A, B, R and S are four positive numbers. Does AS = BR?

Question

Statement #1:  \sqrt{A^2 + B^2} = \sqrt{R^2 + S^2} 
Statement #2: In the x-y plane, the line through (A, B) and (R, S) goes through the origin

0ld · Accepted Answer

(1)  \sqrt{A^2 + B^2} = \sqrt{R^2 + S^2}

==>   A^2 + B^2 = R^2 + S^2  ==>  A^2 - S^2 = R^2 - B^2  ==> (A-S)(A+S)=(R-B)(R+B)   -- Not sufficient.

(2) In the x-y plane, the line through (A, B) and (R, S) goes through the origin

Slope of line between Origin and point (A,B) =  \frac{(B-0)}{(A-0)} =  \frac{B}{A}

Slope of line between Origin and point (R,S) =   \frac{(S-0)}{(R-0)} =  \frac{S}{R}

As the line passes through (A, B) and (R, S) slope will always be same.

\frac{B}{A}  =  \frac{S}{R}

BR = SA      --Sufficient.

Ans B.

chetan2u · Answer

Hi,
Statement I is clearly insuff..
Let me concentrate on statement II..
Few points..
Since numbers are positive, all four numbers are in I quadrant..
Since they lie on the same line and line passes thru origin, the equation of line is y=mx....
So A=mB and R = mS..
But m or the slope is equal as they lie on same line..
So A/B = R/S..
Or AS= BR...

B

rohit8865 · Answer

(1) Let A=B=R=S=\sqrt{50}---------Yes
Let A=B=\sqrt{50}
R=8 and S=6--------------------------No
insuff

(2) as we are making lines with unique points (A, B) and (R, S) , having same slope
both distance AS and BR must be equal.....suff..

Ans B

KrishnakumarKA1 · Answer

Prompt analysis
A, B, R and S are positive numbers.

Superset
The answer will be either yes or no

Translation
To find the answer, we need:
1# exact value of the four numbers
2# 4 equations to find all 4 numbers
3# any relation or property that will lead us to find the answer

Statement analysis

St 1: cannot be said anything as we don't know anything about A, B, R and S individually
St 2: the equation of line passing through (A,B) and origin will be Ay =Bx. Putting (R,S) in the equation we get AS =BR. Answer

Option B

loserunderachiever · Answer

Thank you Chetan for the detailed explanation , I chose B , as the two lines are crossing through the origin , then I thought slopes of both the line will be equal. After going through your explanation , it makes sense .

sanjna2023 · Answer

Statement 1 is insuff. 
Statement 2, since it passes through the origin is a x=y equation line. This implies, ratio of A/R=B/S or AS=BR.

GMATGuruNY · Answer

Statement 1: 
Case 1: A=1, B=2, R=1, and S=2, with the result that  A^2+B^2=5  and  R^2+S^2=5 
In this case, AS=2 and BR=2, so the answer to the question stem is YES.
Case 2: A=1, B=2, R=2, and S=1, with the result that  A^2+B^2 = 5  and  R^2+S^2=5 
In this case, AS=1 and BR=4, so the answer to the question stem is NO.
Since the answer is YES in Case 1 but NO in Case 2, INSUFFICIENT.

Statement 2: 
Slope yielded by (A, B) and (0, 0)  = \frac{B-0}{A-0} = \frac{B}{A} 
Slope yielded by (R, S) and (0, 0)  = \frac{S-0}{R-0} = \frac{S}{R} 
Since the slope in each case must be THE SAME, we get:
 \frac{B}{A} = \frac{S}{R} 
 AS=BR 
Thus, the answer to the question stem is YES.
SUFFICIENT.

B

Fdambro294 · Answer

All four variables must be positive numbers

The statement information should clue you in to the idea of the Distance Formula and the Coordinate Plane, in which each point in the coordinate plane is given by a unique “run” along the X Axis and a unique “rise” up the Y Axis

Does: AS = BR?

S1: if there were Points (A , B) and (R , S) on the coordinate plane, statement 1 is essentially telling us that the Distance of each point to the Origin (0 , 0) is EQUAL

Since the distance formula is essentially an offshoot of the Pythagorean Theorem, you can use easy triplets to find cases and disprove statement 1

Case 1: (A , B) = 3 , 4 —- and —- (R , S) = 4 , 3

(3)^2 + (4)^2 = (4)^2 + (3)^2

25 = 25 ——- satisfies statement 1

However:

AS = 3 * 3 = 9

BR = 4 * 4 = 16

NO they are NOT equal

Case 2: can have A = R and B = S

When all four variables are positive —— A * S = B * R

Answers question YES

Yes and No - s1 NOT sufficient alone

S2: essentially tells us that points (A , B) and (R , S) are on the same line that passes through the origin (0 , 0)

In other words, all three points are on the same line and all 3 points will satisfy a particular line equation y = mx + b, a straight line which will pass through the Origin (0, 0)

Which means the SLOPE, calculated from each Point - to - the Origin should be EQUAL

m = (B - 0) / (A - 0) = (S - 0) / (R - 0)

B/A = S/R

-cross multiply-

BR = AS

Definite yes to question for every positive value of (A , B) and (R , S)

S2 sufficient alone
Answer (B)

Posted from my mobile device

Crytiocanalyst · Answer

The key to solving the question is getting all the possiblities right

Statement #1: √R^2+S^2=√A2+B2
This can yield results in 2 ways let us assume A=3 and b=4 and r=3 and s=4 
Then AS=BR is satisfied 
However if R=4 ans S=3 AS not equal to BR 
Hence Statement 1 clearly insuff

Statement #2: In the x-y plane, the line through (A, B) and (R, S) goes through the origin
y=x can be the equation of the line 
in that A=B and R=S since these lie on the number line 
AS =BR clearly suff 
Hence IMO B

bumpbot · Answer

Automated notice from GMAT Club BumpBot: 

A member just gave Kudos to this thread, showing it’s still useful. I’ve bumped it to the top so more people can benefit. Feel free to add your own questions or solutions.

 This post was generated automatically.

A, B, R and S are four positive numbers. Does AS = BR?

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Data Sufficiency (DS) Questions

A, B, R and S are four positive numbers. Does AS = BR?

Prep Toolkit

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards