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# If 0 < a < b is a < 10?

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Math Expert
Joined: 02 Sep 2009
Posts: 55266
If 0 < a < b is a < 10?  [#permalink]

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21 Aug 2017, 06:41
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Difficulty:

75% (hard)

Question Stats:

50% (02:15) correct 50% (01:45) wrong based on 95 sessions

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If 0 < a < b is a < 10?

(1) 1/a + 1/b = 5

(2) 1/a > 1/10

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Joined: 02 Aug 2009
Posts: 7684
If 0 < a < b is a < 10?  [#permalink]

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21 Aug 2017, 09:40
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1
Bunuel wrote:
If 0 < a < b is a < 10?

(1) 1/a + 1/b = 5

(2) 1/a > 1/10

hi...
0<a<b so $$\frac{1}{a}>\frac{1}{b}$$

statement I
$$\frac{1}{a} + \frac{1}{b} = 5$$....
both a and b are positive and a<b, so $$\frac{1}{a}>\frac{1}{b}$$..

lets find slightly MORE than the MAX possible value of a........ so let a=b
so $$\frac{1}{a} + \frac{1}{a}>\frac{1}{a} + \frac{1}{b}.............\frac{2}{a} > 5......a<\frac{2}{5}$$
so a HAS TO BE less than 10.... a<10
sufficient

statement II
$$\frac{1}{a}>\frac{1}{10}$$
cross multiply..
$$a<10$$
sufficient

D
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If 0 < a < b is a < 10?  [#permalink]

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21 Aug 2017, 10:09
2
Bunuel wrote:
If 0 < a < b is a < 10?

(1) 1/a + 1/b = 5

(2) 1/a > 1/10

Statement 1: as $$a$$ & $$b$$ are positive and $$a<b$$, so reciprocal of $$a$$ & $$b$$ will be $$\frac{1}{a}>\frac{1}{b}$$ ------(1)
Now adding $$\frac{1}{a}$$ to both sides of eq.1

therefore $$\frac{1}{a}$$ $$+$$ $$\frac{1}{a}$$$$>$$$$\frac{1}{b}$$ $$+$$ $$\frac{1}{a}$$, or

$$\frac{2}{a}>5$$ (as $$\frac{1}{a} +\frac{1}{b} =5$$). Hence $$a<\frac{2}{5}$$
Therefore $$a<10$$. Sufficient

Statement 2: $$\frac{1}{a}>\frac{1}{10}$$, cross multiplying
$$a<10$$. Sufficient

Option $$D$$
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Re: If 0 < a < b is a < 10?  [#permalink]

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18 Mar 2018, 05:44
Bunuel wrote:
If 0 < a < b is a < 10?

(1) 1/a + 1/b = 5

(2) 1/a > 1/10

(2) 1/a > 1/10

We can flip the fractions and hence flip the the sign

a < 10...Answer is always Yes

Sufficient

(1) 1/a + 1/b = 5

Let's see a pattern

Let b = 1/2......1/a = 5 - 2 =3......a =1/3..............b>a & a<10...........Answer is Yes

Let b = 1...........1/a= 5 - 1 =4.......a =1/4.............b>a & a<10...........Answer is Yes

Let b = 5...........1/a = 5 - 1/5 =24/5......5/24..........b>a & a<10...........Answer is Yes

We can see a pattern here

Let b = 10.........1/a = 5 - 1/10 =49/5......5/49..........b>a & a<10...........Answer is Yes......More smaller

As B gttes higher ......a gets smaller............So a<10 is always Yes

Sufficient

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Re: If 0 < a < b is a < 10?  [#permalink]

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20 Mar 2018, 22:58
Bunuel wrote:
If 0 < a < b is a < 10?

(1) 1/a + 1/b = 5

(2) 1/a > 1/10

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

If a quetions is an inequality, inequalities in the original conditions are considered as equations.
Since we have 2 variables ( a and b ) and 1 equation, D is most likely to be the answer. So, we should consider each of the conditions on their own first.

Since 0 < a < b, we have 1/a > 1/b > 0

Condition 1)
Since 1/a + 1/b = 5 and 1/a > 1/b, we have 1/a > 5/2 or a < 2/5.
Thus a < 2/5 < 10.
Condition 1) is sufficient.

Condition 2)
Since a > 0 and 1/a > 1/10, we have a < 10.
Condition 2) is sufficient.

If the original condition includes “1 variable”, or “2 variables and 1 equation”, or “3 variables and 2 equations” etc., one more equation is required to answer the question. If each of conditions 1) and 2) provide an additional equation, there is a 59% chance that D is the answer, a 38% chance that A or B is the answer, and a 3% chance that the answer is C or E. Thus, answer D (conditions 1) and 2), when applied separately, are sufficient to answer the question) is most likely, but there may be cases where the answer is A,B,C or E.
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If 0 < a < b is a < 10?  [#permalink]

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24 Mar 2018, 14:44

Solution

• We are given that 0 < a < b, hence, 1/a> 1/b and both are positive numbers.

Statement-1$$\frac{1}{a} + \frac{1}{b} = 5$$

• $$\frac{1}{a}$$ + $$\frac{1}{b}$$ = $$5$$
• On adding $$\frac{1}{a}$$ to both sides of the inequality $$\frac{1}{a}> \frac{1}{b}$$, we get:
o $$\frac{1}{a} +\frac{1}{a} > \frac{1}{a} + \frac{1}{b}$$
o $$\frac{2}{a} > 5$$
o Since $$a$$ is a positive number, then, multiplying both sides of the inequality by$$a$$, we get:
 $$2 > 5a$$
 $$a < \frac{2}{5}$$, which is less than $$10$$
.

Hence, Statement 1 alone is sufficient to answer the question.

Statement-2$$\frac{1}{a} > \frac{1}{10}$$”.

• If $$\frac{1}{a}$$> $$\frac{1}{10}$$ then $$a<10$$.

Hence, Statement 2 alone is sufficient to answer the question.

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If 0 < a < b is a < 10?   [#permalink] 24 Mar 2018, 14:44
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