Manipulating Reciprocals

Algebra Level 4

p p , q q , and r r are all positive numbers, and they all satisfy the following equations: p − 1 q = 3 p - \frac1q = 3 , q − 1 r = 4 q - \frac1r = 4 , r − 1 p = 5 r - \frac1p = 5 . Compute the value of p q r − 1 p q r pqr - \frac1{pqr} .


The answer is 72.

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Cody Johnson by Cody Johnson , 24, USA

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2 solutions

Ben Habeahan
Aug 19, 2015

p − 1 q = 3 … ( 1 ) q − 1 r = 4 … ( 2 ) r − 1 p = 5 … ( 3 ) p- \frac{ 1}{ q}=3 \dots (1) \\ q- \frac{ 1}{ r}=4 \dots (2) \\r- \frac{ 1}{ p}=5 \dots (3) \\ from ( 1 ) × ( 2 ) × ( 3 ) (1) \times (2) \times (3) we have, ( p − 1 q ) ( q − 1 r ) ( r − 1 p ) = 3.4.5 ( p q r − 1 p q r ) − ( p − 1 q ) − ( q − 1 r ) − ( r − 1 p ) = 60 … ( 4 ) (p- \frac{ 1}{ q}) (q- \frac{ 1}{ r}) (r- \frac{ 1}{ p}) = 3.4.5 \\ (pqr- \frac{ 1}{ pqr})-(p- \frac{ 1}{ q}) -(q- \frac{ 1}{ r})-(r- \frac{ 1}{ p}) =60 \dots(4) \\ substitution ( 1 ) , ( 2 ) a n d ( 3 ) t o ( 4 ) (1), (2) and (3) to (4) \\ p q r − 1 p q r − 3 − 4 − 5 = 60 p q r − 1 p q r = 72 pqr- \frac{ 1}{ pqr} -3-4-5=60 \\ pqr- \frac{ 1}{ pqr} = \boxed{72}

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add all 3:
p + q + r - 1/p - 1/q - 1/r = 12

multiply all 3:
pqr - 1/pqr - (12) = 60

Answer: 60+12 = 72

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