Leo can buy a certain computer for p_1 dollars in State A, where the sales tax is t_1 percent, or...

Understanding the Question

Let's break down what we're being asked. We need to determine whether the total cost of a computer is greater in State A than in State B.

Given Information

• Computer costs \(\mathrm{p_1}\) dollars (before tax) in State A
• Same computer costs \(\mathrm{p_2}\) dollars (before tax) in State B
• State A has sales tax of \(\mathrm{t_1}\) percent
• State B has sales tax of \(\mathrm{t_2}\) percent

What We Need to Determine

The total costs including tax are:

• State A: \(\mathrm{p_1} + (\mathrm{t_1}/100)\mathrm{p_1} = \mathrm{p_1}(1 + \mathrm{t_1}/100)\)
• State B: \(\mathrm{p_2} + (\mathrm{t_2}/100)\mathrm{p_2} = \mathrm{p_2}(1 + \mathrm{t_2}/100)\)

So we need to know: Is \(\mathrm{p_1}(1 + \mathrm{t_1}/100) > \mathrm{p_2}(1 + \mathrm{t_2}/100)\)?

This is a yes/no question. To get a definitive answer, we need to be able to say either:

• YES, State A is always more expensive, or
• NO, State A is not always more expensive

Key Insight

Notice we have two competing factors here: the base price and the tax rate. Even if one state has higher taxes, it might have a lower base price, making the total cost comparison uncertain.

Analyzing Statement 1

Statement 1 tells us: \(\mathrm{t_1} > \mathrm{t_2}\)

This means State A has a higher tax rate than State B. But here's the challenge - we know nothing about how the base prices \(\mathrm{p_1}\) and \(\mathrm{p_2}\) compare.

Let's test different scenarios to see if we can get different answers:

Scenario 1: State A has much lower base price

• \(\mathrm{p_1} = \$100, \mathrm{t_1} = 10\%\)
• \(\mathrm{p_2} = \$200, \mathrm{t_2} = 5\%\)
• State A total: \(\$100(1.10) = \$110\)
• State B total: \(\$200(1.05) = \$210\)
• Answer: NO (State B is more expensive)

Scenario 2: State A has higher base price

• \(\mathrm{p_1} = \$200, \mathrm{t_1} = 10\%\)
• \(\mathrm{p_2} = \$100, \mathrm{t_2} = 5\%\)
• State A total: \(\$200(1.10) = \$220\)
• State B total: \(\$100(1.05) = \$105\)
• Answer: YES (State A is more expensive)

Since we can get both YES and NO answers, 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: \(\mathrm{p_1t_1} > \mathrm{p_2t_2}\)

This means the product of price and tax rate is higher in State A. But this doesn't directly tell us about the total costs. Remember, total cost depends on \(\mathrm{p_1}(1 + \mathrm{t_1}/100)\), not just \(\mathrm{p_1t_1}\).

Let's test scenarios that satisfy \(\mathrm{p_1t_1} > \mathrm{p_2t_2}\):

Scenario 1: High price with moderate tax in State A

• \(\mathrm{p_1} = \$200, \mathrm{t_1} = 10\%\)
• \(\mathrm{p_2} = \$100, \mathrm{t_2} = 15\%\)
• Check: \(\mathrm{p_1t_1} = 200 \times 10 = 2000 > \mathrm{p_2t_2} = 100 \times 15 = 1500\) ✓
• State A total: \(\$200(1.10) = \$220\)
• State B total: \(\$100(1.15) = \$115\)
• Answer: YES (State A is more expensive)

Scenario 2: Lower price with high tax in State A

• \(\mathrm{p_1} = \$100, \mathrm{t_1} = 20\%\)
• \(\mathrm{p_2} = \$110, \mathrm{t_2} = 18\%\)
• Check: \(\mathrm{p_1t_1} = 100 \times 20 = 2000 > \mathrm{p_2t_2} = 110 \times 18 = 1980\) ✓
• State A total: \(\$100(1.20) = \$120\)
• State B total: \(\$110(1.18) = \$129.80\)
• Answer: NO (State B is more expensive)

Again, we can get both YES and NO answers, so Statement 2 is NOT sufficient.

[STOP - Not Sufficient!] This eliminates choice B.

Combining Both Statements

Now we know both:

• Statement 1: \(\mathrm{t_1} > \mathrm{t_2}\) (State A has higher tax rate)
• Statement 2: \(\mathrm{p_1t_1} > \mathrm{p_2t_2}\) (Product of price and tax is higher in State A)

Let's see if together these give us a definitive answer.

Testing Scenario 1:

• \(\mathrm{p_1} = \$100, \mathrm{t_1} = 10\%, \mathrm{p_2} = \$90, \mathrm{t_2} = 5\%\)
• Check Statement 1: \(10\% > 5\%\) ✓
• Check Statement 2: \(100 \times 10 = 1000 > 90 \times 5 = 450\) ✓
• State A total: \(\$100(1.10) = \$110\)
• State B total: \(\$90(1.05) = \$94.50\)
• Answer: YES (State A is more expensive)

Testing Scenario 2:

• \(\mathrm{p_1} = \$100, \mathrm{t_1} = 6\%, \mathrm{p_2} = \$200, \mathrm{t_2} = 2\%\)
• Check Statement 1: \(6\% > 2\%\) ✓
• Check Statement 2: \(100 \times 6 = 600 > 200 \times 2 = 400\) ✓
• State A total: \(\$100(1.06) = \$106\)
• State B total: \(\$200(1.02) = \$204\)
• Answer: NO (State B is more expensive)

Even with both statements combined, we can still construct scenarios that give different answers.

[STOP - Still Not Sufficient!] The statements together are NOT sufficient.

The Answer: E

Since neither statement alone nor both statements together provide sufficient information to determine whether the total cost is greater in State A than State B, the answer is E.

Why E? Both statements give us information about how prices and taxes relate, but neither alone nor together can tell us definitively which state has the higher total cost. The key insight is that knowing \(\mathrm{t_1} > \mathrm{t_2}\) and \(\mathrm{p_1t_1} > \mathrm{p_2t_2}\) still allows for scenarios where either state could have the higher total cost.