A basketball player is practising free throws. Their current success rate (ratio of successful throws to total) is exactly 70% (or 0.7 in terms of ratio). After a few more throws that success rate is 90%.
Show that at some point the success rate was exactly 80%.
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Look at (misses, hits) on the integer lattice and x-y coordinate plane. If a miss occurs, we increment x coordinate, else we increment y coordinate.
Initially we are on the line $3y = 7x$ (70% hit rate) and reach the line $y = 9x$ (90% hit rate), crossing the line $y = 4x$ (80% hit rate) at some point, with some hit (i.e by incrementing the y-coordinate).,
The only way to cross the line $y = 4x$ vertically is to actually land on it first (every x = N line intersects y = 4x at (N, 4N) which is part of the integer lattice. Thus we achieve exactly 80%,
