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# If a and b are both positive, is a < 5 < b?

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Math Expert
Joined: 02 Sep 2009
Posts: 51280
If a and b are both positive, is a < 5 < b?  [#permalink]

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07 May 2018, 02:03
00:00

Difficulty:

45% (medium)

Question Stats:

69% (01:38) correct 31% (01:14) wrong based on 61 sessions

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If a and b are both positive, is a < 5 < b?

(1) a < b and ab = 25
(2) a^2 < 25 < b^2

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Joined: 08 Aug 2017
Posts: 279
Re: If a and b are both positive, is a < 5 < b?  [#permalink]

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07 May 2018, 02:31
both a and b are positive.

Statement 1
ab=25 or ab=5x5 but a<b so a<5<b
Sufficient

Statement 2
a2<25<b2
a<5 & b >5
So a<5<b

Please let me know if I am wrong.
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Re: If a and b are both positive, is a < 5 < b?  [#permalink]

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07 May 2018, 10:42
1
If a and b are both positive, is a < 5 < b?

(1) a < b and ab = 25
(2) a^2 < 25 < b^2

Solution :

a & b are positive. It can be any positive number

Statement 1 :- for simplicity of calculation I am using fraction.

a<b & ab = 25 ......

Few possible values of a and b are -
1)
a= 5/7 and b = 35. it satisfies the condition of statement 1.
Lets evaluate if it answers our question.

5/7 <25 < 35....yes it does....

2)
a= 5/2 and b = 10. it satisfies the condition of statement 1.
Lets evaluate if it answers our question.

5/2 <25< 10....yes it does.

Hence Statement 1 satisfies.

Statement 2:-

We can use the same number that we used for statement 1. We will get the same result.

Statement 2 also satisfies.

Hence D.

Please let me know if I am missing any important aspect to solve this correctly.

Thanks.
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Re: If a and b are both positive, is a < 5 < b?  [#permalink]

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07 Aug 2018, 00:27
Bunuel wrote:
If a and b are both positive, is a < 5 < b?

(1) a < b and ab = 25
(2) a^2 < 25 < b^2

Given, a>0 and b>0

Question stem:- Is a < 5 < b?

St1:- a < b and ab = 25

We have , a<b
Or, a*b < b*b
Or, $$ab < b^2$$
Or, 25 < b^2[/m]------------(1)

Similarly, a < b
Or, a*a < b*a
Or, $$a^2 < ab$$
Or, $$a^2 < 25$$--------(2)

Note:- We were able to multiply variables on both sides91 of the inequality since a and b are positive.
On the same reasoning, we will take the square root of equation(1) and (2).

From (1) and (2), we have,
$$a^2 < 25 < b^2$$-----------(3)
Or, $$\sqrt{a^2} < \sqrt{25} < \sqrt{b^2}$$
Or, a < 5 < b

Sufficient.

St2:- Given, $$a^2 < 25 < b^2$$
Applying square root operation on both sides.
$$\sqrt{a^2} < \sqrt{25} < \sqrt{b^2}$$
Or, a < 5 < b

Sufficient.

Ans. (D)

P.S:- We could solve it without number plug-in.
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PKN

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Re: If a and b are both positive, is a < 5 < b? &nbs [#permalink] 07 Aug 2018, 00:27
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