Problem 20 · 2014 Math Kangaroo
Medium
Algebra & Patterns
substitutionwork-backward
p, q and r are positive whole numbers with \(p+\cfrac{1}{q+\cfrac{1}{r}}=\dfrac{25}{19}\). What is the value of the product pqr?
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Answer: C — 18
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Hint 1 of 2
Read the nested fraction as a continued fraction equal to 25/19.
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Hint 2 of 2
Peel off the whole-number part at each layer.
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Approach: unfold the continued fraction layer by layer
- 25/19 = 1 + 6/19, so p = 1 and 1/(q + 1/r) = 6/19, i.e. q + 1/r = 19/6.
- 19/6 = 3 + 1/6, so q = 3 and r = 6.
- Then pqr = 1 · 3 · 6 = 18.
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