Word Problems: Rates, Work, Mixtures, and Translating Language
What you must know
Word problems test your ability to translate English into equations. Core setups include distance = rate × time, combined work rates (add rates, not times), weighted averages and mixtures, profit and cost, and consecutive-integer or age problems. Define a variable for the unknown, express every other quantity in terms of it, and write one equation per relationship.
Traps
- Adding times instead of rates in work problems: if A finishes in 4 hours and B in 6, together they finish in 1/(1/4 + 1/6) hours, not 5.
- Average speed over two equal distances is the harmonic-style average 2ab/(a+b), not (a+b)/2.
- Mixture concentration: track the amount of solute, not just percentages.
- Unit mismatches - minutes versus hours, cents versus dollars.
- Misreading "how many more" as "how many times."
Worked example
Pipe A fills a tank in 6 hours; Pipe B fills it in 3 hours. Working together, their combined rate is 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2 tank per hour, so they fill it in 2 hours. The trap answer, 4.5 hours, comes from averaging the times.
Rule of thumb
For rate problems, convert everything to a common unit first, then add or subtract rates. Write the units next to each number so mismatches become visible.