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Counterfeit bag in one weighing

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There are $12$ bags numbered $1$ to $12$ , each with plenty of coins. A genuine coin weighs $10$ g. Exactly $1$ bag contains only fake coins of $9$ g each, and the rest contain only genuine coins. You take $k$ coins from bag $k$ for every $k$ , put all of them on a digital scale, and read $770$ g. What is the number of the fake bag?

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Hint

Compare the reading with the weight you would get if every coin were genuine.

Solution

The number of coins taken is $1 + 2 + \dots + 12 = 78$ .
If all were genuine, the scale would read $10 \times 78 = 780$ g.
The reading is $770$ g, so the shortfall is $780 - 770 = 10$ g.
Each fake coin is $1$ g lighter, so exactly $10$ fake coins were weighed.
Bag $k$ contributes $k$ coins, so the fake bag is number $10$ .

Answer: $10$

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