Time, speed, distance and work: everything is one ratio

Time and speed questions look varied — cars, trains, boats, taps, workers — but they are all the same equation seen from different sides, plus a habit of spotting inverse proportion. Learn the one relationship and the handful of standard adjustments, and the topic stops being about setting up equations and becomes about reading which quantity is fixed.

The one relationship, and the proportions inside it

Everything starts from speed = distance ÷ time, rearranged as needed. What makes it quick is reading which of the three is constant:

• Distance fixed: speed and time are inversely proportional. If speed rises in the ratio 3 : 4, time falls in the ratio 4 : 3. Most “if he walks faster he saves x minutes” questions are this and nothing more.
• Time fixed: distance is proportional to speed.
• Speed fixed: distance is proportional to time.

The unit conversion you will use constantly is km/h to m/s by multiplying by 5/18, and the reverse by 18/5. Do this before anything else so the units never fight you.

Average speed: the harmonic trap

The single most tested subtlety is average speed over equal distances at different speeds. Because more time is spent at the slower speed, the average is not (a + b)/2 but the harmonic mean 2ab/(a + b). Go from home to office at 40 and back at 60, and the average is 2 × 40 × 60 / 100 = 48, not 50. Equal times at each speed is the only case where the ordinary average is right — and questions are worded carefully to tell you which it is.

Trains, boats and relative speed

These are the same equation with the distance or the speed adjusted:

• Trains. Crossing a pole covers the train’s own length; crossing a platform covers train length plus platform length. Two trains passing use relative speed — the sum of speeds moving in opposite directions, the difference moving the same way.
• Boats and streams. Downstream speed is boat + stream, upstream is boat − stream. So boat speed is the average of the two, and stream speed is half their difference — two facts that answer most river questions on sight.

Relative speed is just the observation that from one moving object’s frame, only the gap between speeds matters.

Time and work: assume the total is the LCM

The clean method is to stop using fractions. Take the total work as the LCM of the times.

• A finishes in 12 days, B in 18. Take the job as 36 units. Then A does 3 units/day, B does 2 units/day, together 5 units/day, so they finish in 36 ÷ 5 = 7.2 days.
• This handles workers joining and leaving, alternate days, and “A works then B takes over” without ever adding 1/12 to 1/18.

Time and work is also the same shape as pipes and cisterns (a leak is negative work) and as distance (work done = rate × time is speed = distance ÷ time rearranged), so the habits carry straight across.

A worked reading

A train 180 m long passes a 120 m platform in 20 seconds; how fast is it in km/h? The distance is length + platform = 300 m, so speed = 300 ÷ 20 = 15 m/s, and 15 × 18/5 = 54 km/h. Three steps — fix the distance, divide, convert — for a question written to look like it needs an equation.

Where the marks leak

Averaging speeds directly. For equal distances use 2ab/(a + b); reaching for (a + b)/2 is the most common wrong answer in the topic.

Wrong crossing distance. Pole means train length only; platform, bridge or another train means add the second length.

Relative-speed sign. Opposite directions add, same direction subtract — decide before you compute.

Adding times as if they were speeds. You can add times and add distances, but never add speeds unless the times spent are equal.

Practice that transfers

Because trains, boats, pipes and workers are one relationship in costume, practise them mixed rather than by sub-type. Work through the time, speed, distance and work practice set, and keep the arithmetic underneath quick with calculation speed so the ratios resolve without slowing you down.