Problem 4 · Growing discounts
The total saving is $1+2+\cdots+63$ percent of one unit.
Integer answer, at most 4 digitsA shop runs a promotion on an item with a fixed initial price. Buying one unit gets a 1% discount. Buying two, the first gets 1% and the second 2% (off the initial price). Buying three: 1%, 2% and 3%. With more units the discount continues likewise, so the $n$-th unit gets an $n\%$ discount (with 100 units the last one would be free). If we buy 63 units, what percentage of the order do we save with this promotion?
Copa Cangur · SCM
Medium
Closed answer
Reasoned solution
Key idea: if one unit costs $p$, the saving on the $n$-th unit is $\tfrac{n}{100}p$.
$$\text{Saving} = \frac{p}{100}(1 + 2 + \cdots + 63) = \frac{p}{100} \cdot \frac{63 \cdot 64}{2} = \frac{2016\,p}{100}.$$
The undiscounted order costs $63p$, so the saved fraction is
$$\frac{2016p/100}{63p} = \frac{2016}{6300} = \frac{32}{100} = 32\%.$$
Answer: 32