Word Problem Practice
Rate, work, mixture, age and distance problems with the equation shown — and the tempting wrong answer named beside the right one.
The answers almost everybody gives
- Average speed is not the average of the speeds. Sixty miles an hour there and thirty back over the same road averages 40, not 45 — because you spend twice as long on the slow leg, so it counts for twice as much. Averaging the two numbers assumes equal TIME at each speed when the problem states equal DISTANCE.
- The right figure is the harmonic mean, and it is ALWAYS lower than the arithmetic one. That is not a coincidence of the numbers: any time you cover equal distances at different speeds, the slow stretch takes longer and drags the average down.
- Rates add; times do not. One tap filling a tank in 3 hours and another in 6 take 2 hours together — not 4.5. What adds is the share of the job done per hour: a third plus a sixth is a half. And there is a sanity check that catches the error every time — two people working together must finish faster than the quicker one alone.
- Percentages do not add either. Mixing 20% with 60% gives 40% only when the volumes are equal. What adds is the actual amount of the active ingredient, which is why a mixture equation is written in litres rather than in percentages.
- Meeting problems are the exception worth noticing: approaching speeds genuinely do add, because the gap closes at their sum. Knowing WHICH quantity adds in each situation is the whole skill, and it is not the same quantity every time.
- Every problem here shows the tempting wrong answer beside the right one, because being told a rule is much less convincing than watching your own first instinct miss.
How to use
- Pick which kinds of problem you want.
- Answer, then read the equation and the working.
- Check the tempting wrong answer, and why it is wrong.
Frequently asked questions
Why is the average of 60 mph and 30 mph not 45?
Because over the same distance you spend twice as long at the slow speed, so it counts for twice as much. The answer is 40 mph. Averaging the two numbers would be right if equal TIME were spent at each, but the problem gives equal DISTANCE — and the true figure, the harmonic mean, is always lower than the arithmetic one.
If one person takes 3 hours and another 6, why is it not 4.5 together?
Because rates add and times do not. One does a third of the job per hour and the other a sixth; together that is a half, so two hours. There is a sanity check that catches this every time — two people working together must finish faster than the quicker one alone, and 4.5 is slower than 3.
Do percentages work the same way?
No, and that is the mixture trap. Mixing 20% with 60% gives 40% only when the volumes are equal. What adds is the actual amount of the active ingredient, which is why a mixture equation is written in litres rather than in percentages.
Is adding ever the right move?
Yes, in meeting problems — two people riding towards each other close the gap at the SUM of their speeds. That is genuinely different from the work and speed problems, where adding the obvious quantity is the error, and knowing which quantity adds in each situation is the whole skill.
What changes in an age problem?
The ratio, never the difference. Both people age at the same rate, so the gap between them is fixed for ever while the ratio drifts steadily towards one. Tracking the difference rather than the ratio is what makes these problems easy.
Could it mark a right answer wrong?
It is built hard against that. Every problem is verified by reading its own prompt back and substituting the answer into the situation described — 15,000 problems across five kinds were put through that before shipping, including 5,000 checks each that average speed really is distance over time and that the two work rates really do add to exactly one job.
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