An alloy weighing 80 kg contains 35% copper by weight. How many kilograms of pure copper must be added so that the alloy becomes 48% copper?

The alloy holds $0.35\times 80=28$ kg of copper and 52 kg of other metal. Adding $x$ kg of copper: $(28+x)/(80+x)=0.48$ → $28+x=38.4+0.48x$ → $0.52x=10.4$ → $x=20$ kg. (Check: 48 kg of copper in 100 kg of alloy.) Taking 13% of 80 = 10.4 kg forgets that the total weight grows too.