Two kinds of real problems lead straight to rational equations: people (or machines) sharing a job, and costs spread over many items.
Key idea
Work rate: if a job takes $a$ hours, the rate is $\frac{1}{a}$ job per hour, and rates working together add: $\frac{1}{a} + \frac{1}{b} = \frac{1}{t}$. Average cost: with fixed cost $F$ and cost $c$ per item, $\bar{C}(x) = \frac{F + cx}{x}$; set it equal to a target and solve.
Why it works
Rates add because the amounts of work done in one hour add: if one painter does $\frac{1}{3}$ of the fence each hour and the other does $\frac{1}{6}$, together they do $\frac{1}{2}$ of the fence per hour, so they need 2 hours. Times don't add: two people can't take longer than either one alone. For average cost, the fixed cost is shared by more items as $x$ grows, so $\bar{C}(x) = \frac{F}{x} + c$ gets close to $c$ but never reaches it.
Worked examples
Worked example: Working together
Ana paints a fence in 3 hours; Ben takes 6 hours. How long together?
- Rates: $\frac{1}{3}$ and $\frac{1}{6}$ fence per hour.
- Together: $\frac{1}{3} + \frac{1}{6} = \frac{1}{2}$ fence per hour.
- Time $= 1 \div \frac{1}{2} = 2$ hours.
Answer: 2 hours
Worked example: Finding one worker's time
Two pumps empty a tank in 4 hours together. The first alone takes 6 hours. How long does the second take alone?
- $\frac{1}{6} + \frac{1}{t} = \frac{1}{4}$.
- Multiply by $12t$: $2t + 12 = 3t$, so $t = 12$.
Answer: 12 hours
Worked example: Average cost
Printing costs \$500 to set up plus \$3 per book. How many books make the average cost \$5?
- $\frac{500 + 3x}{x} = 5$, so $500 + 3x = 5x$.
- $500 = 2x$ and $x = 250$.
Answer: 250 books
Common mistake
Adding the times. If Ana takes 3 hours and Ben takes 6, "9 hours together" can't be right; together they must be faster than Ana alone. Add the rates $\frac{1}{3} + \frac{1}{6}$, then flip. Check: in 2 hours Ana paints $\frac{2}{3}$ and Ben paints $\frac{1}{3}$, one whole fence.