Rates, work and travel

One relationship, three forms

Travel questions rest on $\text{distance} = \text{rate} \times \text{time}$, and work questions on $\text{work} = \text{rate} \times \text{time}$. Every question gives two of the three quantities, sometimes in disguise, and asks for the third. A small table with a row for each traveller or worker and columns for rate, time and distance keeps the facts straight.

Units first

Rates in kilometres per hour and times in minutes do not mix. Convert before computing: 45 minutes is $\frac{3}{4}$ hour, and 1.9 hours is 1 hour and $0.9 \times 60 = 54$ minutes, not 1 hour 9 minutes. When answer choices are clock times, convert the decimal part of an hour carefully at the end.

Average speed is total distance over total time

$$\text{average speed} = \frac{\text{total distance}}{\text{total time}}$$

It is not the average of the speeds, unless equal times are spent at each speed. Driving 120 km out at 60 km/h and back at 40 km/h takes 2 hours plus 3 hours, so the average speed is $\frac{240}{5} = 48$ km/h, not 50. More time is spent at the slower speed, which pulls the average towards it. For two equal distances at speeds $a$ and $b$ there is a shortcut:

$$\text{average speed} = \frac{2ab}{a + b}$$

and $\frac{2 \times 60 \times 40}{100} = 48$ agrees.

Relative speed

When two objects move towards each other, or away from each other, the gap between them changes at the sum of their speeds. When one follows the other in the same direction, the gap changes at the difference. So two cyclists 63 km apart riding towards each other at 15 and 13 km/h close the gap at 28 km/h and meet after $\frac{63}{28} = 2.25$ hours.

For a chase, first find the head start: the distance the leader covers before the follower sets off. The catch-up time is the head start divided by the closing speed.

Work: add rates, not times

If a job takes a worker $a$ hours, the worker's rate is $\frac{1}{a}$ of the job per hour. Rates of people working together add; times do not. A drain, or anyone undoing the work, contributes a negative rate. For two workers,

$$\frac{1}{T} = \frac{1}{a} + \frac{1}{b}, \qquad T = \frac{ab}{a + b}$$

A useful device without a calculator: pretend the job has a size equal to the LCM of the individual times. If one printer takes 10 hours and another 15, call the job 30 pages; the printers do 3 and 2 pages an hour, 5 together, so the job takes 6 hours. Fractions become whole numbers.

Sanity checks

• Working together must be faster than the fastest worker alone and, with two equal workers, exactly half of one worker's time.
• An average speed lies between the slowest and fastest speeds.
• When two travellers set off towards each other at the same moment, they meet closer to the slower one's start.

These checks take seconds and often eliminate two or three answer choices before any arithmetic.