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Esther bought a computer at a price of \(\mathrm{p}\) dollars, and she paid a sales tax of \(\mathrm{t}\) percent. If Esther had less than \(\mathrm{r}\) dollars left of the $1,000 that she budgeted for the computer, was the price of the computer, excluding the sales tax, greater than $800?
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 \(\mathrm{p} \times (1 + \mathrm{t}/100)\).
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 \(\mathrm{p} > \$800\):
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.
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:
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.
With both statements together, we know:
This means:
So we have: \(\mathrm{p} \times 1.06 > \$800\)
To find the minimum possible price:
Critical question: Does knowing that \(\mathrm{p} > \$755\) tell us whether \(\mathrm{p} > \$800\)?
Let's check with concrete examples:
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.
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."