Maria and her companion had a $20 gift certificate to a restaurant. They applied the certificate to the cost of...

Phase 1: Owning the Dataset

Visualization Selection

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)

Starting with Concrete Numbers

Let's use \(\$100\) as our original meal cost to test our understanding:

• Original cost: \(\$100\)
• Sales tax: \(7\% \times \$100 = \$7\)
• Tip: \(18\% \times \$100 = \$18\)
• Total before gift certificate: \(\$100 + \$7 + \$18 = \$125\)
• Total paid after gift certificate: \(\$125 - \$20 = \$105\)

Phase 2: Understanding the Question

We need to find expressions for:

1. Original cost of the meal
2. Tip amount paid

Both should be expressed in terms of P (the total amount paid after applying the gift certificate).

Setting Up the General Equations

Let \(\mathrm{O}\) = original cost of the meal

• Sales tax = \(0.07 \times \mathrm{O}\)
• Tip = \(0.18 \times \mathrm{O}\)
• Total before gift certificate = \(\mathrm{O} + 0.07\mathrm{O} + 0.18\mathrm{O} = \mathrm{O}(1 + 0.07 + 0.18) = 1.25\mathrm{O}\)
• Total paid \(\mathrm{P} = 1.25\mathrm{O} - 20\)

Key Insight

From \(\mathrm{P} = 1.25\mathrm{O} - 20\), we can solve for O:

• \(1.25\mathrm{O} = \mathrm{P} + 20\)
• \(\mathrm{O} = \frac{\mathrm{P} + 20}{1.25}\)

And the tip amount = \(0.18 \times \mathrm{O} = 0.18 \times \frac{\mathrm{P} + 20}{1.25}\)

Phase 3: Finding the Answer

For Original Cost

We derived: \(\mathrm{O} = \frac{\mathrm{P} + 20}{1.25}\)

Looking at our answer choices, this matches: \(\frac{\mathrm{P}+20}{1.25}\)

For Tip Amount

We need: Tip = \(0.18 \times \frac{\mathrm{P} + 20}{1.25}\)

Let's simplify this expression:

• \(0.18 \times \frac{\mathrm{P} + 20}{1.25} = \frac{0.18\mathrm{P} + 0.18 \times 20}{1.25} = \frac{0.18\mathrm{P} + 3.6}{1.25}\)

This matches: \(\frac{0.18\mathrm{P}+3.6}{1.25}\)

Verification

Using our concrete example where \(\mathrm{P} = \$105\):

• Original cost: \(\frac{105 + 20}{1.25} = \frac{125}{1.25} = \$100\) ✓
• Tip: \(\frac{0.18 \times 105 + 3.6}{1.25} = \frac{18.9 + 3.6}{1.25} = \frac{22.5}{1.25} = \$18\) ✓

Phase 4: Solution

Final Answer:

• Original cost: \(\frac{\mathrm{P}+20}{1.25}\)
• Tip: \(\frac{0.18\mathrm{P}+3.6}{1.25}\)

These expressions correctly represent the original meal cost and tip amount in terms of the total paid P after applying the \(\$20\) gift certificate.