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If a, b, c, and d are positive numbers and a^(1/2) + b^(1/2) = c^(1/2)

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Bunuel
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If a, b, c, and d are positive numbers and a^(1/2) + b^(1/2) = c^(1/2) [#permalink] New post   24 Feb 2020, 02:14
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If a, b, c, d, and e are positive numbers and \(\sqrt{a} + \sqrt{b} = \sqrt{c} + \sqrt{d} + \sqrt{e}\), is a < c?

(1) c = d

(2) \(\sqrt{b}+\sqrt{d}<\sqrt{e}\)


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Re: If a, b, c, and d are positive numbers and a^(1/2) + b^(1/2) = c^(1/2) [#permalink] New post   24 Feb 2020, 04:35
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a,b,c,d,e are all positive
Q. is a < c?

(1) We can only deduce from c=d that √a+√b=2√c+√e. We cannot determine whether a < c
NOT SUFFICIENT

(2) √a+√b +√d = √c+√d+√e +√d
If √b+√d < √e, then √a > √c + 2√d --> 2√d is positive, thus √a > {√c + positive}.
We can determine that it must be true that √a > √c and therefore, a > c
SUFFICIENT

FINAL ANSWER IS (B)
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Re: If a, b, c, and d are positive numbers and a^(1/2) + b^(1/2) = c^(1/2) [#permalink] New post   24 Feb 2020, 09:15
given
√a+√b=√c+√d+√e determine whether a<c
#1
c=d
we can have
√9+√4= √4+√4+√1
3+2= 2+2+1
no
and
√4+√9= √4+√4+√1
we get yes
insufficient
#2
√b+√d=√e
in that case √e>√d
so
for any value of b,d ,e
to get √a+√b=√c+√d+√e
a has to be greater than c is not >a
IMO B; sufficient

If a, b, c, d, and e are positive numbers and a‾√+b‾√=c√+d‾‾√+e√a+b=c+d+e, is a < c?

(1) c = d

(2) b‾√+d‾‾√<e√
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Re: If a, b, c, and d are positive numbers and a^(1/2) + b^(1/2) = c^(1/2) [#permalink] New post   24 Feb 2020, 10:47
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Quote:
If a, b, c, d, and e are positive numbers and √a+√b=√c+√d+√e, is a < c?

(1) c = d

(2) √b+√d<√e


(1) insufic
c=d
√a+√b=2√c+√e

(2) sufic
√b+√d<√e
if √b<√e, then √a>√b
√a+√b>√c+√d+(√b+√d)
√a>√c+2√d
√a>√c
|a|>|c|
a>c

Ans (B)
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Re: If a, b, c, and d are positive numbers and a^(1/2) + b^(1/2) = c^(1/2) [#permalink] New post   24 Feb 2020, 14:28
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If a, b, c, d, and e are positive numbers and a√+b√=c√+d√+e√a+b=c+d+e, is a < c?

(1) c = d

(2) b√+d√<e√

1) In this case whether a < c depends on the values of b and e. If e > b , then a will be greater than c and vice versa. Again, if these are fractions, then the opposite case is possible as square root of a fraction is greater than the fraction itself. so not sufficient.

2) since \(b^(1/2)\) is less than e^(1/2), so to make the equation , a^(1/2) will be greater than c^(1/2). so a > c. sufficiemt.

B is the correct answer.
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If a, b, c, and d are positive numbers and a^(1/2) + b^(1/2) = c^(1/2) [#permalink] New post   25 Feb 2020, 00:00
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If a, b, c, d, and e are positive numbers and √a + √b=√c +√d +√e, is a < c ?

(Statement1): c = d
--> √a +√b=√c +√c +√e --> √a +√b=2√c +√e
√1 +√16=2√4 +√1 (1 < 2 YES)
√16 +√1=2√4 +√1 (4 < 2 NO)
Insufficient

(Statement2): √b +√d < √e
√b +√d < √a +√b —√c —√d
√c < √a —2√d --> √c + 2√d < √a
--> Since a, c and d are positive numbers, --> a < c.
Sufficient

The answer is B.
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Re: If a, b, c, and d are positive numbers and a^(1/2) + b^(1/2) = c^(1/2) [#permalink] New post   23 Jun 2022, 08:26
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Re: If a, b, c, and d are positive numbers and a^(1/2) + b^(1/2) = c^(1/2) [#permalink]
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