Idea: Run the film in reverse
Reverse the order of the steps and invert each one.
- Doubling becomes halving
- Subtracting becomes adding
- Write the inverse step once with a letter, then repeat it
Example: Undo a trader's day
She doubled her position, sold lots, doubled again, sold more, and holds . What did she start with?
- Before the last sale: , then
- Before the first sale:
- Start:
Formula: Add rates, not times
Turn each time into a share of the job per hour, add the shares, invert. A drain counts as minus.
Example: Filling a tank together
One worker fills a tank in hours, another in . How long do they take together?
- Rates: of the tank per hour
- Together: hours, shorter than either
Common mistake: Averaging or adding times
Saying and hours together take or hours.
Instead Times combine through their reciprocals. Together they take hours, shorter than either.
Quick check: A pipe against a drain
A pipe fills a tank in hours; a drain empties it in . With both open, how many hours to fill it?
Show the answer
Answer:
of the tank per hour, so it takes hours.
Idea: Speeds add the same way
Walkers km apart at and km/h close the gap at km/h.
- They meet after hours
- For a shift, find the share done before the change
- Then divide the rest by the new rate
Example: Ages in one equation
A mother is times her son's age; in years she will be times his age. How old is the son?
- Let the son be :
- Age gaps never change, so solve:
Quick check: Undo the rounds
A number goes through rounds of "double, then subtract " and ends at . What was it?
Show the answer
Answer:
Undo each round with : gives , then , then .
Idea: Say which way you go
Start with "I will undo the steps from the end."
- For rates, say "per hour" with every fraction
- For ages, name one unknown and the gap that never changes
- A fraction where the story needs whole numbers means no solution
Courses / Brainteaser patterns