Mia purchases 8 identical glass vases from an online shop, each at the same original price.She applies a coupon for...

1. TRANSLATE the problem information

• Given information:
  • 8 identical vases at same original price \(\mathrm{p}\)
  • $36 coupon applied to entire order
  • $12 flat shipping fee added
  • Total charge: $60

• What this tells us: We need to track the cost modifications in the correct order

2. TRANSLATE to set up the equation

• Start with base cost: \(\mathrm{8p}\) (8 vases × price per vase)
• Apply coupon discount: \(\mathrm{8p - 36}\)
• Add shipping: \(\mathrm{8p - 36 + 12 = 8p - 24}\)
• Set equal to total paid: \(\mathrm{8p - 24 = 60}\)

3. SIMPLIFY by solving the linear equation

• Add 24 to both sides: \(\mathrm{8p = 60 + 24 = 84}\)
• Divide both sides by 8: \(\mathrm{p = 84 ÷ 8 = 10.5}\)

4. Verify the answer

• Check: \(\mathrm{8(10.5) = 84}\), then \(\mathrm{84 - 36 = 48}\), then \(\mathrm{48 + 12 = 60}\) ✓

Answer: 10.5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the order of cost modifications, particularly applying shipping before the coupon discount.

They might set up: \(\mathrm{8p + 12 - 36 = 60}\), leading to \(\mathrm{8p - 24 = 60}\) (which happens to be correct) OR \(\mathrm{8p - 36 + 12 ≠ 8p + 12 - 36}\), creating confusion about which operations to perform first. More critically, they might think the coupon applies per vase rather than to the total order, setting up: \(\mathrm{8(p - 36) + 12 = 60}\), which gives \(\mathrm{8p - 288 + 12 = 60}\), leading to \(\mathrm{8p = 336}\), so \(\mathrm{p = 42}\).

This leads to confusion and potentially guessing or selecting an incorrect answer.

Second Most Common Error:

Inadequate SIMPLIFY execution: Students make arithmetic errors when combining terms or solving the equation.

For example, when going from \(\mathrm{8p - 24 = 60}\) to \(\mathrm{8p = 84}\), they might add incorrectly (\(\mathrm{60 - 24}\) instead of \(\mathrm{60 + 24}\)) or make division errors when calculating \(\mathrm{84 ÷ 8}\).

This causes them to arrive at incorrect numerical answers even with the right setup.

The Bottom Line:

This problem requires careful attention to the sequence of cost modifications and systematic algebraic manipulation. Success depends on accurately translating the real-world scenario into mathematical language while maintaining precision in calculations.