If x and y are positive integers and y = 9 - x, what is the

You should read the stem carefully: "If x and y are positive integers..."

Also \(y^2=9-x\) --> as given that \(y\) is a positive integer then \(y^2\) is a positive perfect square less than 9 (perfect square is a square of an integer): so \(y^2=4=2^2\) if \(x=5\) or \(y^2=1=1^2\) if \(x=8\).

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For the given expression we have x,y are positive integers and therefore squaring both sides

y^2= 9-x
From St 1 we have x< 8 -----> so we have possible value of in the range 0<x<8
Only for x=5, we have y=2

So options B,C and E ruled out

St 2 says y>1 so we have y^2=9-x
Now let us say y=2 so x=5
y=3, we get 9=9-x ----> x=0 which is not possible as "x" is a positive integer
hence y=2

Ans should be D

I messed it with choosing A as ans trying to beat the average time and not carefully solving the st2

\(y=\sqrt{9 - x}\)

x and y are positive integers.
If x could be 0, y = 3
If x = 9, y = 0 (not allowed)

So y can be either 1 (x = 8) or 2 (x = 5)

(1) x < 8
So x cannot be 8. It must be 5 and y must be 2.
Sufficient alone.

(2) y > 1
So y cannot be 1. It must be 2.
Sufficient alone.

Answer (D)

Forget the conventional way to solve DS questions.

We will solve this DS question using the variable approach.

DS question with 2 variables and 1 Equation: Let the original condition in a DS question contain 2 variables and 1 Equation. Now, 2 variables and 1 Equation would generally require 1 more equation for us to be able to solve for the value of the variable.

We know that each condition would usually give us an equation, and Since we need 1 more equation to match the numbers of variables and equations in the original condition, the logical answer is D.

To master the Variable Approach, visit https://www.mathrevolution.com and check our lessons and proven techniques to score high in DS questions.

Let’s apply the 3 steps suggested previously. [Watch lessons on our website to master these 3 steps]

Step 1 of the Variable Approach: Modifying and rechecking the original condition and the question.

We have to find the value of 'y'.

=> 'x' and 'y' are positive integers and y = \(\sqrt{9 - x}\)

=> y = \(\sqrt{9 - x}\) = \(y^2\) = 9 - x [\(y^2\) is a perfect square less than '9' for a value of 'x']

Second and the third step of Variable Approach: From the original condition, we have 2 variables (x and y) and 1 Equation (y = \(\sqrt{9 - x}\)).To match the number of variables with the number of equations, we need 1 more equation. Since conditions (1) and (2) will provide 1 equation each, D would most likely be the answer.

Let’s take look at each condition separately.

Condition(1) tells us that x < 8.

=> For x = 5: \(y^2\) = 9 - 5 = 4 and hence y = 2

Since the answer is unique , Condition(1) is alone sufficient by CMT 2.

Condition(2) tells us that y > 1.

=> For y = 2: \(y^2\) = 9 - 5 = 4 [a perfect square]

Since the answer is unique , Condition(2) is alone sufficient by CMT 2.


Each condition alone is sufficient.

So, D is the correct answer.

Answer: D


SAVE TIME: By Variable Approach, when you know that value of Con(1) = Con(2), then 'D' is the correct answer.

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