v = d/(T - t)The given equation is used to relate the average velocity v of an object to its...

1. INFER the approach needed

• Given: \(\mathrm{v = \frac{d}{(T - t)}}\)
• Goal: Solve for T (isolate T on one side)
• Strategy: Since T is in a denominator, we need to clear that denominator first

2. SIMPLIFY by clearing the denominator

• Multiply both sides by the entire denominator (T - t):
  \(\mathrm{v(T - t) = d}\)
• This eliminates the fraction and gets T out of the denominator

3. SIMPLIFY by isolating the T term

• Divide both sides by v to get the parenthetical expression alone:
  \(\mathrm{T - t = \frac{d}{v}}\)
• Now we have T with just a subtracted term

4. SIMPLIFY to solve for T completely

• Add t to both sides to isolate T:
  \(\mathrm{T = t + \frac{d}{v}}\)

Answer: D. T = t + d/v

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students often make mistakes when clearing denominators, especially with more complex expressions.

Many students might try to "flip" the equation incorrectly, writing something like \(\mathrm{T - t = \frac{v}{d}}\) instead of properly multiplying both sides by (T - t). Or they might forget to apply the multiplication to both the v and the entire denominator, leading to incorrect intermediate steps.

This leads to confusion and often results in selecting Choice A (\(\mathrm{T = t - \frac{d}{v}}\)) or Choice B (\(\mathrm{T = \frac{d}{v} - t}\)) due to sign errors.

Second Most Common Error:

Poor INFER strategy: Students may not recognize that clearing the denominator is the most efficient first step.

Some students attempt to work directly with the fraction, trying to manipulate it without first eliminating the denominator. This makes the problem unnecessarily complex and prone to errors. They might attempt to move terms around while T is still in the denominator, creating confusion about proper algebraic procedures.

This causes them to get stuck and guess among the answer choices.

The Bottom Line:

This problem tests systematic algebraic thinking. Success depends on recognizing that denominators should be cleared first, then following through with careful algebraic manipulation. The key insight is treating this as a multi-step process where each operation serves a specific purpose in isolating the target variable.