Question

CSATHardPrelims 2024Reasoning

If P means ‘greater than ( $>$ )’; Q means ‘less than ( $<$ )’; R means ‘not greater than ( $less$ )’; S means ‘not less than ( $geq$ )’ and T means ‘equal to ( $=$ )’, then consider the following statements :

If $2x (S) 3y$ and $3x (T) 4z$ , then $9y (P) 8z$ .
If $x (Q) 2y$ and $y (R) z$ , then $x (R) z$ .

Which of the statements given above is/are correct?

Explanation

Statement 1: If $2x \ge 3y$ and $3x = 4z$ , then $9y > 8z$ .

Interpretation:
1. $2x \ge 3y$
2. $3x = 4z \Longrightarrow x = \frac{4z}{3}$
Substituting $x = \frac{4z}{3}$ in $2x \ge 3y$ :
- $2(\frac{4z}{3}) \ge 3y$
- $\frac{8z}{3} \ge 3y \Longrightarrow 8z \ge 9y$
This implies $9y \le 8z$ , which contradicts $9y > 8z$ .
Conclusion: Statement 1 is incorrect.

Statement 2: If $x < 2y$ and $y \le z$ , then $x \le z$ .

Interpretation:
1. $x < 2y$
2. $y \le z$
From the above, $x < 2y$ does not guarantee $x \le z$ because $z$ depends on $y$ , and $x$ could still be greater than $z$ .
Conclusion: Statement 2 is incorrect.