Leo can buy a certain computer for p1 dollars in State
A, where the sales tax is t1 percent, or he can buy the
same computer for p2 dollars in State B, where the
sales tax is t2 percent. Is the total cost of the
computer greater in State A than in State B ?
(1) t1 > t2
(2) p1t1 > p2t2
To save time, try to plug in combinations that satisfy both statements.
Case 1:
Let p� = 100 and t� = 20, so that p�t� = 100*20 = 2000.
Let pâ‚‚ = 100 and tâ‚‚ = 10, so that pâ‚‚tâ‚‚ = 100*10 = 1000.
A = p� + (t�% of p�) = 100 + (20% of 100) = 120.
B = pâ‚‚ + (tâ‚‚% of pâ‚‚) = 100 + (10% of 100) = 110.
Result:
A>B.
Since A>B in Case 1, the goal in Case 2 is to find values such that A<B.
In Case 1, p�t� = 2000.
To make A<B, try to increase pâ‚‚tâ‚‚ to a value just below 2000.
Case 2:
To goal is to find a combination such that A
Let p� = 100 and t� = 20, so that p�t� = 100*20 = 2000.
Let pâ‚‚ = 190 and tâ‚‚ = 10, so that pâ‚‚tâ‚‚ = 190*10 = 1900.
A = p� + (t�% of p�) = 100 + (20% of 100) = 120.
B = pâ‚‚ + (tâ‚‚% of pâ‚‚) = 190 + (10% of 190) = 209.
Result:
A<B.
Since A>B in Case 1, but A<B in Case 2, the two statements combined are INSUFFICIENT.
The correct answer is
E.
Algebraically:
Total cost in A = p� + (t�% of p�) = p� + (t�/100)p�
Total cost in B = pâ‚‚ + (tâ‚‚% of pâ‚‚) = pâ‚‚ + (tâ‚‚/100)pâ‚‚
Question stem:
p� + (t�/100)p� > p₂ + (t₂/100)p₂ ?
p� + (p�t�)/100 >
pâ‚‚ + (pâ‚‚tâ‚‚)/100 ?
When the statements are combined, we know that (p�t�)/100 > (p₂t₂)/100, but we cannot determine the relationship between the portions in red.
Thus, the two statements combined are INSUFFICIENT.
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: Is there a logical conclusion we can come to without testing numbers? 
