Esther bought a computer at a price of p dollars, and she paid a sales tax of t percent. If...

Understanding the Question

Esther budgeted \(\$1{,}000\) for a computer purchase. She bought a computer at price p dollars plus sales tax of t percent. After the purchase, she had less than r dollars remaining from her budget.

What we need to determine: Is \(\mathrm{p} > \$800\)?

This is a yes/no question. To be sufficient, we need to definitively answer either "Yes, p must be greater than \(\$800\)" or "No, p cannot be greater than \(\$800\)" in all possible scenarios.

Key insight: Since Esther has less than r dollars left from her \(\$1{,}000\) budget, she spent more than \(\$(1{,}000 - \mathrm{r})\) in total. This total spending equals the price p plus the tax, which is \(\mathrm{p} \times (1 + \mathrm{t}/100)\).

Analyzing Statement 1

Statement 1 tells us: \(\mathrm{r} = 200\)

This means Esther has less than \(\$200\) remaining, so she spent more than \(\$800\) total (including tax).

What we still don't know: The tax rate t

Let's test different scenarios to see if we can determine whether \(\mathrm{p} > \$800\):

• Scenario 1: If tax rate = 0%
  - Total spent > \(\$800\)
  - Since there's no tax, price \(\mathrm{p}\) = total spent > \(\$800\)
  - Answer: YES, \(\mathrm{p} > \$800\)
• Scenario 2: If tax rate = 25%
  - Total spent > \(\$800\)
  - With 25% tax, this means \(\mathrm{p} \times 1.25 > \$800\)
  - So \(\mathrm{p} > \$640\)
  - But this only tells us p is greater than \(\$640\)
  - Example: p could be \(\$700\) (which is < \(\$800\)) or \(\$900\) (which is > \(\$800\))
  - Answer: Can't determine if \(\mathrm{p} > \$800\)

Since different tax rates lead to different conclusions about whether \(\mathrm{p} > \$800\), Statement 1 alone cannot answer our question.

Statement 1 is 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: \(\mathrm{t} = 6\)

This means the sales tax rate is 6%.

What we still don't know: The value of r (how much money Esther has left)

Let's test different scenarios:

• Scenario 1: If \(\mathrm{r} = 50\) (she has less than \(\$50\) left)
  - She spent more than \(\$950\) total
  - With 6% tax: \(\mathrm{p} \times 1.06 > \$950\)
  - So \(\mathrm{p} > \$950 \div 1.06 \approx \$896\)
  - Since \(\$896 > \$800\), we know \(\mathrm{p} > \$800\)
  - Answer: YES, \(\mathrm{p} > \$800\)
• Scenario 2: If \(\mathrm{r} = 250\) (she has less than \(\$250\) left)
  - She spent more than \(\$750\) total
  - With 6% tax: \(\mathrm{p} \times 1.06 > \$750\)
  - So \(\mathrm{p} > \$750 \div 1.06 \approx \$708\)
  - But knowing \(\mathrm{p} > \$708\) doesn't tell us if \(\mathrm{p} > \$800\)
  - Example: p could be \(\$720\) (which is < \(\$800\)) or \(\$850\) (which is > \(\$800\))
  - Answer: Can't determine if \(\mathrm{p} > \$800\)

Since different values of r lead to different conclusions about whether \(\mathrm{p} > \$800\), Statement 2 alone cannot answer our question.

Statement 2 is NOT sufficient.

This eliminates choice B.

Combining Both Statements

With both statements together, we know:

• \(\mathrm{r} = 200\) (from Statement 1)
• \(\mathrm{t} = 6\) (from Statement 2)

This means:

• Esther has less than \(\$200\) left
• She spent more than \(\$800\) total
• The total includes 6% tax

So we have: \(\mathrm{p} \times 1.06 > \$800\)

To find the minimum possible price:

• If total spent = exactly \(\$800\), then \(\mathrm{p} = \$800 \div 1.06 \approx \$755\)
• But she spent MORE than \(\$800\), so \(\mathrm{p} > \$755\)

Critical question: Does knowing that \(\mathrm{p} > \$755\) tell us whether \(\mathrm{p} > \$800\)?

Let's check with concrete examples:

• Possible scenario 1: \(\mathrm{p} = \$760\)
  - This satisfies \(\mathrm{p} > \$755\) ✓
  - Total spent: \(\$760 \times 1.06 = \$805.60\) (which is > \(\$800\)) ✓
  - But \(\$760 < \$800\)
  - Answer: NO, p is not greater than \(\$800\)
• Possible scenario 2: \(\mathrm{p} = \$850\)
  - This satisfies \(\mathrm{p} > \$755\) ✓
  - Total spent: \(\$850 \times 1.06 = \$901\) (which is > \(\$800\)) ✓
  - And \(\$850 > \$800\)
  - Answer: YES, p is greater than \(\$800\)

Since we can construct valid scenarios with different answers to our yes/no question, even the combined statements cannot definitively answer whether \(\mathrm{p} > \$800\).

The combined statements are NOT sufficient.

This eliminates choice C.

The Answer: E

Neither statement alone nor both statements together provide enough information to determine whether the pre-tax price exceeded \(\$800\). We can only establish that \(\mathrm{p} > \$755\), but this leaves open the possibility that p could be either below or above \(\$800\).

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