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: Look for a simple pattern in the paired values first.
When a table of paired values is given like this, the fastest check is to try multiplying each pair together, since a constant product is one of the most common hidden relationships.

Step 2: Multiply each X-Y pair.
\(3 \times 24 = 72\)
\(6 \times 12 = 72\)
\(9 \times 8 = 72\)
\(12 \times 6 = 72\)
\(24 \times 3 = 72\)
Every single pair gives the same product, 72. So \(XY = 72\), a constant, for every row of the table.

Step 3: Turn the constant product into a proportionality statement.
If \(XY = k\) for a constant \(k\), then \(X = \dfrac{k}{Y}\), which is exactly what it means to say \(X \propto \dfrac{1}{Y}\): X and Y vary inversely with each other.

Step 4: Check why the other options fail.
For option (C), \(X \propto Y\) would need \(X/Y\) to stay constant. But \(3/24 = 0.125\) while \(6/12 = 0.5\), so this ratio is not constant, ruling out (C).
For option (A), \((X+Y) \propto (X-Y)\) would need \(\dfrac{X+Y}{X-Y}\) to stay constant. Taking the first two rows, \(\dfrac{3+24}{3-24} = \dfrac{27}{-21} \approx -1.29\), while \(\dfrac{6+12}{6-12} = \dfrac{18}{-6} = -3\). These are not equal, so (A) fails.
Option (B) fails for the same reason as (A), since it is really the reciprocal form of the same broken relationship, and reciprocals of unequal numbers are still unequal.

Final Answer:
Since XY stays fixed at 72 across every pair, X and Y are inversely proportional. \[ \boxed{X \propto \dfrac{1}{Y}} \]