How to Solve: Speed, Distance, Time (GRE/GMAT) – Complete Guide

How to Solve: Speed, Distance, Time (GRE/GMAT) – Complete Guide

Score Impact: This question type appears 4-6 times per GRE/GMAT—mastering it can boost your Quant score by 5-7 points (enough to move from the 70th to the 90th percentile).

WHAT THIS QUESTION TYPE IS ACTUALLY TESTING

The exam isn’t testing your ability to plug numbers into formulas. It’s testing: ✅ Unit consistency – Can you keep track of hours vs. minutes, miles vs. kilometers? ✅ Relative motion – Can you model two moving objects (trains, cars, planes) interacting? ✅ Hidden assumptions – Are you missing a key detail (e.g., "stops for 10 minutes," "round trip")?

ANATOMY OF THE QUESTION

Structure Breakdown

1. Stem – Sets up the scenario (e.g., "A train leaves City A at 60 mph…").
2. Conditions – Adds complexity (e.g., "A second train leaves City B 30 minutes later at 80 mph…").
3. Question – Asks for time, distance, or speed (e.g., "When will they meet?").
4. Answer Choices – Usually 4-5 options, with 1-2 traps (e.g., wrong units, misapplied relative speed).

Representative Example (Full Question)

"A car travels from Town X to Town Y at 60 mph. On the return trip, it travels at 40 mph due to traffic. If the total time for the round trip is 5 hours, what is the distance between Town X and Town Y?"

Answer Choices: A) 100 miles B) 120 miles C) 150 miles D) 200 miles E) 240 miles

What to Ignore: - Irrelevant details (e.g., "The car is red," "Town X is north of Town Y"). - Overcomplicating (e.g., assuming acceleration/deceleration unless stated).

THE DECISION FRAMEWORK (Step-by-Step)

Run this process for every Speed-Distance-Time (SDT) question.

Step 1: Identify What’s Given and What’s Asked

• Given: Speeds, times, or distances (explicit or implied).
• Asked: Usually one of three things (pick one):
• Distance = Speed × Time
• Time = Distance / Speed
• Speed = Distance / Time

Example: In the question above, we’re given two speeds (60 mph, 40 mph) and total time (5 hours). We’re asked for distance (one-way).

Step 2: Assign Variables (If Needed)

• Let D = one-way distance (since round trip is 2D).
• Let T₁ = time to go from X to Y, T₂ = time to return.

Step 3: Write Equations

• X to Y: D = 60 × T₁ → T₁ = D / 60
• Y to X: D = 40 × T₂ → T₂ = D / 40
• Total time: T₁ + T₂ = 5

Step 4: Substitute and Solve

• (D / 60) + (D / 40) = 5
• Find common denominator (120): (2D / 120) + (3D / 120) = 5 5D / 120 = 5 → D = 120

Step 5: Check Units and Logic

• Units: mph × hours = miles (correct).
• Logic: 120 miles at 60 mph = 2 hours; 120 miles at 40 mph = 3 hours → Total = 5 hours (matches).

Step 6: Eliminate Wrong Answers

• A) 100 miles → Total time = (100/60) + (100/40) = 1.67 + 2.5 = 4.17 hours (❌)
• B) 120 miles → Total time = 5 hours (✅)
• C-E → Too large (❌)

Answer: B) 120 miles

Worked Examples

Example 1 – Straightforward (Relative Speed)

"Train A leaves Station X at 8 AM traveling at 60 mph. Train B leaves Station Y at 9 AM traveling toward Station X at 40 mph. If the stations are 300 miles apart, at what time will the trains meet?"

Step 1: Identify Given/Asked - Given: Two speeds, departure times, distance apart. - Asked: Time they meet (relative to 8 AM).

Step 2: Assign Variables - Let T = time after 8 AM when they meet. - Train A travels for T hours. - Train B travels for T - 1 hours (since it leaves at 9 AM).

Step 3: Write Equations - Distance covered by Train A: 60T - Distance covered by Train B: 40(T - 1) - Total distance: 60T + 40(T - 1) = 300

Step 4: Solve 60T + 40T - 40 = 300 → 100T = 340 → T = 3.4 hours 3.4 hours = 3 hours 24 minutes after 8 AM → 11:24 AM

Step 5: Check Logic - Train A: 60 × 3.4 = 204 miles - Train B: 40 × 2.4 = 96 miles - Total = 300 miles (✅)

Answer: 11:24 AM

Example 2 – Common Trap (Average Speed)

"A cyclist travels uphill at 10 mph and downhill at 20 mph. What is the average speed for the entire trip?"

Trap: Students assume average speed = (10 + 20) / 2 = 15 mph (❌).

Step 1: Identify Given/Asked - Given: Two speeds (uphill/downhill). - Asked: Average speed for entire trip (not arithmetic mean!).

Step 2: Assign Variables - Let D = one-way distance. - Total distance = 2D. - Time uphill = D / 10. - Time downhill = D / 20. - Total time = (D / 10) + (D / 20) = 3D / 20.

Step 3: Average Speed = Total Distance / Total Time - Average speed = 2D / (3D / 20) = (2D × 20) / 3D = 40 / 3 ≈ 13.33 mph

Step 4: Eliminate Wrong Answers - 15 mph (arithmetic mean, ❌) - 13.33 mph (correct, ✅)

Answer: 40/3 mph (or ~13.33 mph)

Example 3 – Hard Variant (Multiple Trips with Stops)

"A bus travels from Town A to Town B at 50 mph. It stops for 30 minutes, then returns at 60 mph. If the total time from departure to return is 5 hours, what is the distance between Town A and Town B?"

Step 1: Identify Given/Asked - Given: Two speeds, stop time (30 min = 0.5 hours), total time (5 hours). - Asked: One-way distance (D).

Step 2: Assign Variables - Time to B: T₁ = D / 50 - Time to A: T₂ = D / 60 - Total time: T₁ + 0.5 + T₂ = 5

Step 3: Write Equation (D / 50) + 0.5 + (D / 60) = 5 (D / 50) + (D / 60) = 4.5

Step 4: Solve Common denominator = 300: (6D / 300) + (5D / 300) = 4.5 → 11D / 300 = 4.5 → D = (4.5 × 300) / 11 ≈ 122.73 miles

Step 5: Check Logic - D ≈ 122.73 miles - Time to B: 122.73 / 50 ≈ 2.45 hours - Time to A: 122.73 / 60 ≈ 2.05 hours - Total time: 2.45 + 0.5 + 2.05 ≈ 5 hours (✅)

Answer: ~123 miles (closest option may be 120 or 125, depending on choices)

WRONG ANSWER PATTERNS

1. Unit Mismatch

• Why it looks right: Uses correct numbers but wrong units (e.g., minutes vs. hours).
• Why it’s wrong: 30 minutes ≠ 0.3 hours (it’s 0.5 hours).

2. Arithmetic Mean for Average Speed

• Why it looks right: "Average of 50 and 60 is 55 mph."
• Why it’s wrong: Average speed = total distance / total time, not (S₁ + S₂)/2.

3. Ignoring Stop/Wait Times

• Why it looks right: Calculates travel time but forgets stops.
• Why it’s wrong: Total time includes stops (e.g., "30-minute break").

4. Relative Speed Errors

• Why it looks right: Adds speeds when objects are moving toward each other (correct), but subtracts when moving away (incorrect).
• Why it’s wrong: Relative speed is sum for closing distance, difference for increasing distance.

Common Mistakes

1. Forgetting to Convert Units

• Why it happens: Mixing mph with km/h or minutes with hours.
• Correct approach: Always convert to consistent units first (e.g., 30 minutes = 0.5 hours).

2. Assuming Equal Distances for Average Speed

• Why it happens: Using (S₁ + S₂)/2 without checking if distances are equal.
• Correct approach: Only use (S₁ + S₂)/2 if distances are equal (e.g., round trip with same distance).

3. Misapplying Relative Speed

• Why it happens: Adding speeds when objects are moving in the same direction.
• Correct approach:
• Toward each other? Add speeds.
• Same direction? Subtract speeds.

4. Overcomplicating with Unnecessary Variables

• Why it happens: Assigning variables to every unknown (e.g., T₁, T₂, D₁, D₂).
• Correct approach: Let D = one-way distance (simplifies round trips).

5. Skipping the "Check Logic" Step

• Why it happens: Rushing to pick an answer without verifying.
• Correct approach: Plug answer back into the scenario (e.g., "Does 120 miles at 60 mph take 2 hours?").

TIME STRATEGY

• Target time: 1.5 - 2 minutes per question.
• When to skip: If you’re stuck after Step 3 (writing equations), flag and return.
• Minimum work needed:
• Identify given/asked (10 sec).
• Write 1-2 equations (30 sec).
• Solve and eliminate (1 min).

BACKSOLVING AND SHORTCUTS

1. Backsolving (Plugging in Answers)

• When to use: If the question asks for distance or time and answer choices are numbers.
• How:
• Start with B or D (middle values).
• Plug into the scenario and check if it fits.
• Example: In the round-trip question, test B) 120 miles:
  • Time to Y: 120 / 60 = 2 hours
  • Time to X: 120 / 40 = 3 hours
  • Total: 5 hours (✅)

2. Shortcut: Harmonic Mean for Average Speed (Round Trip)

• When to use: Same distance, two speeds (e.g., round trip).
• Formula: Average speed = 2 × (S₁ × S₂) / (S₁ + S₂)
• Example: 60 mph and 40 mph → 2 × (60 × 40) / (60 + 40) = 4800 / 100 = 48 mph

3. Elimination First

• When to use: If you’re unsure, eliminate impossible answers.
• Example: If a question asks for time and one answer is negative, cross it out immediately.

1-Minute Recap

"Here’s the exact process to solve any Speed-Distance-Time question in under 2 minutes:

1. Identify what’s given and what’s asked – Is it distance, time, or speed? Write it down.
2. Assign one variable – Let D = distance (if round trip, 2D). Keep it simple.
3. Write 1-2 equations – Use D = S × T for each leg of the trip.
4. Solve and check units – Did you mix minutes and hours? Fix it now.
5. Eliminate wrong answers – Plug your answer back in. If it doesn’t fit, try another.
6. Watch for traps – Average speed ≠ arithmetic mean. Stops count as time.

Most students lose points on unit errors or relative speed. Slow down, write it out, and you’ll get it right every time."

Final Tip:

Practice with a timer. The GRE/GMAT rewards speed + accuracy—not just math skills. Use this framework on 10-15 questions, and you’ll see your score climb. ?