In a box of pens, the ratio of black pens to red pens is 8:1. There are 40 black pens...

1. TRANSLATE the problem information

• Given information:
  • Ratio of black pens to red pens is \(8:1\)
  • There are 40 black pens
  • Need to find: number of red pens

• What this tells us: For every 8 black pens, there is 1 red pen in the box

2. INFER the solution approach

• Since we know the ratio relationship and the actual number of black pens, we can set up a proportion to find the red pens
• Key insight: We need to find what number of red pens corresponds to 40 black pens, given the \(8:1\) ratio

3. TRANSLATE the ratio into a proportion

• Set up: \(40:x = 8:1\)
• Written as fractions: \(\frac{40}{x} = \frac{8}{1}\)

4. SIMPLIFY to solve for x

• Cross multiply: \(8x = 40 \times 1 = 40\)
• Divide both sides by 8: \(x = 40 \div 8 = 5\)

Answer: A. 5

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret the ratio direction, thinking "8 to 1" means 8 red pens for every 1 black pen instead of 8 black pens for every 1 red pen.

Following this incorrect interpretation, they calculate \(40 \times 8 = 320\) red pens.

This leads them to select Choice D (320).

Second Most Common Error:

Inadequate INFER reasoning: Students recognize the ratio correctly but don't know how to connect it to the given information about 40 black pens. They might just grab numbers from the ratio itself.

Without understanding how to apply the ratio, they might select Choice B (8) thinking this is the answer because 8 appears in the ratio.

The Bottom Line:

Ratio problems require careful attention to which quantity corresponds to which part of the ratio. Students must translate the ratio relationship correctly and then apply proportional reasoning to find the unknown quantity.