Ken bought a shirt at a price of D dollars, to which a sales tax of p percent was added....

Understanding the Question

Ken bought a shirt and we need to determine if its base price was more than $15. Let's extract what we know:

• Shirt price: \(\mathrm{D}\) dollars (before tax)
• Sales tax: \(\mathrm{p}\) percent added to the price
• Total amount paid: \(\mathrm{D} \times (1 + \mathrm{p}/100)\)
• Payment method: $20 bill
• Change received: Less than C dollars

Since Ken received less than C dollars in change:
\(20 - \mathrm{D}(1 + \mathrm{p}/100) < \mathrm{C}\)

Rearranging: \(\mathrm{D}(1 + \mathrm{p}/100) > 20 - \mathrm{C}\)

What We Need to Determine: Is \(\mathrm{D} > 15\)?

This is a yes/no question requiring a definitive answer about whether the shirt's base price exceeds $15.

Analyzing Statement 1

Statement 1 tells us: \(\mathrm{p} = 6\)

With a 6% tax rate, our constraint becomes:
\(\mathrm{D}(1.06) > 20 - \mathrm{C}\)

But here's the critical issue: we don't know C!

Let's test different scenarios to see why this matters:

• If \(\mathrm{C} = 4\): Then \(\mathrm{D}(1.06) > 16\), so \(\mathrm{D} > 15.09\)
  → Yes, \(\mathrm{D} > 15\)
• If \(\mathrm{C} = 6\): Then \(\mathrm{D}(1.06) > 14\), so \(\mathrm{D} > 13.21\)
  → Cannot determine if \(\mathrm{D} > 15\) (could be $14, could be $16)

Different values of C lead to different answers about whether \(\mathrm{D} > 15\).

Statement 1 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A and D.

Analyzing Statement 2

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

Statement 2 tells us: \(\mathrm{C} = 5\)

This means Ken received less than $5 in change, so:
\(\mathrm{D}(1 + \mathrm{p}/100) > 15\)

But now we don't know p!

Let's explore what happens with different tax rates:

• If \(\mathrm{p} = 0\) (no tax): Then \(\mathrm{D} > 15\)
  → Yes, \(\mathrm{D} > 15\)
• If \(\mathrm{p} = 50\) (50% tax): Then \(\mathrm{D}(1.5) > 15\), so \(\mathrm{D} > 10\)
  → Cannot determine if \(\mathrm{D} > 15\) (could be $11, could be $16)

Without knowing the tax rate, we cannot determine whether \(\mathrm{D} > 15\).

Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices B and D (already eliminated).

Combining Statements

With both statements, we know \(\mathrm{p} = 6\) and \(\mathrm{C} = 5\). This gives us:
\(\mathrm{D}(1.06) > 15\)

Solving for D: \(\mathrm{D} > 15/1.06 \approx 14.15\)

Here's the crucial insight: We know the shirt costs more than $14.15, but we're asked if it costs more than $15. There's an uncertainty zone between $14.15 and $15!

Let's verify with concrete examples:

• Scenario 1: Shirt costs $14.50
  • Total with tax: \($14.50 \times 1.06 = $15.37\)
  • Change: \($20 - $15.37 = $4.63\) (less than $5) ✓
  • Answer to "Is \(\mathrm{D} > 15\)?": NO
• Scenario 2: Shirt costs $15.50
  • Total with tax: \($15.50 \times 1.06 = $16.43\)
  • Change: \($20 - $16.43 = $3.57\) (less than $5) ✓
  • Answer to "Is \(\mathrm{D} > 15\)?": YES

Both scenarios satisfy all our constraints but give opposite answers to the question!

The statements together are NOT sufficient.

[STOP - Not Sufficient!] This eliminates choices A, B, C, and D.

The Answer: E

Even with both pieces of information, we cannot definitively determine whether the shirt's base price exceeded $15. The shirt definitely costs more than $14.15, but that's not enough to answer whether it costs more than $15.

Answer Choice E: "The statements together are not sufficient."