8j = k + 15m The given equation relates the distinct positive numbers j, k, and m. Which equation correctly...

1. INFER what the problem is asking

• Given: \(\mathrm{8j = k + 15m}\)
• Goal: Express \(\mathrm{j}\) in terms of \(\mathrm{k}\) and \(\mathrm{m}\)
• What this means: Solve for \(\mathrm{j}\) (get \(\mathrm{j}\) by itself on one side)

2. SIMPLIFY by using algebraic manipulation

• Start with: \(\mathrm{8j = k + 15m}\)
• To isolate \(\mathrm{j}\), divide both sides by 8:
  • Left side: \(\mathrm{8j ÷ 8 = j}\)
  • Right side: \(\mathrm{(k + 15m) ÷ 8 = \frac{k + 15m}{8}}\)
• Result: \(\mathrm{j = \frac{k + 15m}{8}}\)

Answer: D. j = (k+15m)/8

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students incorrectly apply the division to only part of the right side, writing \(\mathrm{j = \frac{k}{8} + 15m}\) instead of \(\mathrm{j = \frac{k + 15m}{8}}\).

They think: "I need to divide by 8, so I'll divide the first term: \(\mathrm{k ÷ 8 = \frac{k}{8}}\), and keep \(\mathrm{15m}\) as is."

This violates order of operations - when dividing an entire expression by a number, the whole expression must be in parentheses. This leads them to select Choice A (j = k/8 + 15m).

Second Most Common Error:

Poor INFER reasoning: Students misunderstand what "express j in terms of k and m" means and think they need to multiply both sides by something instead of dividing.

They might think: "To get j by itself, I need to multiply," leading to \(\mathrm{j = 8(k + 15m)}\). This makes \(\mathrm{j}\) much larger than it should be and leads them to select Choice C (j = 8(k + 15m)).

The Bottom Line:

This problem tests whether students can correctly apply the division property of equality while maintaining proper order of operations. The key insight is that dividing both sides by \(\mathrm{8}\) means the entire right side expression needs parentheses.