Speed, Time & Distance: One Formula, Three Faces

Every trip you have ever taken — the jeepney to school, the bus to the province, the MRT across the city — obeys one simple relationship between how fast you go, how far you travel, and how long it takes. Master that single relationship and its handful of twists (two vehicles meeting, trains passing platforms, boats fighting a current), and you unlock one of the most reliably tested families of questions on the Civil Service Exam. Let's build that mastery step by step.

1. Lesson Overview

Speed, time, and distance problems all spring from one formula — Distance = Speed × Time — rearranged to find whichever quantity is missing. The exam's real challenge lies in the variations: two objects moving toward or away from each other, one catching up to another, a train crossing a platform, or a boat traveling with or against a current.

On the Civil Service Exam, these questions are a dependable part of Numerical Reasoning, and they connect closely to Ratio (inverse proportion) and Averages (average speed). Counting the variations, this topic touches 4 to 7 items on a typical exam.

The difficulty is intermediate. The formula is trivial; the exam tests whether you can pick the right relative speed (add or subtract?), handle a train's own length, and — above all — keep your units consistent. Get those three habits right and this becomes a strong scoring area.

2. Learning Objectives

After completing this lesson you will be able to:

• Apply Distance = Speed × Time in all three rearranged forms.
• Convert fluently between km/h and m/s.
• Solve "moving toward / away" problems by adding speeds.
• Solve "catching up / overtaking" problems by subtracting speeds.
• Compute average speed correctly for equal-distance trips.
• Handle trains crossing poles, platforms, and other trains.
• Solve boats-and-streams (upstream/downstream) problems.
• Avoid the unit-mixing and add-vs-subtract mistakes that cost marks.

3. Prerequisites

You should be comfortable with:

• Multiplication and division, including with decimals and fractions.
• Unit awareness — knowing that km/h and m/s measure the same thing differently.
• Basic proportion — speed and time are inversely proportional for a fixed distance (see Ratio & Proportion).

A quick refresher on the habit that saves the most marks here: unit consistency. If speed is in km/h, time must be in hours and distance in km. If the problem mixes minutes with km/h, convert first. The two conversions you'll use constantly: to change km/h to m/s multiply by 5/18, and to change m/s to km/h multiply by 18/5 (i.e., 3.6). Keep these two factors at your fingertips.

4. Why This Topic Matters

Speed, time, and distance are the math of getting around, which every Filipino does daily:

• Commuting. Estimating whether the MRT or a Grab will get you to work on time; how long the bus to Baguio really takes.
• Travel planning. Fuel stops on a road trip; catching a ferry or flight with time to spare.
• Logistics and delivery. Riders and couriers estimating arrival times across the metro.
• Sports. A runner's pace, a swimmer's lap time, a cyclist's average speed.
• Public safety. Speed limits, stopping distances, and travel-time advisories.
• Rivers and boats. Bangka operators judging how the current helps going downstream and fights them coming back.

Understanding this topic makes you a sharper planner in daily life — and it's exactly the practical reasoning the exam rewards.

5. Core Concepts

We start with the one formula and add its variations one at a time.

The one formula, three ways

Distance = Speed × Time. Rearranged: Speed = Distance ÷ Time, and Time = Distance ÷ Speed.

A car travels at 60 km/h for 2.5 hours. Distance = 60 × 2.5 = 150 km.

Analogy: Picture a triangle with D on top and S and T below it (the same "cover the unknown" trick as the percentage triangle). Cover D → S × T. Cover S → D ÷ T. Cover T → D ÷ S. One picture, all three formulas.

Converting units — the make-or-break habit

Because speeds come in km/h and lengths often in meters (especially for trains), you must convert.

To go from km/h to m/s, multiply by 5/18. To go from m/s to km/h, multiply by 18/5 (= 3.6).

72 km/h × 5/18 = 20 m/s. And 12.5 m/s × 18/5 = 45 km/h.

Where does 5/18 come from? 1 km = 1,000 m and 1 hour = 3,600 s, so 1 km/h = 1,000/3,600 m/s = 5/18 m/s. You don't need to re-derive it — just multiply by 5/18 (slowing the number down) or 18/5 (speeding it up).

Two objects moving toward or away from each other

When two objects move toward each other, the gap between them closes at the sum of their speeds. When they move apart, the gap grows at the sum of their speeds too. Either way, for a shrinking or growing gap between oppositely-directed objects, add the speeds.

Two buses start 300 km apart and drive toward each other at 50 km/h and 70 km/h. When do they meet? Closing speed = 50 + 70 = 120 km/h. Time = 300 ÷ 120 = 2.5 hours.

Same-direction (overtaking) problems

When one object chases another moving the same way, the gap closes at the difference of their speeds (the faster minus the slower).

A cyclist at 15 km/h is 10 km ahead of a car going 45 km/h in the same direction. When does the car catch up? Relative speed = 45 − 15 = 30 km/h. Time = 10 ÷ 30 = 1/3 hour = 20 minutes.

Memory tip: Opposite directions → add; same direction → subtract. Toward/away combines effort; chasing cancels part of it.

Average speed is not the average of the speeds

For a trip covering equal distances at two speeds, more time is spent at the slower speed, so the average dips below the midpoint:

Average speed (equal distances at a and b) = (2 × a × b) ÷ (a + b).

Half a trip at 40 km/h and half at 60 km/h: (2 × 40 × 60) ÷ (40 + 60) = 4,800 ÷ 100 = 48 km/h, not 50.

But if the two legs take equal time (not equal distance), the simple average does work. Read carefully whether the problem splits by distance or by time.

Trains crossing objects

A train has length, so "crossing" something means the train travels its own length plus the length of the object.

• Crossing a pole or a person (no width): distance = the train's length.
• Crossing a platform or bridge: distance = train length + platform length.

A 150-meter train crosses a 100-meter platform in 20 seconds. Its speed = (150 + 100) ÷ 20 = 250 ÷ 20 = 12.5 m/s (= 45 km/h).

When two trains pass each other, the distance is the sum of both lengths, and their relative speed is added (opposite directions) or subtracted (same direction), exactly like the meeting/overtaking rules.

Boats and streams

A current helps a boat one way and fights it the other:

• Downstream (with the current): effective speed = boat's still-water speed + current speed.
• Upstream (against the current): effective speed = boat's still-water speed − current speed.

A boat does 12 km/h in still water; the current is 3 km/h. Downstream = 15 km/h; upstream = 9 km/h.

And working backward: if you know the downstream and upstream speeds, the still-water speed = (down + up) ÷ 2 and the current = (down − up) ÷ 2.

With every variation covered, let's picture, tabulate, and drill.

6. Visual Learning Suggestions

• [Illustration Suggestion] The D-S-T triangle. Draw a triangle with D on top, S and T below; label how covering each letter reveals its formula. Pin it to your reviewer wall.
• [Illustration Suggestion] Two arrows meeting. Draw two vehicles with arrows pointing toward each other and the gap labeled; write "closing speed = sum." Then a second picture with both arrows the same way and "gap closes at the difference."
• [Illustration Suggestion] Train over a platform. Draw the train just starting to enter and just fully leaving the platform; the front travels (train length + platform length). Seeing both endpoints explains the added lengths.
• [Illustration Suggestion] River with a boat. Arrows for the current and the boat; downstream arrows point the same way (add), upstream arrows oppose (subtract).

7. Formula Library

Why speeds add or subtract: it's all about the relative speed — how fast the distance between the two objects changes. Facing each other, both closings add up; chasing, only the surplus speed of the faster one matters, so you subtract.

8. Pattern Recognition

9. Problem-Solving Framework

Step 1 — Make the units consistent. Pick km/h + hours + km, or m/s + seconds + meters, and convert everything into one system. ↓ Step 2 — Identify the scenario: one object, two meeting, one overtaking, a train with length, or a boat in a current. ↓ Step 3 — Choose the relative speed: add for opposite directions, subtract for the same direction; add lengths for trains; add/subtract the current for boats. ↓ Step 4 — Apply D = S × T (or a rearrangement) and solve. ↓ Step 5 — Check units and reasonableness. Convert the answer to the unit the question wants, and confirm the magnitude makes sense.

Why Step 1 matters most: unit mixing (km/h with seconds, or meters with km) is the single biggest source of wrong answers here. Fix units before anything else.

10. Worked Examples

Beginner

Example 1. A runner covers 12 km in 1.5 hours. Find the speed. Solution: Speed = 12 ÷ 1.5 = 8 km/h. Difficulty: ★☆☆☆☆

Example 2. Convert 90 km/h to m/s. Solution: 90 × 5/18 = 25 m/s. Difficulty: ★☆☆☆☆

Intermediate

Example 3 (meeting). Two trains 480 km apart move toward each other at 60 km/h and 100 km/h. When do they meet? Solution: Closing speed = 160 km/h. Time = 480 ÷ 160 = 3 hours. Difficulty: ★★☆☆☆

Example 4 (overtaking). A truck at 40 km/h passes a point; 1 hour later a car at 60 km/h starts from the same point in the same direction. When does the car catch the truck? Thinking: In that 1 hour the truck goes 40 km ahead. The car closes the gap at 60 − 40 = 20 km/h. Solution: Time = 40 ÷ 20 = 2 hours after the car starts. Difficulty: ★★★☆☆

Example 5 (train crossing a pole). A 240-meter train crosses a pole in 12 seconds. Find its speed in km/h. Solution: Speed = 240 ÷ 12 = 20 m/s = 20 × 18/5 = 72 km/h. Difficulty: ★★☆☆☆

Advanced

Example 6 (train crossing a platform). A 180-meter train traveling at 54 km/h crosses a platform in 20 seconds. Find the platform's length. Thinking: Convert speed to m/s: 54 × 5/18 = 15 m/s. Distance covered = 15 × 20 = 300 m = train + platform. Solution: Platform = 300 − 180 = 120 meters. Difficulty: ★★★☆☆

Example 7 (average speed). A driver goes to a town at 30 km/h and returns along the same road at 45 km/h. Find the average speed. Solution: 2 × 30 × 45 ÷ (30 + 45) = 2,700 ÷ 75 = 36 km/h. Difficulty: ★★★☆☆

Example 8 (boat round trip). A boat's still-water speed is 10 km/h and the current is 2 km/h. It travels 24 km downstream and back. Find the total time. Solution: Downstream 12 km/h → 24 ÷ 12 = 2 h. Upstream 8 km/h → 24 ÷ 8 = 3 h. Total = 5 hours. Difficulty: ★★★★☆

Civil Service Exam Level

Example 9 (two trains crossing). Two trains, 140 m and 160 m long, run in opposite directions at 40 km/h and 50 km/h. How long do they take to completely pass each other? Thinking: Opposite directions → add speeds; distance = sum of lengths. Solution: Combined speed = 90 km/h = 90 × 5/18 = 25 m/s. Total length = 300 m. Time = 300 ÷ 25 = 12 seconds. Difficulty: ★★★★☆

Example 10 (recover boat and current). A boat goes downstream at 15 km/h and upstream at 9 km/h. Find its still-water speed and the current's speed. Solution: Still speed = (15 + 9) ÷ 2 = 12 km/h; current = (15 − 9) ÷ 2 = 3 km/h. Difficulty: ★★★☆☆

Example 11 (equal time vs equal distance). A commuter travels 2 hours at 30 km/h and then 3 hours at 50 km/h. Find the average speed for the whole journey. Thinking: This splits by time, not distance, so add total distance over total time (do NOT use 2ab/(a+b)). Solution: Distance = 30 × 2 + 50 × 3 = 60 + 150 = 210 km; total time = 5 h. Average = 210 ÷ 5 = 42 km/h. Difficulty: ★★★★☆

Example 12 (catch-up with a head start in distance). A thief runs at 8 km/h. A police officer 200 meters behind starts chasing at 10 km/h. How long (in minutes) to catch the thief? Thinking: Same direction → relative speed 10 − 8 = 2 km/h. Gap 200 m = 0.2 km. Solution: Time = 0.2 ÷ 2 = 0.1 hour = 6 minutes. Difficulty: ★★★★★

11. Exam Tricks

• Unit sabotage. The problem gives speed in km/h but time in minutes, or a train length in meters with speed in km/h. Convert everything to one system first — this is where most errors are planted.
• Add vs. subtract. Opposite directions add; same direction subtracts. Mixing these up flips the answer, and both versions are offered as choices.
• The average-speed trap. For a there-and-back trip, never take the plain average of the two speeds; use 2ab/(a+b). But if the split is by time, the plain (weighted) average is right — read carefully.
• Forgetting the train's length. Crossing a platform covers train + platform; crossing a pole covers just the train. Dropping a length gives a wrong distance.
• Upstream sign error. Going upstream subtracts the current. Adding it (as if downstream) is a classic slip.

12. Common Mistakes

• Mixing units — km/h with minutes, or meters with kilometers — without converting.
• Adding speeds for a same-direction chase instead of subtracting (or vice versa for a meeting).
• Averaging two speeds directly on a round trip instead of using the harmonic-mean formula.
• Forgetting to add both lengths when a train crosses a platform or another train.
• Adding the current when going upstream instead of subtracting it.
• Using 2ab/(a+b) when the trip is split by time, where a plain weighted average is correct.

13. Shortcuts

• 5/18 and 18/5 are your fastest tools — memorize them so unit conversion is instant.
• Relative speed collapses two-object problems into one: work with the closing/gap speed and a single distance.
• Harmonic mean 2ab/(a+b) is the one-line answer for equal-distance average speed.
• Boat sum-and-difference: still speed = (down + up)/2, current = (down − up)/2 — recover both in one step.
• Whole-number distance trick: for meeting problems, distance ÷ (sum of speeds) gives the time directly; no need to track each vehicle separately.

14. Memory Techniques

• "D over S and T." The triangle picture gives all three formulas.
• "Opposite adds, same subtracts." The relative-speed rule in three words.
• "Times five, over eighteen." km/h to m/s (× 5/18); flip it to go back.
• "Train eats the platform." Crossing distance = train length + object length.
• "Down adds, up subtracts." Current helps downstream, fights upstream.

15. Real Civil Service Exam Strategy

• Time: a direct D = S × T item takes under 30 seconds; trains, boats, and relative-speed problems 60–90 seconds.
• Convert units first, always — it prevents the most common and most costly error.
• Name the scenario (meeting, chasing, train, boat) before choosing add or subtract.
• Distinguish equal-distance from equal-time average-speed problems; the wrong formula is a listed choice.
• Sanity-check magnitudes: a jeepney averaging 500 km/h or a 6-second answer for a slow chase should trigger a recheck.

16. Practice Questions

Easy

1. A bus travels 240 km in 4 hours. Find its speed.
2. Convert 36 km/h to m/s.
3. How far does a car at 80 km/h travel in 3 hours?

Medium

4. Two cars 360 km apart drive toward each other at 70 and 50 km/h. When do they meet?
5. A 200-meter train crosses a pole in 8 seconds. Find its speed in km/h.
6. A boat's still-water speed is 15 km/h; the current is 5 km/h. Find its downstream and upstream speeds.

Hard

7. A car covers a distance at 40 km/h and returns at 60 km/h. Find the average speed for the round trip.
8. A 120-meter train at 45 km/h crosses a platform in 16 seconds. Find the platform's length.

Challenge

9. A cyclist starts at 12 km/h. Two hours later a motorbike leaves the same point at 36 km/h in the same direction. How long after the motorbike starts does it catch the cyclist?
10. Two trains, 150 m and 100 m long, move in the same direction at 72 km/h and 54 km/h. How long does the faster train take to completely overtake the slower one?

Answers and Explanations

1. 60 km/h. 240 ÷ 4.
2. 10 m/s. 36 × 5/18.
3. 240 km. 80 × 3.
4. 3 hours. Closing speed 120 km/h; 360 ÷ 120 = 3.
5. 90 km/h. 200 ÷ 8 = 25 m/s = 25 × 18/5 = 90 km/h.
6. Downstream 20 km/h, upstream 10 km/h. 15 + 5 and 15 − 5.
7. 48 km/h. 2 × 40 × 60 ÷ 100 = 48.
8. 80 meters. 45 km/h = 12.5 m/s; distance = 12.5 × 16 = 200 m; platform = 200 − 120 = 80 m.
9. 1 hour. In 2 hours the cyclist is 24 km ahead; relative speed 36 − 12 = 24 km/h; 24 ÷ 24 = 1 hour.
10. 50 seconds. Same direction → relative speed 72 − 54 = 18 km/h = 5 m/s; total length 150 + 100 = 250 m; 250 ÷ 5 = 50 seconds.

17. Summary

Everything here grows from Distance = Speed × Time, rearranged as needed. The habits that turn it into marks are: convert units first (km/h ↔ m/s via 5/18 and 18/5), pick the right relative speed (add for opposite directions, subtract for the same direction), remember a train covers its own length plus the object's, and treat a current as adding downstream and subtracting upstream. For average speed over equal distances, use 2ab/(a+b) — but if the trip splits by equal time, use a plain weighted average. Name the scenario, fix the units, choose add-or-subtract deliberately, and this becomes one of the exam's most dependable topics.

18. Cheat Sheet

19. Frequently Asked Questions

When do I add speeds and when do I subtract? Add when the objects move in opposite directions (toward or away) — the gap changes at the sum of speeds. Subtract when they move the same way and one chases the other — the gap changes at the difference.

Why isn't round-trip average speed just the average of the two speeds? Because you spend more time on the slower leg, so it weighs more. The harmonic mean 2ab/(a+b) accounts for that, giving a value below the midpoint.

How do I remember the 5/18 conversion? 1 km/h = 1000 m per 3600 s = 5/18 m/s. Multiply by 5/18 to slow the number down (km/h → m/s) and by 18/5 to speed it up (m/s → km/h).

What distance does a train cover when crossing a platform? Its own length plus the platform's length — because the whole train must clear the whole platform. Crossing a pole (no width) covers just the train's length.

How do I find a boat's still-water speed? Average the downstream and upstream speeds: still speed = (down + up)/2. The current is half their difference: (down − up)/2.

20. Mastery Checklist

• ☐ I can use D = S × T in all three forms.
• ☐ I convert between km/h and m/s using 5/18 and 18/5.
• ☐ I add speeds for opposite directions and subtract for the same direction.
• ☐ I use 2ab/(a+b) for equal-distance average speed, and a weighted average when split by time.
• ☐ I include the train's length (and the object's) when it crosses something.
• ☐ I add the current downstream and subtract it upstream.
• ☐ I can recover still-water speed and current from downstream/upstream speeds.
• ☐ I always make units consistent before computing.

Tick them all and speed–time–distance becomes a fast, dependable scoring area — and you'll plan real trips more sharply too.