The combined original price for a mirror and a vase is $60. After a 25% discount to the mirror and...

1. TRANSLATE the given information into mathematical expressions

• Given information:
  • Original prices: mirror = m dollars, vase = v dollars
  • Combined original price = $60
  • Mirror discount = 25%, Vase discount = 45%
  • Combined sale price after discounts = $39

• What this tells us: We need two equations to represent these two price relationships

2. TRANSLATE the original price relationship

• The original prices add up to $60:
  \(\mathrm{m + v = 60}\)

This gives us our first equation.

3. INFER what the discounts mean for the sale prices

• A 25% discount on the mirror means you pay \(\mathrm{100\% - 25\% = 75\%}\) of the original price
• A 45% discount on the vase means you pay \(\mathrm{100\% - 45\% = 55\%}\) of the original price

The key insight: discount percentage tells us what we DON'T pay, not what we DO pay.

4. TRANSLATE the sale price relationship

• Mirror sale price = \(\mathrm{75\% \text{ of } m = 0.75m}\)
• Vase sale price = \(\mathrm{55\% \text{ of } v = 0.55v}\)
• Combined sale price = $39

\(\mathrm{0.75m + 0.55v = 39}\)

This gives us our second equation.

5. Identify the complete system

The system of equations is:

• \(\mathrm{m + v = 60}\)
• \(\mathrm{0.75m + 0.55v = 39}\)

Answer: C

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE reasoning: Students confuse the discount percentage with the remaining percentage they actually pay.

For example, seeing "25% discount on mirror," they might think the coefficient should be 0.25 instead of 0.75. This leads them to write equations like:

• \(\mathrm{m + v = 60}\)
• \(\mathrm{0.25m + 0.45v = 39}\)

This may lead them to select Choice D (\(\mathrm{0.25m + 0.45v = 39}\)).

Second Most Common Error:

Poor TRANSLATE execution: Students mix up which item gets which discount when setting up the second equation.

They might correctly understand that discounts create coefficients of 0.75 and 0.55, but assign them to the wrong variables:

• \(\mathrm{m + v = 60}\)
• \(\mathrm{0.55m + 0.75v = 39}\) (backwards assignment)

This may lead them to select Choice A (\(\mathrm{0.55m + 0.75v = 39}\)).

The Bottom Line:

This problem tests whether students can correctly translate percentage discounts into algebraic coefficients. The tricky part is remembering that a discount tells you what percentage you DON'T pay, so you need to subtract from 100% to get the coefficient for what you DO pay.