A problem by Arpit Sah
Instead of walking along two adjacent sides of a rectangular field, a boy took a shortcut along the diagonal and saved a distance equal to half the longer side. Then the ratio of the shorter side to the longer side is . Find (b - a).
The answer is 1.
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1 solution
Let the longer side be x and the shorter side be y. \sqrt { { x }^{ 2 }\quad +{ \quad y }^{ 2 } } =\quad 0.5x\quad +\quad y\ Squaring\quad both\quad sides:\quad -\ { x }^{ 2 }\quad +\quad { y }^{ 2 }\quad =\quad 0.25{ x }^{ 2 }\quad +\quad { y }^{ 2 }\quad +\quad xy\ 0.75{ x }^{ 2 }\quad =\quad xy\ \frac { 3x }{ 4 } \quad =\quad y\ Therefore\quad \frac { y }{ x } \quad =\quad \frac { 3 }{ 4 } \quad =\quad \frac { a }{ b } Therefore b - a = 4 - 3 = 1