Rates, work and mixtures

One equation behind three topics

Speed, work and flow problems all rest on the same relation:

$$\text{amount} = \text{rate} \times \text{time}.$$

For travel, the amount is distance and the rate is speed. For work, the amount is the number of jobs (often 1) and the rate is jobs per hour. For pipes, the amount is tanks or litres. Once you see which is which, the problem becomes a line or two of arithmetic.

Units first. Rates and times must match: km/h with hours, metres per second with seconds. Twenty minutes is $\frac{1}{3}$ of an hour, not 0.2. To turn km/h into m/s, multiply by $\frac{1000}{3600} = \frac{5}{18}$: 72 km/h is $72 \times \frac{5}{18} = 20$ m/s.

Average speed is not the average of the speeds

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

Drive 60 km at 30 km/h and another 60 km at 60 km/h. The first leg takes 2 hours and the second 1 hour, so the whole trip is 120 km in 3 hours: an average of 40 km/h, not 45. The slower leg lasts longer, so it pulls the average towards itself. Only when the two legs take equal times does the simple mean of the speeds give the average speed.

For two legs of equal distance at speeds $u$ and $v$, the average speed is $\frac{2uv}{u + v}$; here $\frac{2 \times 30 \times 60}{90} = 40$. It is quicker, though, to pick a convenient distance and compute times.

Moving towards or apart

Two objects moving towards each other close the gap at the sum of their speeds; one chasing another closes it at the difference. Two trains 300 km apart, heading towards each other at 70 km/h and 80 km/h, close the gap at 150 km/h and meet after $\frac{300}{150} = 2$ hours.

Work: add rates, never times

If Ana paints a room in 4 hours, her rate is $\frac{1}{4}$ room per hour. If Ben takes 6 hours, his rate is $\frac{1}{6}$. Working together, their rates add:

$$\frac{1}{4} + \frac{1}{6} = \frac{3}{12} + \frac{2}{12} = \frac{5}{12} \text{ room per hour},$$

so together they need $\frac{12}{5} = 2.4$ hours, which is 2 hours 24 minutes. Sanity check: the answer must be less than the faster worker's time alone, 4 hours, and more than half of it. A pipe that drains a tank counts as a negative rate.

The same idea gives a quick formula for two workers: together they take $\frac{ab}{a + b}$, here $\frac{4 \times 6}{10} = 2.4$ hours.

Mixtures are weighted averages

When you mix two solutions, the amount of the ingredient you care about (salt, alcohol, money) adds up, even though concentrations do not. Mix 30 litres at 20% salt with $x$ litres at 50% salt to get a 30% mixture:

$$0.2 \times 30 + 0.5x = 0.3(30 + x) \;\Longrightarrow\; 6 + 0.5x = 9 + 0.3x \;\Longrightarrow\; x = 15.$$

The balance method reaches the same answer without an equation. The target, 30%, sits 10 points above the weaker solution and 20 points below the stronger one. The amounts must be in the inverse ratio of those distances: weak to strong $= 20 : 10 = 2 : 1$. With 30 litres of the weak solution, you need 15 litres of the strong one. Check: salt $6 + 7.5 = 13.5$ litres in 45 litres, which is $\frac{13.5}{45} = 30\%$.

The method works for any blend: coffee at two prices, two classes with different average marks, two investments with different returns. The mixture always lies between its ingredients, and it lies closer to the one there is more of.

A routine for any rate problem

1. Write the rate of each participant in the same units.
2. Decide whether the rates add (together, towards each other) or subtract (against each other, chasing, draining).
3. Use $\text{amount} = \text{rate} \times \text{time}$ once, for the combined rate.
4. Check the answer's size: together is faster than either alone; an average lies between its parts.