John and Beth each bought a used car at a purchase price of $12{,}000. John paid a sales tax of...

Understanding the Question

We need to find the exact dollar difference between John's and Beth's sales tax payments.

Let's break this down:

• Both bought cars for $12,000 (same base price)
• John paid \(\mathrm{j\%}\) tax, Beth paid \(\mathrm{k\%}\) tax
• We know \(\mathrm{j > k}\) (John's rate is higher)
• Question asks: How much more did John pay?

Since both cars cost the same, the difference in sales tax depends entirely on the difference in tax rates. Here's the key insight: we need to determine the exact value of \(\mathrm{(j - k)}\) to calculate the dollar difference.

What makes this sufficient: We need a single, specific dollar amount for the difference in their sales tax payments.

Analyzing Statement 1

Statement 1: "j is 3 more than k"

This means \(\mathrm{j = k + 3}\), so the percentage point difference \(\mathrm{j - k = 3}\).

Key insight: No matter what k is, the difference is always 3 percentage points.

Let's verify with examples:

• If \(\mathrm{k = 5\%}\), then \(\mathrm{j = 8\%}\). The difference in tax paid = \(\mathrm{\$12{,}000 \times (8\% - 5\%) = \$12{,}000 \times 3\% = \$360}\)
• If \(\mathrm{k = 10\%}\), then \(\mathrm{j = 13\%}\). The difference in tax paid = \(\mathrm{\$12{,}000 \times (13\% - 10\%) = \$12{,}000 \times 3\% = \$360}\)

Notice that regardless of what k is, the percentage point difference is always 3, which translates to exactly $360 more in sales tax for John.

[STOP - Sufficient!] We can determine the exact dollar difference.

This eliminates choices B, C, and E.

Analyzing Statement 2

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

Statement 2: "j is 8/5 of k"

This means \(\mathrm{j = \frac{8}{5}k}\), which gives us a ratio relationship rather than a fixed difference.

Let's test different scenarios:

• If \(\mathrm{k = 5\%}\), then \(\mathrm{j = \frac{8}{5} \times 5\% = 8\%}\). The difference = \(\mathrm{8\% - 5\% = 3\%}\), which is $360
• If \(\mathrm{k = 10\%}\), then \(\mathrm{j = \frac{8}{5} \times 10\% = 16\%}\). The difference = \(\mathrm{16\% - 10\% = 6\%}\), which is $720

Different values of k lead to different dollar amounts. We cannot determine a single answer.

This is NOT sufficient because multiple answers are possible.

This eliminates choices B and D.

The Answer: A

Statement 1 gives us a fixed difference (3 percentage points → $360), while Statement 2 gives us only a ratio that produces different dollar amounts depending on k's value.

Strategic Insight: In "difference" problems, distinguish between:

• Fixed differences (like Statement 1) → Usually sufficient
• Ratio relationships (like Statement 2) → Usually insufficient unless you know one value

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