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If a + b = x, and a – b = y, then ab =

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If a + b = x, and a – b = y, then ab =  [#permalink]

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New post 25 Feb 2017, 21:58
2
00:00
A
B
C
D
E

Difficulty:

  25% (medium)

Question Stats:

77% (01:28) correct 23% (01:51) wrong based on 102 sessions

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If a + b = x, and a – b = y, then ab =

(A) (x^2 – y^2)/2
(B) (x + y)(x – y)/2
(C) (x^2 – y^2)/4
(D) xy/2
(E) (x^2 + y^2)/4
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Re: If a + b = x, and a – b = y, then ab =  [#permalink]

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New post 25 Feb 2017, 22:56
1
2
vikasp99 wrote:
If a + b = x, and a – b = y, then ab =

(A) x^2 – y^2/2
(B) (x + y)(x – y)/2
(C) x^2 – y^2/4
(D) xy/2
(E) x^2 + y^2/4


a + b = x
\((a+b)^2 = x^2\)
\(a^2 + b^2 + 2ab = x^2\) ---------------I

a - b = y
\((a-b)^2 = y^2\)
\(a^2 + b^2 - 2ab = y^2\) ---------------II

I - II
\(4ab = x^2 - y^2\)
\(ab = (x^2 - y^2)/4\)

vikasp99 : Could you please add brackets to the numerators of the options as it will make the options more clear ?

Hence option C is correct
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Re: If a + b = x, and a – b = y, then ab =  [#permalink]

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New post 26 Feb 2017, 02:33
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Re: If a + b = x, and a – b = y, then ab =  [#permalink]

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New post 26 Feb 2017, 02:34
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Re: If a + b = x, and a – b = y, then ab =  [#permalink]

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New post 26 Feb 2017, 06:46
Point to Note:

a*b = [(a+b)^2 - (a-b)^2 ] /4
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Re: If a + b = x, and a – b = y, then ab =  [#permalink]

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New post 26 Feb 2017, 07:18
vikasp99 wrote:
If a + b = x, and a – b = y, then ab =

(A) (x^2 – y^2)/2
(B) (x + y)(x – y)/2
(C) (x^2 – y^2)/4
(D) xy/2
(E) (x^2 + y^2)/4


Hi,

Squaring on both side

\((a + b)^2 = x^2\), and \((a – b)^2 = y^2\)

On simplification

\(x^2-y^2=4ab\)

\(ab=(x^2-y^2)/4\)

Hence C
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If a + b = x, and a – b = y, then ab =  [#permalink]

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New post 26 Feb 2017, 09:52
vikasp99 wrote:
If a + b = x, and a – b = y, then ab =

(A) (x^2 – y^2)/2
(B) (x + y)(x – y)/2
(C) (x^2 – y^2)/4
(D) xy/2
(E) (x^2 + y^2)/4


adding, 2a=x+y
subtracting, 2b=x-y
multiplying, 4ab=(x+y)(x-y)
ab=(x^2-y^2)/4
C
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Re: If a + b = x, and a – b = y, then ab =  [#permalink]

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New post 12 Mar 2018, 07:08
Does anybody mind explaining to me why I should automatically think to square "a+b=x" & "a-b=y" upon seeing these clues? What about the given information should indicate to a test taker "Ya know what... lets square those clues in order to solve this problem"?
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Re: If a + b = x, and a – b = y, then ab =  [#permalink]

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New post 12 Mar 2018, 11:41
1
aaronhew wrote:
Does anybody mind explaining to me why I should automatically think to square "a+b=x" & "a-b=y" upon seeing these clues? What about the given information should indicate to a test taker "Ya know what... lets square those clues in order to solve this problem"?


[(a+b)^2 - (a-b)^2 ] = 4*a*b

[(a+b)^2 + (a-b)^2 ] = a^2 + b^2

These are two of the relevant and standard formula to be at your fingertips. As far as this problem is concerned, asking for the value of a*b is the obvious clue and there is nothing more to research on this problem.
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Re: If a + b = x, and a – b = y, then ab =  [#permalink]

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New post 12 Mar 2018, 11:46
\(a+b = x\)
\(a-b = y\)

when we square both .. we will have the squared terms in positive and the product of a& b term with opposite signs in the two equations.

We can subtract to get rid of the quadratic terms

so 1 - 2 leads to

\(4ab = x^2 - y^2\)

And the answer is C.
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If a + b = x, and a – b = y, then ab =  [#permalink]

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New post 12 Mar 2018, 13:10
1
vikasp99 wrote:
If a + b = x, and a – b = y, then ab =

(A) (x^2 – y^2)/2
(B) (x + y)(x – y)/2
(C) (x^2 – y^2)/4
(D) xy/2
(E) (x^2 + y^2)/4

aaronhew wrote:
Does anybody mind explaining to me why I should automatically think to square "a+b=x" & "a-b=y" upon seeing these clues? What about the given information should indicate to a test taker "Ya know what... lets square those clues in order to solve this problem"?

aaronhew :lol: :lol: :lol: Hilarious. Levity is good.

You may know the algebraic identities but may not have seen the clue.
Bunuel lists them here, Algebraic Identities (scroll down)

If you draw a blank, another way to answer is to assign values.

Let a = 3, b = 2
x = (a + b) = (3 + 2) = 5
y = (a - b) = (3 - 2) = 1

(ab) = (3*2) = 6

Using x = 5 and y = 1, find the answer that yields 6

(A) (x^2 – y^2)/2: \(\frac{(5^2-1^2)}{2}=\frac{24}{2}=12\). NO

(B) (x + y)(x – y)/2: \(\frac{(5+1)(5-1)}{2}=\frac{24}{2}=12\). NO

(C) (x^2 – y^2)/4: \(\frac{(5^2-1^2)}{4}=\frac{24}{4}=6\). MATCH

(D) xy/2: \(\frac{(5*1)}{2}=\frac{5}{2}\). NO

(E) (x^2 + y^2)/4: \(\frac{(5^2+1^2)}{4}=\frac{26}{4}=\frac{13}{2}\). NO

Answer C
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