Maths Olympiad Prep

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Algebra Difficulty 5.5 AIME, harder Find the answer

11. A, B, and C together have 100 yuan. If A's money becomes 6 times the original, B's money becomes 1/31 / 3 of the original, and C's money remains unchanged, then the three of them still have a total of 100 yuan. C's money does not exceed 30 yuan. Question: How much money do A, B, and C each have?

A number or a short expression. Spacing and $ signs are ignored.

Solution

11. x,y,zx, y, z represent the amounts of money that A, B, and C have, respectively. x+y+z=100,18x+y+3z=300x+y+z=100, 18x+y+3z=300. x=10−2t,y=75−15t,z=15+17t,t=0,1,⋯ ,5.t=0x=10-2t, y=75-15t, z=15+17t, t=0,1, \cdots, 5. t=0 gives the required solution.

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Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic and difficulty added by this site.