Speed, distance and time

Three quantities: speed, distance and time. Know two of them and the third always follows — and this is one of the few formulas people genuinely use every day: planning a journey, reading a timetable, working out whether there is time to get there.

One formula, three forms

Speed says how much distance is covered per unit of time:

Rearranging it gives the other two:

These are not three separate formulas to memorise but one relation written three ways. If you remember that speed is distance over time, you can derive the rest.

The units have to agree

A speed is always a quotient of a distance unit and a time unit — which is exactly what its notation shows:

• kilometres and hours give $\text{km}/\text{h}$,
• metres and seconds give $\text{m}/\text{s}$.

If the distance is in kilometres and the time in minutes, the answer is neither of the two — the quantities have to be brought into matching units first.

Between the two commonest units there is a single factor:

It comes from an hour having $3600$ seconds and a kilometre $1000$ metres: $3600 : 1000 = 3.6$. From m/s to km/h we multiply by $3.6$, the other way we divide. When you just need a value — in knots or miles per hour too — convert it with our speed converter.

A clock does not count in tens

Most slips in time arithmetic come from one habit: the numbers around us run in tens, and time does not.

That is why $2.5$ hours is $2$ hours and 30 minutes, not $2$ hours and $50$ minutes. Decimal notation and clock notation are two different things: $2.5\ \text{h} = 150$ min.

Time differences

To work out how long something between two clock times lasts, the easiest route is to stop at the full hour: count the minutes up to the next full hour, then the whole hours, then the remaining minutes.

When the stretch runs through midnight, split it in two: up to $24{:}00$ and after $0{:}00$. A train leaving at $22{:}40$ and arriving at $6{:}15$ travels $1$ h $20$ min plus $6$ h $15$ min, that is $7$ hours and $35$ minutes. Subtracting $6{:}15$ from $22{:}40$ would give a meaningless answer — time does not run sixteen hours backwards.

If you need hours turned into minutes, seconds or days, our time converter does it.

Calendar arithmetic

Months are not equal, so the number of days between two dates cannot come from a single multiplication:

• 31 days: January, March, May, July, August, October, December,
• 30 days: April, June, September, November,
• 28 days: February — or $29$ in a leap year.

A year is a leap year when it is divisible by $4$, but a full century only when it is divisible by $400$. So $2024$ and $2028$ are leap years, $2000$ was one, and $2100$ will not be. A leap year has $366$ days.

Counting days between dates, work month by month: first the days left in the starting month, then any whole months, then the days into the final month.

Common mistakes

• Mixing units — a distance in kilometres with a time in minutes gives neither km/h nor m/s; make the units agree first.
• Reading 2.5 h as 2 h 50 min — an hour has $60$ minutes, so $2.5$ h is $2$ h $30$ min.
• Subtracting clock times across midnight — the stretch has to be split into the part before and the part after midnight.
• Assuming every month has 30 days — over a deadline that drifts by a day or two; check the length of the actual month.
• Treating every year divisible by 4 as a leap year — full centuries are leap years only when divisible by $400$.
• Confusing average speed with instantaneous speed — $v = s/t$ gives the average over the whole route, not what the speedometer read on any part of it.