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# If x and z are positive integers and x + z = 13 what is the ratio of x

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Math Expert
Joined: 02 Sep 2009
Posts: 49968
If x and z are positive integers and x + z = 13 what is the ratio of x  [#permalink]

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03 Jan 2018, 21:12
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45% (medium)

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67% (01:23) correct 33% (01:04) wrong based on 81 sessions

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If x and z are positive integers and x + z = 13 what is the ratio of x to z?

(1) xz = 40
(2) z > 2

_________________
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Joined: 25 Feb 2013
Posts: 1217
Location: India
GPA: 3.82
If x and z are positive integers and x + z = 13 what is the ratio of x  [#permalink]

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04 Jan 2018, 11:01
Bunuel wrote:
If x and z are positive integers and x + z = 13 what is the ratio of x to z?

(1) xz = 40
(2) z > 2

$$x+z=13 => \frac{x}{z}=\frac{13}{z}-1$$. Hence we need to know the value of $$z$$ to get the ratio. Here $$z<13$$ because $$x$$ & $$z$$ are positive integers

Statement 1: $$z$$ can have multiple values depending upon the value of $$x$$. if $$x=5$$, $$z=8$$ & if $$x=8$$, $$z=5$$. Insufficient

Statement 2: this implies $$2<z<13$$, hence $$z$$ can take multiple integer values. Insufficient

Combining 1 & 2: if $$z=8$$, $$x=5$$ & if $$z=5$$, $$x=8$$. Insufficient

Option E
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Re: If x and z are positive integers and x + z = 13 what is the ratio of x  [#permalink]

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04 Jan 2018, 11:27
Bunuel wrote:
If x and z are positive integers and x + z = 13 what is the ratio of x to z?

(1) xz = 40
(2) z > 2

x + z = 13, so we can say that z = 13 - x

(1) x*z = x(13-x) = 40 Or 13x - x^2 = 40 Or x^2 - 13x + 40 = 0
Factorising this quadratic equation, we get: (x-5)(x-8) = 0. So x can be either 5 (which will give z=8) or x can be 8 (which will give z=5).
Thus ratio of x:z could be either 5:8 or 8:5. So Not sufficient.

(2) This doesnt tell us anything about x, so definitely Not Sufficient.

Combining the two statements, we have two cases for x & z respectively as 5,8 and 8,5. And in both cases z is > 2. So still two cases possible. Hence still Not Sufficient.

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Joined: 06 Jan 2018
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Re: If x and z are positive integers and x + z = 13 what is the ratio of x  [#permalink]

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07 Jan 2018, 09:59
Given: x + z = 13
1. xz = 40
2. z > 2

1.
Multiply x + z = 13 with x
x^2 + xz = 13x
x^2 - 13x + 40 = 0
x = 8 or 5

if x = 5 then z = 8 and ratio = 5/8 OR if x = 8 then z = 5 and ration = 8/5. So stmt 1 is insufficient.
(At this point I am thinking of a clue from stmt 2)

stmt 2 just tells me z > 2, so x + z = 13 will give me many possible values. Can' conclude on stmt 2 alone

Combine 1 + 2 > (x=5,z=8) or (x=8,z=5). And z > 2, which is true for both z=5 and z=8. Can't conclude on stmt 1&2

Re: If x and z are positive integers and x + z = 13 what is the ratio of x &nbs [#permalink] 07 Jan 2018, 09:59
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