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Difference between revisions of "1964 AHSME Problems/Problem 32"

(Created page with "If <math>\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}</math>, then: <math>\textbf{(A) }a \text{ must equal }c\qquad\textbf{(B) }a+b+c+d\text{ must equal zero}\qquad</math> <math>\textbf{(...")
 
(→‎Solution)
 
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==Problem==
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If <math>\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}</math>, then:
 
If <math>\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}</math>, then:
  
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<math> \textbf{(E) }a(b+c+d)=c(a+b+d)</math>
 
<math> \textbf{(E) }a(b+c+d)=c(a+b+d)</math>
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==Solution==
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Cross-multiplying gives:
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<math>(a+b)(a+d) = (b+c)(c+d)</math>
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<math>a^2 + ad + ab + bd = bc + bd + c^2 + cd</math>
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<math>a^2 + ad + ab  - bc - c^2 - cd = 0</math>
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<math>a(a + b + d) - c(b+c+d)= 0</math>
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This looks close to turning into option C, but we don't have a <math>c</math> term in the first parentheses, and we don't have an <math>a</math> term in the second parentheses to allow us to complete the factorization.  However, if we both add <math>ac</math> and subtract <math>ac</math> on the LHS, we get:
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<math>a(a + b + d) + ac - c(b+c+d) - ca= 0</math>
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<math>a(a+b+d +c) - c(b+c+d+a) = 0</math>
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<math>(a-c)(a+b+c+d) = 0</math>
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This is equivalent to <math>\boxed{\textbf{(C)}}</math>
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==See Also==
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{{AHSME 40p box|year=1964|num-b=31|num-a=33}}
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[[Category:Introductory Algebra Problems]]
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{{MAA Notice}}

Latest revision as of 01:27, 25 July 2019

Problem

If $\dfrac{a+b}{b+c}=\dfrac{c+d}{d+a}$, then:

$\textbf{(A) }a \text{ must equal }c\qquad\textbf{(B) }a+b+c+d\text{ must equal zero}\qquad$

$\textbf{(C) }\text{either }a=c\text{ or }a+b+c+d=0\text{, or both}\qquad$

$\textbf{(D) }a+b+c+d\ne 0\text{ if }a=c\qquad$

$\textbf{(E) }a(b+c+d)=c(a+b+d)$

Solution

Cross-multiplying gives:

$(a+b)(a+d) = (b+c)(c+d)$

$a^2 + ad + ab + bd = bc + bd + c^2 + cd$

$a^2 + ad + ab - bc - c^2 - cd = 0$

$a(a + b + d) - c(b+c+d)= 0$

This looks close to turning into option C, but we don't have a $c$ term in the first parentheses, and we don't have an $a$ term in the second parentheses to allow us to complete the factorization. However, if we both add $ac$ and subtract $ac$ on the LHS, we get:

$a(a + b + d) + ac - c(b+c+d) - ca= 0$

$a(a+b+d +c) - c(b+c+d+a) = 0$

$(a-c)(a+b+c+d) = 0$

This is equivalent to $\boxed{\textbf{(C)}}$

See Also

1964 AHSC (Problems • Answer Key • Resources)
Preceded by
Problem 31 		Followed by
Problem 33
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 • 31 • 32 • 33 • 34 • 35 • 36 • 37 • 38 • 39 • 40
All AHSME Problems and Solutions

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