At a fundraiser, the ticket price, in dollars, is set to be x increased by 20% of x, and the...

1. TRANSLATE the problem information

• Given information:
  • Ticket price = x increased by 20% of x
  • Number of tickets sold = \(\mathrm{150 - 5x}\)
  • Revenue \(\mathrm{R(x) = price \times number~of~tickets}\)
  • Need to find \(\mathrm{R(10)}\)

• What this tells us:
  • Price = \(\mathrm{x + 0.20x = 1.20x}\) dollars
  • Quantity = \(\mathrm{150 - 5x}\) tickets

2. INFER what we need to do

• To find \(\mathrm{R(10)}\), we need to substitute \(\mathrm{x = 10}\) into our revenue formula
• This means calculating both the price and quantity when \(\mathrm{x = 10}\)
• Then multiply them together for the revenue

3. SIMPLIFY by substituting x = 10

• Price when \(\mathrm{x = 10}\): \(\mathrm{1.20(10) = 12}\) dollars
• Quantity when \(\mathrm{x = 10}\): \(\mathrm{150 - 5(10) = 150 - 50 = 100}\) tickets
• Revenue: \(\mathrm{R(10) = 12 \times 100 = 1,200}\) dollars

Answer: B. 1,200

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may interpret "x increased by 20% of x" incorrectly as \(\mathrm{x + 20}\) instead of \(\mathrm{x + 0.20x}\).

If they use price = \(\mathrm{x + 20}\), then when \(\mathrm{x = 10}\):

• Price = \(\mathrm{10 + 20 = 30}\) dollars
• Quantity = \(\mathrm{150 - 5(10) = 100}\) tickets
• Revenue = \(\mathrm{30 \times 100 = 3,000}\) dollars

This doesn't match any answer choice, leading to confusion and guessing.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{1.20x}\) but make arithmetic errors in the final calculation.

For example, they might calculate \(\mathrm{1.20 \times 10 = 10.20}\) instead of 12, leading to:

• Revenue = \(\mathrm{10.20 \times 100 = 1,020}\) dollars

This also doesn't match any answer choice, causing them to second-guess their approach.

The Bottom Line:

This problem tests your ability to translate percentage language into mathematical expressions. The key insight is recognizing that "increased by 20%" means multiplying by 1.20, not adding 20.