Vivian bought party hats and cupcakes for $71. Each package of party hats cost $3, and each cupcake cost $1....

1. TRANSLATE the problem information

• Given information:
  • Total amount spent: \(\$71\)
  • Cost per package of party hats: \(\$3\)
  • Cost per cupcake: \(\$1\)
  • Number of party hat packages bought: 10
  • Need to find: Number of cupcakes bought

2. INFER the solution strategy

• Since we know the total spent and can calculate what was spent on party hats, we can find what was spent on cupcakes
• Once we know the cupcake spending, we can divide by the price per cupcake to get the quantity

3. SIMPLIFY to find the cost of party hats

• Cost of party hats = \(10 \times \$3 = \$30\)

4. SIMPLIFY to find the amount spent on cupcakes

• Amount spent on cupcakes = Total spent - Cost of party hats
• Amount spent on cupcakes = \(\$71 - \$30 = \$41\)

5. SIMPLIFY to find the number of cupcakes

• Number of cupcakes = Amount spent on cupcakes ÷ Cost per cupcake
• Number of cupcakes = \(\$41 \div \$1 = 41\)

Answer: 41

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students may try to set up a complex algebraic equation immediately instead of recognizing the straightforward step-by-step approach. They might write something like '\(3(10) + 1\mathrm{x} = 71\)' which is correct but unnecessarily complicated, or they might get confused about how to set up the equation properly. This can lead to calculation errors or getting stuck on the algebra when simple arithmetic would work better.

The Bottom Line:

This problem rewards students who can break down multi-step word problems into logical sequences. The key insight is recognizing that you can work with known quantities first (party hats) to find unknown quantities (cupcakes), rather than trying to solve everything at once.