Angela is playing a video game. In this game, players can score points only by collecting coins and stars. Each...

1. TRANSLATE the problem information

• Given information:
  • Each coin worth \(\mathrm{c}\) points, each star worth \(\mathrm{s}\) points
  • Game 1: 20 coins, 10 stars, 700 total points
  • Game 2: 25 coins, 12 stars, 850 total points

2. INFER the relationship structure

• Total points = Points from coins + Points from stars
• Points from coins = (number of coins) × (points per coin) = (number of coins) × \(\mathrm{c}\)
• Points from stars = (number of stars) × (points per star) = (number of stars) × \(\mathrm{s}\)

3. TRANSLATE each game into an equation

• Game 1: 20 coins × \(\mathrm{c}\) + 10 stars × \(\mathrm{s}\) = 700 total points
  → \(\mathrm{20c + 10s = 700}\)

• Game 2: 25 coins × \(\mathrm{c}\) + 12 stars × \(\mathrm{s}\) = 850 total points
  → \(\mathrm{25c + 12s = 850}\)

4. Form the system

The system of equations is:

\(\mathrm{20c + 10s = 700}\)
\(\mathrm{25c + 12s = 850}\)

Answer: B

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students mix up which numbers represent quantities versus which represent the total.

They might think "20 coins and 700 points" means coins are worth 700 points each, or they might switch the coefficients and constants. This confusion about what multiplies with what leads to equations like \(\mathrm{10c + 20s = 700}\) or \(\mathrm{20c + 700s = 10}\).

This may lead them to select Choice A (\(\mathrm{10c + 20s = 700, 12c + 25s = 850}\)) or Choice C (\(\mathrm{20c + 700s = 10, 25c + 850s = 12}\))

The Bottom Line:

Success depends on carefully identifying what each number represents in the relationship "total = (quantity₁ × value₁) + (quantity₂ × value₂)" before writing any equations.