Lesson 12 of 12 · Algebra Foundations
Rate, Work, Mixture & Motion Problems
Four classic word-problem families, all solved the same way: identify the quantity that's conserved or additive (distance, work completed, amount of active ingredient), write one equation that balances it, and solve. The algebra is rarely hard — the setup is everything.
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Learning objectives
- Apply d = rt to motion problems, including same-direction and opposite-direction relative speed.
- Solve work-rate problems by combining individual rates (as fractions of the job per hour).
- Set up and solve mixture problems using a concentration-balance equation.
- Recognize which quantity stays conserved in a given word problem, and build the equation around it.
The core idea
Every problem in this lesson reduces to the same move: find the quantity that has to balance on both sides of an equation — total distance, total work completed, total amount of the active ingredient — and write that balance down directly. Motion problems balance distance. Work problems balance the fraction of a job completed. Mixture problems balance the amount of a specific ingredient. Once you see which quantity is conserved, the equation almost writes itself.
Distance, rate, time
d = rt
Same direction: relative speed is the difference of the two rates. Opposite directions: relative speed is the sum of the two rates.
Combined work rate
If a worker takes a hours alone, their rate is 1/a job per hour. Combined rate is the sum of individual rates (subtract for something that undoes the work, like a draining pipe). Time together is the reciprocal of the combined rate.
Mixture / concentration balance
(amount₁)(concentration₁) + (amount₂)(concentration₂) = (total amount)(final concentration). The total amount of the active ingredient before mixing must equal the amount after.
Worked example 1 — Motion, opposite directions
Problem
Two trains are 300 miles apart, moving toward each other at 60 mph and 90 mph. When do they meet?
Key insight
Moving toward each other, their combined closing speed is the sum of the two rates.
Solution
t = 300/150 = 2 hours.
Takeaway
You don't need to track each train's position separately — the combined closing speed collapses the problem into a single d=rt equation.
Worked example 2 — Combined work rate
Problem
A can paint a house alone in 6 hours; B can paint it alone in 4 hours. How long does it take working together?
Key insight
Add their rates (jobs per hour), not their times.
Solution
1/6 + 1/4 = 5/12 job per hour, so the time together is 12/5 = 2.4 hours.
Takeaway
Averaging the two times (6 and 4) would give the wrong answer — rates add, times don't.
Worked example 3 — Mixture
Problem
How many liters of a 20% acid solution must be mixed with 30 liters of a 50% acid solution to produce a 40% acid solution?
Key insight
The total amount of pure acid before mixing equals the total amount after.
Solution
Let x = liters of 20% solution: 0.2x + 0.5(30) = 0.4(x+30) → 0.2x+15=0.4x+12 → x=15.
Takeaway
Every mixture problem has exactly this structure — write the "amount of active ingredient" balance first, and the rest is routine algebra.
Worked example 4 — Motion, current/wind
Problem
A boat travels 60 miles downstream in 3 hours. If the current is 5 mph, find the boat's speed in still water.
Key insight
Downstream speed is the boat's still-water speed plus the current.
Solution
Downstream rate: 60/3 = 20 mph. Since downstream rate = boat + current: 20 = b+5 → b = 15 mph.
Takeaway
Always find the actual rate from the given distance and time first — don't try to guess the boat's speed directly.
Strategy notes
Identify the conserved quantity before writing any equation
Ask: what has to be equal on both sides? Total distance covered, total job completed, or total amount of an ingredient. That answer tells you exactly what equation to write.
Work with rates, never with times, when combining
Rates (jobs per hour, or miles per hour) add or subtract directly. Times do not — always convert to a rate, combine, then convert back if the question asks for a time.
Common mistakes
Averaging times instead of adding rates in work problems
"A takes 6 hours, B takes 4 hours, so together they take 5 hours" is wrong. Combine 1/6 and 1/4 as rates, then take the reciprocal of the sum.
Averaging speeds instead of dividing total distance by total time
Average speed for an entire trip is total distance ÷ total time, which is generally not the same as the simple average of the individual leg speeds, unless the two legs take equal time.
Adding rates instead of subtracting for a draining pipe or headwind
Anything that works against the main process (a drain, a current against you, a headwind) subtracts from the combined rate — don't add it by mistake.
Practice
20 questions across four difficulty tiers, each with a single numeric
answer (some are repeating decimals — type as many digits as you like, or
a fraction like 10/3). Up to three tries per question before
the solution is shown.