Competition problem — Algebra — Maths Olympiad Prep

Algebra Difficulty 5.9 AIME, harder Prove it Estonia

Juku discovered that of the things in his satchel, $60\%$ were ugly and $76\%$ were useless. He scrapped all things that were both ugly and useless, and added things that were both beautiful and useful. After this, of the things in Juku's satchel, $25\%$ are ugly and $45\%$ are useless. How many percent of the things in Juku's satchel were both beautiful and useful initially?

Answer: $4\%$ .

Solution

Observe that the amount of things that are beautiful but useless and things that are useful but ugly remained unchanged. The difference between the percentages of these things was $76\% - 60\% = 16\%$ before displacement but is $45\% - 25\% = 20\%$ after that. Hence the overall number of things in the satchel decreased $20 : 16 = 1.25$ times. Things that are beautiful but useless form $45\%$ of all things after the displacement, hence they formed $45\% : 1.25 = 36\%$ before it. As there were $100\% - 60\% = 40\%$ of beautiful things in total, the things that were both beautiful and useful constituted $40\% - 36\% = 4\%$ of the content of the satchel.

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