A car completes a journey consisting of two segments, where the ratio of time (in minutes) spent on the first...

1. TRANSLATE the problem information

• Given information:
  • Original ratio of first segment to second segment = \(5:3\)
  • Second segment time decreases by 12 minutes
  • Need to find change in first segment time to maintain ratio

• What this tells us: We need to set up equations using the ratio relationship

2. TRANSLATE the ratio into algebra

• Let \(\mathrm{t_1}\) = original first segment time, \(\mathrm{t_2}\) = original second segment time
• From the \(5:3\) ratio: \(\mathrm{t_1/t_2 = 5/3}\)
• This means: \(\mathrm{t_1 = (5/3)t_2}\)

3. INFER the approach for maintaining ratios

• After the change: second segment time = \(\mathrm{t_2 - 12}\)
• To maintain the same \(5:3\) ratio, the new first segment time divided by \(\mathrm{(t_2 - 12)}\) must equal \(5/3\)
• Strategy: Set up proportion equation and solve

4. TRANSLATE the ratio maintenance condition

• \(\mathrm{(New\ first\ segment\ time)/(t_2 - 12) = 5/3}\)
• Therefore: \(\mathrm{New\ first\ segment\ time = (5/3)(t_2 - 12)}\)

5. SIMPLIFY to find the new first segment time

• New first segment time = \(\mathrm{(5/3)(t_2 - 12)}\)
• = \(\mathrm{(5/3)t_2 - (5/3)(12)}\)
• = \(\mathrm{(5/3)t_2 - 20}\)

6. INFER the change needed

• Change = New time - Original time
• = \(\mathrm{[(5/3)t_2 - 20] - (5/3)t_2}\)
• = \(\mathrm{-20}\)

Answer: (A) It must decrease by 20 minutes.

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students often misinterpret 'maintain the ratio' and think both segments should change by the same amount (12 minutes).

They reason: 'If the second segment decreases by 12, the first should also decrease by 12 to keep things proportional.' This leads them to select Choice (C) (decrease by 12 minutes).

Second Most Common Error:

Poor SIMPLIFY execution: Students correctly set up the proportion but make sign errors when expanding \(\mathrm{(5/3)(t_2 - 12)}\), getting \(\mathrm{(5/3)t_2 + 20}\) instead of \(\mathrm{(5/3)t_2 - 20}\).

This calculation error makes them think the first segment increases by 20 minutes, leading them to select Choice (B) (increase by 20 minutes).

The Bottom Line:

This problem requires understanding that maintaining a ratio doesn't mean making equal changes to both parts - the changes must be proportional to the original ratio itself.