yields <math>a^2+2ab+b^2=ab</math>. 
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Subtracting <math>ab</math> from both sides yields <math>a^2+ab+b^2=0</math>.
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Suppose <math>b</math> is constant and <math>a</math> is variable.  Then the discriminant of this quadratic equation is <math>b^2 - 4 \cdot 1 \cdot b^2 = -3b^2</math>, which is negative if <math>b \neq 0</math>.  Therefore, for each <math>b \neq 0</math>, this equation has no real solutions, and the answer is <math>\boxed{\text{(\textbf{A})\ none}}</math>.
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== See also ==
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{{AHSME box|year=1980|num-b=7|num-a=9}}
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{{MAA Notice}}

Latest revision as of 01:36, 23 July 2025

Problem

How many pairs $(a,b)$ of non-zero real numbers satisfy the equation

\[\frac{1}{a} + \frac{1}{b} = \frac{1}{a+b}\] $\text{(A)} \ \text{none} \qquad \text{(B)} \ 1 \qquad \text{(C)} \ 2 \qquad \text{(D)} \ \text{one pair for each} ~b \neq 0$ $\text{(E)} \ \text{two pairs for each} ~b \neq 0$

Solution

Adding the two fractions on the left side yields $\frac{a+b}{ab}=\frac{1}{a+b}$.

Cross-multiplying yields $a^2+2ab+b^2=ab$.

Subtracting $ab$ from both sides yields $a^2+ab+b^2=0$.

Suppose $b$ is constant and $a$ is variable. Then the discriminant of this quadratic equation is $b^2 - 4 \cdot 1 \cdot b^2 = -3b^2$, which is negative if $b \neq 0$. Therefore, for each $b \neq 0$, this equation has no real solutions, and the answer is $\boxed{\text{(\textbf{A})\ none}}$.

See also

1980 AHSME (Problems • Answer Key • Resources)
Preceded by
Problem 7 		Followed by
Problem 9
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 • 26 • 27 • 28 • 29 • 30
All AHSME Problems and Solutions

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