The equation r = d/t + 5 relates rate r, distance d, and time t. Which equation correctly expresses t...

1. TRANSLATE the problem information

• Given equation: \(\mathrm{r = \frac{d}{t} + 5}\)
• Goal: Solve for t in terms of r and d

2. INFER the solving strategy

• Need to isolate t on one side
• First get the fraction \(\mathrm{\frac{d}{t}}\) by itself, then manipulate to solve for t
• Strategy: subtract 5, then use reciprocals

3. SIMPLIFY by removing the constant term

• Subtract 5 from both sides: \(\mathrm{r - 5 = \frac{d}{t}}\)
• Now we have the fraction \(\mathrm{\frac{d}{t}}\) isolated

4. SIMPLIFY by taking reciprocals

• Take reciprocal of both sides: \(\mathrm{\frac{1}{r - 5} = \frac{t}{d}}\)
• This flips both fractions

5. SIMPLIFY to solve for t

• Multiply both sides by d: \(\mathrm{t = \frac{d}{r - 5}}\)

Answer: (A) \(\mathrm{t = \frac{d}{r - 5}}\)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY skill: Students make sign errors when isolating the fraction \(\mathrm{\frac{d}{t}}\), adding 5 instead of subtracting 5 from both sides.

Starting incorrectly with \(\mathrm{r + 5 = \frac{d}{t}}\) instead of \(\mathrm{r - 5 = \frac{d}{t}}\), they eventually arrive at \(\mathrm{t = \frac{d}{r + 5}}\).

This leads them to select Choice (C) \(\mathrm{t = \frac{d}{r + 5}}\)

Second Most Common Error:

Poor SIMPLIFY execution: Students struggle with taking reciprocals correctly, confusing which terms go in numerator versus denominator.

After correctly getting \(\mathrm{r - 5 = \frac{d}{t}}\), they incorrectly take reciprocals to get \(\mathrm{\frac{r - 5}{1} = \frac{d}{t}}\), leading to \(\mathrm{t = \frac{r - 5}{d}}\).

This may lead them to select Choice (B) \(\mathrm{t = \frac{r - 5}{d}}\)

The Bottom Line:

This problem tests systematic algebraic manipulation skills. Success requires careful attention to signs when moving terms and proper understanding of reciprocals.