If x – q = s – y, what is the value of z?

Question

Tough and Tricky questions: Geometry. https://gmatclub.com/forum/download/file.php?id=24537 If x – q = s – y, what is the value of z? (1) xq + sy + sx + yq = zr (2) zq – ry = rx – zs 2014-10-28_1925.png

vietnammba · Accepted Answer

x-q = s-y => x+y = q+s => z=r
St1: (x+y)(q+s) = rz => (x+y) square = z square => (180-z) square = z square => z = 90. efficient
St2: r ( x+y) = z ( q+s) => inefficient

navkaran · Answer

Answer a)

The question x-q = s-y
Thus x+y = q + s
Thus Z = r too
1) take x and y common and and the. Take q+s common.
We get (x+y)(q+s)=z*r
Putting the above 2 equations from the question in this we get
(X+y)^2= z^2
X+y=z
180-z=z
Z=90
Thus 1) is sufficient

2)
Simplifying 2) we get
Z(q+s)=r(x+y)
Only using this we can get no answer.
This is no additional information also. Although can be used with 1) to get the answer, This we can easily conclude from the data in the question also.
Thus this information is not needed to find the answer.
2) is not needed

WoundedTiger · Answer

On simplifying the given equation we have x+y=s+q Also z=180-(x+y) or z=180-(s+q) or z=180-(r) St 1 says : q(x+y)+s(x+y)=z*r or (s+q)^2=zr or r^2=zr---> either r=0 or r=z..... Now r/neq{0} so r=z...sufficient St 2 says z(s+q)=r(x+y) or z=r or s+q=0... Now s+q /neq{0}....so z=r... Ans should be D...

Bunuel....Explanation please

MathRevolution · Answer

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem. Remember equal number of variables and independent equations ensures a solution.

If x – q = s – y, what is the value of z?

(1) xq + sy + sx + yq = zr 
(2) zq – ry = rx – zs
 
When you modify the original condition and the question, x-q=s-y becomes x+y=s+q, 180-z=180-r, z=r, and z=r=?. However, for 1), (q+s)+y(q+s)=zr=r^2, (x+y)(q+s)=r^2, (180-z)(180-r)=r^2, (180-r)^2=r^2 -> 180-r=r. 180-r=-r becomes 180=0, which is impossible. In 180-r=r, r=90, which is unique and sufficient.
For 2), it becomes z(q+s)=r(x+y), z=r. However, since this is only a restatement to the original condition, it is not sufficient. Therefore, the answer is A.

 Once we modify the original condition and the question according to the variable approach method 1, we can solve approximately 30% of DS questions.

shringi87 · Answer

On simplifying the given equation we have x+y=s+q Also z=180-(x+y) or z=180-(s+q) or z=180-(r) St 1 says : q(x+y)+s(x+y)=z*r or (s+q)^2=zr or r^2=zr---> either r=0 or r=z..... Now r/neq{0} so r=z...sufficient St 2 says z(s+q)=r(x+y) or z=r or s+q=0... Now s+q /neq{0}....so z=r... Ans should be D...

Bunuel can you explain why the D will not be the right if we go by above approach ?

Bunuel · Answer

For (2): z/(x + y) = r/(q + s).

Consider the following cases:

z = r = 20 and x + y = q + s = 160.
z = r = 30 and x + y = q + s = 150

Raksat · Answer

x – q = s – y
=> x+y =s+q
=>180-z=180-r
=> z=r
statement 1
xq + sy + sx + yq = zr 
(x+y)(s+q)=zr
(s+q)^2=r^2
(180-r)^2=r ^2
=>r=90 (sufficient)
statement 2
zq – ry = rx – zs
z(s+q)=r(x+y)
=> z=r not sufficient 
Hence option A is correct

Mo2men · Answer

Analyzing question stem:

x – q = s – y

x + y = s + q

But in every triangle the sum of it angles = 180

x + y = 180 - z

s + q = 180 - r

Hence,
 
180 - z = 180 - r

z = r

What is value of z??

(1) xq + sy + sx + yq = zr

xq + yq + sy + sx = zr

q (x + y ) + s (x + y ) = zr

(x + y ) (q +s) = zr

From question stem: x +y = s + q & r = z

(x + y ) (x + y ) = zr

(x + y)^2 = z^2...Take square root for both sides

x + y = z

But 180 - z = x + y...then

180 - z = z

Z =90

Sufficient

(2) zq – ry = rx – zs

zq + zs = rx + ry

z (q + s) = r ( x + y).........but z = r, so cancel out

q + s = x + y...same info in question stem

Insuffeicnet

Answer: A

MrJglass · Answer

In the second statement, Why can't we say that since 
z/r = (x+y)/(s+q)
Then, z=(x+y)
???

Please help, I need to resolve this. 
Thanks!

Posted from my mobile device

amanvermagmat · Answer

Hello

If z/r = (x+y)/(s+q) then how did you arrive at z = (x+y)? 
Basically you have cancelled 'r' from one side and 's+q' from the other side. So that means you have assumed r = s+q. But why? All we know is that s+q = 180-r, since sum of angles of a triangle is 180 degrees.

MrJglass · Answer

Oooh yeeeeaa, now I understand.
The last time I solved this, I was so sure of why statement 1 was correct, but today I became confused. Thanks for the clarification.

My maths knowledge just increased by 0.00000000001%, I guess.

Thanks a lot!

Fdambro294 · Answer

The key is at the beginning to figure out that it must be the case that Angle R = Angle Z

X - Q = S - Y

Using the property that the 3 Interior Angles of each triangle SUM to equal = 180 degrees

X = 180 - Y - Z

Q = 180 - S - R

Subtract the 2 Equations to get (X - Q)

X - Q = -Y - Z + S + R

X - Q = (S - Y) + R - Z

Since we are Given that: X - Q = S - Y

the only way this Given can be True is if the Expression at the End -----> R - Z = 0

If R - Z = 0

then: R = Z

What is the Value of Z (or R) = ?

S1: XQ + SY + SX + YQ = ZR

XQ + SX + SY + YQ = ZR

X(Q + S) + Y(S + Q) = ZR

(X + Y) (S + Q) = ZR --- eq1

X + Y + Z = 180
X + Y = 180 - Z ---- eq2

S + Q + R = 180
S + Q = 180 - R ---- eq3

Substituting eq2 and eq3 into eq1:

(180 - Z)(180 - R) = ZR

Since we found that Z = R above, we can substitute in Z for every instance of R:

(180 - Z)(180 - Z) = (Z)^2

(180)^2 - 360Z + (Z)^2 = (Z)^2

---subtract (Z)^2 from both sides----

(180)(180) = 360Z

180 = 2Z

Z = 90 degrees

S1 Sufficient

S2: After performing the Algebra, since we know that Z = R, S2 does not give us any new information to use

ZQ + ZS = RX + RY

Z(Q + S) = R(X + Y)

Q + S = 180 - R

X + Y = 180 - Z

so we have:

Z(180 - R) = R(180 - Z)

and again because Z = R

Z(180 - Z) = Z(180 - Z)

S2 is NOT sufficient to answer the Value of Z

(A) S1 Alone is sufficient

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If x – q = s – y, what is the value of z?

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If x – q = s – y, what is the value of z?

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