Problem 15 · 2013 Math Kangaroo
Hard
Algebra & Patterns
substitution
The sides of rectangle ABCD are parallel to the coordinate axes. The rectangle lies below the x-axis and to the right of the y-axis, as shown. For each of the points A, B, C, D the quotient (y-coordinate) : (x-coordinate) is calculated. For which point will you obtain the smallest quotient?
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Answer: D — D
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Hint 1 of 2
For every corner the quotient is (negative y) over (negative x), so it is positive; you want the smallest positive value.
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Hint 2 of 2
Smallest quotient = smallest |y| paired with largest |x| — find that corner.
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Approach: compare y/x at each corner
- In the diagram the rectangle has both coordinates negative, so each quotient y : x is positive.
- The smallest quotient comes from the corner closest to the x-axis (smallest |y|) and farthest from the y-axis (largest |x|).
- That is the top-left corner, point D.
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