Reduce everyone to per-unit (work per minute, water per second), then add. Combined rate equals the sum of individual rates.
Method · Rate Invariance
Intro
Many word problems hide a one-line calculation behind a sentence about workers, machines, pipes, or swimmers. The recipe: (i) compute each actor’s per-unit rate (widgets per machine-minute, lawns per hour, litres per minute), (ii) combine — rates add in parallel, signed-add for fill-vs-drain or stream currents, (iii) divide work by combined rate to get time. The deepest trap is rate-invariance itself: parallel scaling multiplies throughput but does not change per-unit time, so “5 machines, 5 minutes, 5 widgets → 100 machines, 100 widgets” is still 5 minutes.
✓ Intro · expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine — being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
✓ Try first · expand
Worked example
If 5 machines take 5 minutes to produce 5 widgets, how long do 100 machines take to produce 100 widgets, assuming each machine works independently and at the same rate?
✓ Worked example · expand
Practice 1 of 3Type a fraction, decimal, or expression — mathjs parses it.
✓ Practice · expand
Reflection
Why does <em>parallel</em> work add rates while <em>serial</em> work adds times? Where in the problem statement is the cue that tells you which regime you are in — and what is a real-world scenario where the wrong regime would lead to a wrong answer?