Time, Speed and Distance: Every Formula Your Child Needs

Time, speed and distance is the topic where word problems suddenly get long. Trains crossing platforms, boats going upstream, two cars leaving different cities — the arithmetic is simple, but the setup confuses students.

The good news is that the whole topic rests on one relationship and about six standard situations. Learn those and the rest is reading comprehension.

The one relationship

> Distance = Speed × Time

Everything else is this rearranged:

• Speed = Distance ÷ Time
• Time = Distance ÷ Speed

A useful way to remember it is the triangle: D on top, S and T below. Cover the one you want and the arrangement of the other two tells you what to do.

Unit conversion: the step most marks are lost on

Speeds are given in km/h; lengths in metres; times in seconds. Mixing them is the single most common error.

> km/h → m/s: multiply by 5/18 > m/s → km/h: multiply by 18/5

Worth memorising: 36 km/h = 10 m/s, 54 km/h = 15 m/s, 72 km/h = 20 m/s.

Teach the habit of converting everything to one unit system before touching the question.

Situation 1: average speed

If a car travels at 60 km/h for one hour and 40 km/h for one hour, the average is 50 km/h. But if it covers the same distance at each speed, the average is not 50.

For equal distances at speeds a and b:

> Average speed = 2ab ÷ (a + b)

For 60 and 40: (2 × 60 × 40) ÷ 100 = 48 km/h

This is the harmonic mean, and it is always lower than the simple average. Exams test this constantly — remember that more time is spent at the slower speed, so the average leans slower.

Situation 2: relative speed

When two objects move, what matters is their speed relative to each other.

• Same direction: subtract the speeds
• Opposite directions: add the speeds

Two trains, 60 km/h and 40 km/h, moving towards each other, 200 km apart. When do they meet?

• Relative speed = 100 km/h → time = 200 ÷ 100 = 2 hours

Same problem but same direction:

• Relative speed = 20 km/h → time = 200 ÷ 20 = 10 hours

Situation 3: trains crossing things

This is the classic, and the only difficulty is knowing what distance to use.

A 200 m train at 72 km/h crosses a 100 m platform.

• 72 km/h = 20 m/s
• Distance = 200 + 100 = 300 m
• Time = 300 ÷ 20 = 15 seconds

The rule to remember: a pole has no length, a platform does.

Situation 4: boats and streams

• Downstream speed = boat speed + stream speed
• Upstream speed = boat speed − stream speed

And working backwards from the two:

• Boat speed = (downstream + upstream) ÷ 2
• Stream speed = (downstream − upstream) ÷ 2

A boat goes 30 km downstream in 2 hours and returns in 3 hours.

• Downstream = 15 km/h, upstream = 10 km/h
• Boat speed = 12.5 km/h, stream = 2.5 km/h

Situation 5: proportionality

For a fixed distance, speed and time are inversely proportional.

If a bus increases speed by 25%, by what percentage does the journey time fall?

Speed becomes 5/4 of before, so time becomes 4/5 — a fall of 1/5 = 20%.

This shortcut avoids setting up equations for a whole class of questions.

Situation 6: two-part journeys

A man walks part of the way at 4 km/h and cycles the rest at 12 km/h, covering 24 km in 4 hours. How far did he walk?

Let walking distance = x, so cycling = 24 − x.

• x/4 + (24 − x)/12 = 4
• Multiply by 12: 3x + 24 − x = 48 → 2x = 24 → x = 12 km

Setting up one equation in one unknown handles almost every mixed-journey question.

A practice order that works

1. Unit conversion drills until 5/18 is automatic
2. Simple D = S × T questions
3. Relative speed, both directions
4. Trains crossing poles, then platforms, then other trains
5. Boats and streams
6. Mixed and two-part journeys

Fifteen minutes daily for three weeks covers this comfortably. Our free worksheets include the kind of timed arithmetic practice that keeps the divisions quick.

Frequently asked questions

What class is this topic for? It begins around Class 7 and is a staple of every competitive exam afterwards.

Why is the average speed formula different from a normal average? Because the two halves of the journey take different amounts of time. The slower stretch occupies more of the total time, so it pulls the average down. Only when the times are equal does the simple average apply.

What is the most common mistake? Forgetting to convert km/h to m/s in train questions, and using the train's length alone when a platform is involved.

Do I need to memorise all these formulas? Only D = S × T and the 5/18 conversion. Everything else can be derived, though knowing the relative-speed rules by heart saves time.

Is this useful outside exams? Yes — estimating journey times, fuel stops and arrival times all use exactly this reasoning.

The arithmetic has to be quick

Every question here ends in a division. When those divisions are instant, the topic feels easy; when they are not, it feels impossible.

Our Quantitative Aptitude course builds both the speed and the setups, for Class 6 upwards.

Book a free demo class and see how your child approaches a word problem.

Cover image: Acroterion — CC BY-SA 4.0.