Find the 3-Digit Values A, B and CDate: 08/08/99 at 22:03:30 From: Brad Goorman Subject: Stuck on a problem Dear Dr. Math, I have a math problem that I am stuck on. Where do I start? Q: The digits of the 3-digit integers a, b, and c are the nine nonzero digits 1, 2, 3, ..., 9, each of them appearing exactly once. Given that the ratio a:b:c is 1:3:5, determine a, b and c. Sincerely, Brad Goorman Date: 08/08/99 at 22:50:38 From: Doctor Ian Subject: Re: Stuck on a problem Hi Brad, Whenever you don't know what else to do, start by writing down what you know. You're going to have three 3-digit numbers whose values are in a particular ratio: a : b : c = _ _ _ : _ _ _ : _ _ _ = 1 : 3 : 5 When you've done that, start looking for ways to rule out bogus solutions. For example, you know that if a ends in 1, then b must end in 3 and c must end in 5. Try all the other possibilities: _ _ 1 : _ _ 3 : _ _ 5 2 6 0 0 can't appear 3 9 5 4 2 0 0 can't appear 5 5 5 5 can't appear three times 6 8 0 0 can't appear 7 1 5 8 4 0 0 can't appear 9 7 5 In other words, the only possible combinations for the final digits are: _ _ 1 : _ _ 3 : _ _ 5 3 9 5 7 1 5 9 7 5 You can apply similar constraints to the initial digits. In fact, the initial digit of a can only be 1 (do you see why?), so this rules out two of the four possibilities: 1 _ 3 : _ _ 9 : _ _ 5 1 9 7 5 And the initial digit of b can only be 3 or 4. Can you take it from here? - Doctor Ian, The Math Forum http://mathforum.org/dr.math/ |
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