Problem 20 · 1994 AJHSME
Stretch
Fractions, Decimals & Percents
minimize-fractions
Let W, X, Y, and Z be four different digits selected from the set {1, 2, 3, 4, 5, 6, 7, 8, 9}.
If the sum WX + YZ is to be as small as possible, then WX + YZ must equal
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Answer: D — 25/72.
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Hint 1 of 2
A fraction shrinks when its top is small and its bottom is big. With four digits to place, send your two SMALLEST digits (1, 2) up top and your two LARGEST (8, 9) to the bottoms.
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Hint 2 of 2
That's not the whole answer — you still choose how to pair them. There are only two pairings of {1,2} over {8,9}, so test both. Watch out: the 'obvious' choice isn't the winner.
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Approach: smallest tops over largest bottoms, paired well
- Smallest tops, largest bottoms ⇒ numerators 1 and 2, denominators 8 and 9.
- Two ways to pair them, both over a common 72: (a) 1/9 + 2/8 = 8/72 + 18/72 = 26/72; (b) 1/8 + 2/9 = 9/72 + 16/72 = 25/72. Option (b) is smaller.
- So the minimum sum is 25/72.
- Why the better pairing puts the bigger numerator over the bigger denominator: the '2' does the most damage, so park it over the biggest bottom (9) to shrink its effect. Lesson — for these extremes, 'pick the right digits' is only half the job; how you MATCH them up is the tiebreaker, so always check the few pairings.
Mark:
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