Question

X and Y are two variable quantities. The corresponding values of X and Y are given below:

X: 3, 6, 9, 12, 24
Y: 24, 12, 8, 6, 3

Then the relationship between X and Y is given by:

• A. \(X + Y \propto X - Y\)
• B. \(X + Y \propto \dfrac{1}{X-Y}\)
• C. \(X \propto Y\)
• D. \(X \propto \dfrac{1}{Y}\)

Hint:

Try multiplying each X-Y pair together first; if the product stays the same across every row, X and Y are inversely proportional.

Answer 1

The Correct Option is D

Step 1: Test the options directly instead of guessing a pattern first.
A quicker route than spotting a pattern by eye is to plug the actual numbers into each option and see which one gives a genuinely constant ratio across every pair.

Step 2: Try option (C), $X \propto Y$.
This needs $X/Y$ constant. Row 1: $3/24 = 0.125$. Row 3: $9/8 = 1.125$. These are very different, so (C) is eliminated right away.

Step 3: Try option (A) and option (B) together, since they both hinge on $(X-Y)$.
Row 1: $X - Y = 3 - 24 = -21$, and $X + Y = 27$. Row 2: $X - Y = 6 - 12 = -6$, and $X + Y = 18$. For option (A), $(X+Y)/(X-Y)$ should be constant: $27/(-21) \approx -1.29$ against $18/(-6) = -3$. These do not match, so (A) is out, and (B), which is just the reciprocal of the same broken ratio, fails for the identical reason.

Step 4: Try option (D), $X \propto 1/Y$, which means $XY$ should be constant.
$3 \times 24 = 72$
$6 \times 12 = 72$
$9 \times 8 = 72$
$12 \times 6 = 72$
$24 \times 3 = 72$
Every row gives exactly 72, with no exceptions. This is the only option that survives being checked against all five data points.

Final Answer:
Since the product XY is fixed at 72 throughout the table, X and Y vary inversely, confirming option (D).
\[ \boxed{XY = 72 \implies X \propto \dfrac{1}{Y}} \]