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# Does a^2 + b^2 = 2ab?

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Does a^2 + b^2 = 2ab?  [#permalink]

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19 Sep 2017, 01:22
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55% (hard)

Question Stats:

58% (01:30) correct 42% (01:43) wrong based on 149 sessions

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Does a^2 + b^2 = 2ab?

(1) a + b = 3
(2) ab > 0

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Does a^2 + b^2 = 2ab?  [#permalink]

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Updated on: 19 Sep 2017, 03:48
Bunuel wrote:
Does a^2 + b^2 = 2ab?

(1) a + b = 3
(2) ab > 0

a^2 + b^2 = 2ab
a^2 + b^2 - 2ab = 0
(a-b)^2 = 0
Is a equal to b?

(1) a,b can be 3/2,3/2 or 2,1..not sufficient

(2) ab can be equal (1,1) and cannot be equal (3,2) same sign
not sufficient

on combining 3/2,3/2 or 2,1 give different answers

E

P.S. Second attempt
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Luckisnoexcuse

Originally posted by Luckisnoexcuse on 19 Sep 2017, 01:30.
Last edited by Luckisnoexcuse on 19 Sep 2017, 03:48, edited 1 time in total.
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Does a^2 + b^2 = 2ab?  [#permalink]

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19 Sep 2017, 02:46
1
Luckisnoexcuse wrote:
Bunuel wrote:
Does a^2 + b^2 = 2ab?

(1) a + b = 3
(2) ab > 0

a^2 + b^2 = 2ab
a^2 + b^2 - 2ab = 0
(a-b)^2 = 0
Is a equal to b?

(1) a is not equal to b..sufficient

(2) ab can be equal (1,1) and cannot be equal (3,2)
not sufficient

A

The ques basically asks whether a=b..

with 1) a+b=3, we can have it as yes ( a=1.5 and b=1.5) or no (a=1 and b=2)
So A can not be the answer.

with 2, both a and b have the same sign. Not sufficient.

With 1 and 2, we know that a+b=3 and a & b are of the same sign.

a = 1 and b=2 (same sign) we get a NO
a=1.5 and b=1.5 (same sign) we get a YES.

i think the answer should be E.

chetan2u, any help would be appreciated.
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Re: Does a^2 + b^2 = 2ab?  [#permalink]

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23 Sep 2017, 22:04
Bunuel wrote:
Does a^2 + b^2 = 2ab?

(1) a + b = 3
(2) ab > 0

I will go with E

because ...
(1)
(a+b)^2 = a^2 + 2ab + b^2
9 = a^2 + b^2 + 2ab

different values will satisfy this equation..
not sufficient

(2) ab > 0
so what? not sufficient

(1) + (2)
even 2 statements together do not help much...
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Does a^2 + b^2 = 2ab?  [#permalink]

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20 Feb 2018, 10:32
Bunuel wrote:
Does a^2 + b^2 = 2ab?

(1) a + b = 3
(2) ab > 0

Simplying the question stem,
$$a^2 + b^2 - 2ab = 0$$ -> $$a^2 + b^2 + 2ab - 4ab = 0$$ -> $$(a + b)^2 - 4ab = 0$$

The information given in either statement is not enough separately, but when we combine the
information from both the statements,

the question stem becomes $$9 - 4ab = 0$$ where ab > 0

However, 2 cases are possible

Case 1: ab = 2.25, then the equation is true!
Case 2: ab = 3, the equation is not true (Insufficient - Option E)
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Does a^2 + b^2 = 2ab?  [#permalink]

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20 Feb 2018, 17:01
2
Top Contributor
Bunuel wrote:
Does a² + b²= 2ab?

(1) a + b = 3
(2) ab > 0

Target question: Does a² + b²= 2ab?

This is a good candidate for REPHRASING the target question.
Take a² + b²= 2ab and subtract 2ab from both sides to get: a² - 2ab + b²= 0
Factor to get: (a - b)(a - b) = 0
Solve to get: a - b = 0
In other words, a = b

So, we get....
REPHRASED target question: Does a = b?

NOTE: Once we REPHRASE the target question, the statements should be easy to analyze.

Statement 1: a + b = 3
There are several values of a and b that satisfy statement 1. Here are two:
Case a: a = 1 and b = 2. In this case, the answer to the REPHRASED target question is NO, a does NOT equal b
Case b: a = 1.5 and b = 1.5. In this case, the answer to the REPHRASED target question is YES, a DOES equal b
Since we cannot answer the REPHRASED target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: ab > 0
There are several values of a and b that satisfy statement 2. Here are two:
Case a: a = 1 and b = 2 (these values satisfy the condition that ab > 0). In this case, the answer to the REPHRASED target question is NO, a does NOT equal b
Case b: a = 1.5 and b = 1.5. In this case, the answer to the REPHRASED target question is YES, a DOES equal b
Since we cannot answer the REPHRASED target question with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined
IMPORTANT: Notice that I was able to use the same counter-examples to show that each statement ALONE is not sufficient. So, the same counter-examples will satisfy the two statements COMBINED.
In other words,
Case a: a = 1 and b = 2. In this case, the answer to the REPHRASED target question is NO, a does NOT equal b
Case b: a = 1.5 and b = 1.5. In this case, the answer to the REPHRASED target question is YES, a DOES equal b
Since we cannot answer the REPHRASED target question with certainty, the combined statements are NOT SUFFICIENT

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Re: Does a^2 + b^2 = 2ab?  [#permalink]

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23 Feb 2018, 10:21
1
Bunuel wrote:
Does a^2 + b^2 = 2ab?

(1) a + b = 3
(2) ab > 0

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.

The first step of the VA (Variable Approach) method is to modify the original condition and the question, and then recheck the question.

We can modify the question in the following way.

$$a^2 + b^2 = 2ab$$
$$⇔ a^2 - 2ab + b^2 = 0$$
$$⇔ (a-b)^2 = 0$$
$$⇔ a = b$$

Since we have 2 variables (x and y) and 0 equations, C is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) & 2) :
$$a = 1.5$$, $$b = 1.5$$ : The answer is Yes
$$a = 1$$, $$b = 2$$: The answer is No.

Since we don't have a unique solution, both conditions together are not sufficient.

Normally, in problems which require 2 equations, such as those in which the original conditions include 2 variables, or 3 variables and 1 equation, or 4 variables and 2 equations, each of conditions 1) and 2) provide an additional equation. In these problems, the two key possibilities are that C is the answer (with probability 70%), and E is the answer (with probability 25%). Thus, there is only a 5% chance that A, B or D is the answer. This occurs in common mistake types 3 and 4. Since C (both conditions together are sufficient) is the most likely answer, we save time by first checking whether conditions 1) and 2) are sufficient, when taken together. Obviously, there may be cases in which the answer is A, B, D or E, but if conditions 1) and 2) are NOT sufficient when taken together, the answer must be E.
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Re: Does a^2 + b^2 = 2ab?  [#permalink]

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13 Sep 2019, 03:17
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