Bunuel wrote:
A certain customer at a restaurant calculates his tip by adding a constant dollar amount to another sum that is directly proportional to the total bill for the meal. If his total bill for the meal was $24.00, what will the be the dollar amount of his tip?
(1) If the total bill for his meal had been four dollars greater, the customer would've calculated a tip of $4.80.
(2) If the total bill for his meal had been six dollars less, the customer would've calculated a tip of $3.80.
Set the cost of the meal itself as x and the tip percentage as y. We know x*(1 + y) = 24.
Statement 1:Now we have (x + 4)*y = 4.8. Then we have \(y = \frac{4.8}{x + 4}\). Plug in above to get:
\(x*(1 + \frac{4.8}{x + 4}) = 24\)
\(x*\frac{x + 8.8}{x + 4} = 24\)
\(x^2 + 8.8x = 24x + 96\)
\(x^2 - 15.2x - 96 = 0\)
Note we don't have to solve this. Since the constant is negative, there will be only one positive solution for x, so sufficient.
Statement 2:Similar to above, we have \((x - 6)y = 3.8\) and \(y = \frac{3.8}{x - 6}\).
\(x*(1 + \frac{3.8}{x - 6}) = 24\)
\(x*\frac{x - 2.2}{x - 6} = 24\)
\(x^2 - 2.2x = 24x - 144\)
\(x^2 - 26.2x + 144 = 0\)
The constant is positive while the x term is negative, so there are two possible positive solutions. Insufficient.
Ans: ANote: Statement B should be "a tip of 2.8" as from A we know x = 20 and y = 20%.
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