Whenever martin has a restaurant bill with an amount between $10 and $99, he calculates the dollar amount of the...

Understanding the Question

Let's decode Martin's unusual tipping system. For any restaurant bill between \(\$10\) and \(\$99\), Martin calculates his tip as 2 times the tens digit of the bill amount.

What We Need to Determine: Is Martin's tip greater than 15% of the bill amount?

Here's the key insight: Martin's system creates "brackets" where all bills in the same tens range receive the exact same tip. For example:

• Bills from \(\$40\)-\(\$49\) all get an \(\$8\) tip \((2 \times 4)\)
• Bills from \(\$30\)-\(\$39\) all get a \(\$6\) tip \((2 \times 3)\)
• And so on...

Within each bracket, the tip percentage is highest for the smallest bill and lowest for the largest bill. This makes intuitive sense - an \(\$8\) tip on a \(\$40\) bill (20%) is a higher percentage than an \(\$8\) tip on a \(\$49\) bill (about 16.3%).

For sufficiency: We need to determine with certainty whether the tip exceeds 15% - the answer must be definitively YES or definitively NO.

Analyzing Statement 1

Statement 1: The bill was between \(\$15\) and \(\$50\).

This range spans multiple brackets with different fixed tips:

• \(\$15\)-\(\$19\): tip = \(\$2\) (since tens digit = 1)
• \(\$20\)-\(\$29\): tip = \(\$4\) (since tens digit = 2)
• \(\$30\)-\(\$39\): tip = \(\$6\) (since tens digit = 3)
• \(\$40\)-\(\$49\): tip = \(\$8\) (since tens digit = 4)
• \(\$50\): tip = \(\$10\) (since tens digit = 5)

Let's check the extremes within the lowest bracket (\(\$15\)-\(\$19\)):

• For a \(\$15\) bill: tip = \(\$2\), which is \(\$2/\$15 \approx 13.3\%\) (less than 15%)
• For a \(\$19\) bill: tip = \(\$2\), which is \(\$2/\$19 \approx 10.5\%\) (less than 15%)

Now let's check a higher bracket (\(\$40\)-\(\$49\)):

• For a \(\$40\) bill: tip = \(\$8\), which is \(\$8/\$40 = 20\%\) (greater than 15%)

Since we get different answers - NO for bills in the teens, YES for bills in the forties - we cannot determine a definitive answer. Statement 1 is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Now let's analyze Statement 2 independently, forgetting Statement 1 completely.

Statement 2: The tip calculated by Martin was \(\$8\).

Since tip = 2 × tens digit, and the tip is \(\$8\), we can determine:

• Tens digit = \(\$8 \div 2 = 4\)
• Therefore, the bill must be between \(\$40\) and \(\$49\)

Here's the elegant insight: Even for the largest possible bill in this range (\(\$49\)), is an \(\$8\) tip more than 15%?

Quick check: 15% of \(\$50\) would be \(\$7.50\), so 15% of \(\$49\) is slightly less than \(\$7.50\). Since \(\$8 > \$7.50\), the tip exceeds 15% even in this worst-case scenario.

For any bill from \(\$40\) to \(\$49\), the \(\$8\) tip will always exceed 15% of the bill amount.

[STOP - Sufficient!] Statement 2 is sufficient.

This eliminates choices C and E.

The Answer: B

Statement 2 alone tells us the exact tip amount, which pins down the bill to a specific range (\(\$40\)-\(\$49\)) where the tip always exceeds 15%.

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