The values of x and y vary with the value of z so that each additive increase of 2 in...

Understanding the Question

Let's break down what we're being asked. The question gives us a specific relationship: as z increases by 2, x doubles and y triples. We need to find the value of \(\mathrm{x/(x+y)}\) when \(\mathrm{z=12}\).

What We Need to Determine: Can we find a unique value for the expression \(\mathrm{x/(x+y)}\) at \(\mathrm{z=12}\)?

Key Insight: Since x and y grow at different rates (x doubles while y triples for every increase of 2 in z), the ratio between them changes as z changes. This means that to find \(\mathrm{x/(x+y)}\) at any specific z-value, we need to know the starting ratio between x and y.

Think of it this way: if we know how x and y relate at ANY point, we can work forward or backward using the growth factors to find their relationship at \(\mathrm{z=12}\).

Analyzing Statement 1

Statement 1 tells us: When \(\mathrm{z=6}\), \(\mathrm{x=5y}\)

This gives us the exact ratio between x and y at a specific point (\(\mathrm{z=6}\)). Since we know the growth pattern, knowing their ratio at one point allows us to determine their ratio at any other point.

Here's why this works: If \(\mathrm{x=5y}\) at \(\mathrm{z=6}\), we can track how this ratio changes:

- From \(\mathrm{z=6}\) to \(\mathrm{z=8}\): x doubles, y triples → the ratio \(\mathrm{x/y}\) goes from \(\mathrm{5/1}\) to \(\mathrm{(5×2)/(1×3) = 10/3}\)

- From \(\mathrm{z=8}\) to \(\mathrm{z=10}\): x doubles again, y triples again → the ratio becomes \(\mathrm{(10/3)×(2/3) = 20/9}\)

- From \(\mathrm{z=10}\) to \(\mathrm{z=12}\): x doubles once more, y triples once more → the ratio becomes \(\mathrm{(20/9)×(2/3) = 40/27}\)

Since we can determine the exact ratio at \(\mathrm{z=12}\), we can calculate \(\mathrm{x/(x+y)}\) uniquely.

[STOP - Statement 1 is Sufficient!]

This eliminates choices B, C, and E.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us: When \(\mathrm{z=0}\), \(\mathrm{x = y+1}\)

This gives us a relationship between x and y at the starting point, but not their actual values or ratio.

Let's test different scenarios to see if we always get the same answer:

Scenario 1: If \(\mathrm{y=1}\) at \(\mathrm{z=0}\), then \(\mathrm{x=2}\)

- Starting ratio: \(\mathrm{x/y = 2/1 = 2}\)

- After 6 increases of 2 (to reach \(\mathrm{z=12}\)): \(\mathrm{x/y = 2×(2/3)^6}\)

Scenario 2: If \(\mathrm{y=2}\) at \(\mathrm{z=0}\), then \(\mathrm{x=3}\)

- Starting ratio: \(\mathrm{x/y = 3/2 = 1.5}\)

- After 6 increases of 2 (to reach \(\mathrm{z=12}\)): \(\mathrm{x/y = 1.5×(2/3)^6}\)

Since we start with different ratios (2 vs 1.5), and the ratio \(\mathrm{x/y}\) determines \(\mathrm{x/(x+y)}\), we'll get different final values. For instance:

- If \(\mathrm{x/y = 2}\), then \(\mathrm{x/(x+y) = 2/(2+1) = 2/3}\)

- If \(\mathrm{x/y = 1.5}\), then \(\mathrm{x/(x+y) = 1.5/(1.5+1) = 3/5}\)

Different starting values of y lead to different answers at \(\mathrm{z=12}\).

Statement 2 is NOT sufficient.

This eliminates choices B and D.

The Answer: A

Statement 1 alone gives us enough information to determine a unique value for \(\mathrm{x/(x+y)}\) at \(\mathrm{z=12}\), while Statement 2 alone does not.

Answer Choice A: "Statement 1 alone is sufficient, but Statement 2 alone is not sufficient."