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Maria and her companion had a \(\$20\) gift certificate to a restaurant. They applied the certificate to the cost of their meal but left an \(18\%\) tip based on the original cost of the meal. They also paid \(7\%\) sales tax that was applied to the original cost of the meal.
Let \(\mathrm{P}\) denote the total amount Maria and her companion paid, in dollars, after the gift certificate had been applied to the bill. Select Original cost for the expression that represents the original cost of the meal, and select Tip for the expression that represents the tip they paid, in dollars. Make only two selections, one in each column.
\(\frac{\mathrm{P}-20}{1.25}\)
\(\frac{\mathrm{P}+20}{1.25}\)
\(\frac{0.25\mathrm{P}+5}{1.25}\)
\(0.18 \times (\frac{\mathrm{P}-20}{1.25})\)
\(\frac{0.18\mathrm{P}+3.6}{1.25}\)
This is a calculation problem involving cost breakdowns. Let's use an Equation Format with a supporting flow diagram:
Original Cost (O)
|
+---> Sales Tax (7% of O)
|
+---> Tip (18% of O)
|
v
Total Before Gift Certificate
|
- $20 Gift Certificate
|
v
Total Paid (P)
Let's use \(\$100\) as our original meal cost to test our understanding:
We need to find expressions for:
Both should be expressed in terms of P (the total amount paid after applying the gift certificate).
Let \(\mathrm{O}\) = original cost of the meal
From \(\mathrm{P} = 1.25\mathrm{O} - 20\), we can solve for O:
And the tip amount = \(0.18 \times \mathrm{O} = 0.18 \times \frac{\mathrm{P} + 20}{1.25}\)
We derived: \(\mathrm{O} = \frac{\mathrm{P} + 20}{1.25}\)
Looking at our answer choices, this matches: \(\frac{\mathrm{P}+20}{1.25}\)
We need: Tip = \(0.18 \times \frac{\mathrm{P} + 20}{1.25}\)
Let's simplify this expression:
This matches: \(\frac{0.18\mathrm{P}+3.6}{1.25}\)
Using our concrete example where \(\mathrm{P} = \$105\):
Final Answer:
These expressions correctly represent the original meal cost and tip amount in terms of the total paid P after applying the \(\$20\) gift certificate.