A shipment consists of 5-pound boxes and 10-pound boxes with a total weight of 220 pounds. There are 13 10-pound...

1. TRANSLATE the problem information

• Given information:
  • Shipment has 5-pound boxes and 10-pound boxes
  • Total weight is 220 pounds
  • There are 13 10-pound boxes
  • We need to find the number of 5-pound boxes

• Let \(\mathrm{x}\) = number of 5-pound boxes
• Let \(\mathrm{y}\) = number of 10-pound boxes
• Weight equation: \(\mathrm{5x + 10y = 220}\)

2. INFER the solving approach

• We have one equation with two variables, but we know \(\mathrm{y = 13}\)
• This means we can substitute the known value and solve for \(\mathrm{x}\)
• Strategy: Substitute \(\mathrm{y = 13}\) into our equation

3. SIMPLIFY through substitution and algebraic steps

• Substitute \(\mathrm{y = 13}\):

\(\mathrm{5x + 10(13) = 220}\)

• Multiply:

\(\mathrm{5x + 130 = 220}\)

• Subtract 130 from both sides:

\(\mathrm{5x = 90}\)

• Divide by 5:

\(\mathrm{x = 18}\)

Answer: D. 18

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students may set up the wrong equation or confuse what each variable represents. For example, they might write \(\mathrm{x + y = 220}\) (forgetting about the different weights) or mix up which variable represents which type of box.

This leads to an entirely wrong equation and incorrect final answer, causing them to get stuck and guess randomly.

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up \(\mathrm{5x + 10(13) = 220}\) but make arithmetic mistakes like calculating \(\mathrm{10 × 13 = 120}\) instead of \(\mathrm{130}\), or incorrectly solving \(\mathrm{5x = 90}\).

This may lead them to select Choice B (10) if they miscalculate and get \(\mathrm{x = 10}\), or other incorrect choices based on their arithmetic errors.

The Bottom Line:

This problem tests whether students can translate a real-world scenario into a mathematical equation and then apply basic substitution. The key challenge is recognizing that even though there are two types of boxes, knowing the quantity of one type allows you to solve for the other using the total weight constraint.