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# Two different positive numbers a and b each differ from their reciproc

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Two different positive numbers a and b each differ from their reciproc  [#permalink]

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17 Nov 2018, 10:01
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95% (hard)

Question Stats:

14% (02:43) correct 86% (02:01) wrong based on 57 sessions

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GMATbuster's Weekly Quant Quiz#9 Ques #2

Two different positive numbers a and b each differ from their reciprocals by 1. What is a+b ?
A) 1
B) 2
C) $$\sqrt{5}$$
D) $$\sqrt{6}$$
E) 3

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Re: Two different positive numbers a and b each differ from their reciproc  [#permalink]

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Updated on: 17 Nov 2018, 12:18
a - (1/a) = 1 or (1/a) - a = 1, since the question says the numbers differ by 1.

which gives, a^2 - 1 = a or a^2 - a - 1 = 0 OR 1 - a^2 = a or a^2 + a - 1 = 0

Solutions to the 1st quadratic equation are: [1 + sqrt(5)]/2 and [1 - sqrt(5)]/2

Solutions to the 2nd quadratic equation are: [-1 + sqrt(5)]/2 and [-1 - sqrt(5)]/2

Question says that a and b are positive integers, so out of the above 4 solutions, the positive ones are [1 + sqrt(5)]/2 and [-1 + sqrt(5)]/2, which must be a and b.

Hence a + b = 0.5 + sqrt(5)/2 - 0.5 + sqrt(5)/2 = 2*sqrt(5)/2 = sqrt(5)

Option C!
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Originally posted by nkin on 17 Nov 2018, 10:08.
Last edited by nkin on 17 Nov 2018, 12:18, edited 1 time in total.
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Re: Two different positive numbers a and b each differ from their reciproc  [#permalink]

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17 Nov 2018, 10:22
Each number a and b differs from its reciprocal by 1.
this implies,
a-(1/a)=1 ------ eq 1
b-(1/b)=1 -------eq 2
on solving eq 1
(a^2)-1=a
a=(1+\sqrt{5})/2 or (1+\sqrt{5})/2
similarly b=(1+\sqrt{5})/2 or (1-\sqrt{5})/2
when trying all 4 possibilities for a+b
(i.e) a+b=(1+\sqrt{5})/2+(1+\sqrt{5})/2 = 1+\sqrt{5}
or a+b= (1+\sqrt{5})/2+(1-\sqrt{5})/2=\sqrt{5}
or a+b=(1-\sqrt{5})/2+(1-\sqrt{5})/2=1-\sqrt{5}
or a+b=(1-\sqrt{5})/2+(1+\sqrt{5})/2=\sqrt{5}
The only option which satisfies one of the possibilities is C
IMO: OPTION C

NOTE: EVEN IF EQUATION ONE IS TAKEN AS (1/a)-a=1 AND EQUATION TWO AS (1/b)-b=1, the answer remains the same
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Re: Two different positive numbers a and b each differ from their reciproc  [#permalink]

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17 Nov 2018, 10:32
a-(1/a)=1
a^2-a=1
b-(1/b)=1
b^2-b=1
Subtracting both
(a+b)(a-b)-(a-b)= 0
(a-b)(a+b-1)=0
Since a and b are different numbers thus (a-b) cant be 0
Hence (a+b-1)=0
Thus (a+b)= 1
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Re: Two different positive numbers a and b each differ from their reciproc  [#permalink]

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17 Nov 2018, 22:32
We have -

a^2 - a = b^2 - b
(a-b)(a+b-1) = 0

As a,b are different numbers
a+b = 1

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Re: Two different positive numbers a and b each differ from their reciproc  [#permalink]

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19 Nov 2018, 07:07

Official Solution

Attachment:

Sol2.jpg [ 62.3 KiB | Viewed 512 times ]

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Re: Two different positive numbers a and b each differ from their reciproc  [#permalink]

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19 Nov 2018, 08:11
Similar language was also used in a question in which difference between two quantities was written as |x-y|.
So, now I think whenever we are asked to difference between x and y, write it as |x-y|, not as x-y.

Please experts, appraise my understanding to get correct answer in future based on this word problems.
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Two different positive numbers a and b each differ from their reciproc  [#permalink]

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19 Nov 2018, 08:15
Hi gvij2017

You are right.
The difference between two quantities x and y is written as |x-y|
Difference is always non negative.

gvij2017 wrote:
Similar language was also used in a question in which difference between two quantities was written as |x-y|.
So, now I think whenever we are asked to difference between x and y, write it as |x-y|, not as x-y.

Please experts, appraise my understanding to get correct answer in future based on this word problems.

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Two different positive numbers a and b each differ from their reciproc   [#permalink] 19 Nov 2018, 08:15
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