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How to solve these type of problems?

Dear All, I need your help..... Please tell me how to solve these types of problems LOGICALLY..!!! QUE. Can you arrange 9 numerals - 1, 2, 3, 4, 5, 6, 7, 8 and 9 - (using each numeral just once) above and below a division line, to create a fraction equaling to 1/3 (one third)? Expecting your valuable comments......!!!!!!!

Note by Akhil K
4 years, 5 months ago

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Congratulations on being my 300th follower! Great problem! I have no idea, however. · 3 years, 5 months ago

5832/17496 · 4 years, 5 months ago

5832/17496 · 4 years, 5 months ago

thanks for valuable comment.Please let me know HOW you solved it. · 4 years, 5 months ago

5823/17469 · 4 years, 5 months ago

Dear Yonatan, Please share me HOW you solved it? · 4 years, 5 months ago

It's easy to see the numerator will have four digits, the denominator five. Call the digits abcd/efghi. Also, abcd is divisible by 3, since a+b+c+d+e+f+g+h+i is divisible by 3, and e+f+g+h+i is divisible by 3 (since efghi is) so a+b+c+d must be as well. . Because efghi > 12345, we know abcd >= 4115 (so also >= 4123). This immediately eliminates a=1,2, or 3.

Now I made an assumption that if I will be able to solve this problem quickly, there should be relatively few carries, If there are no carries (besides the last digit), {b,c,d} = {1,2,3}. It's easy to check that there are no solutions of this form, so we must make a weaker assumption, that there is just 1 carry. The options for {c,d} are {{1,2}, {2,3}, {1,3}}. The first pair is eliminated because e is either 1 or 2, so let's try {2,3}. ab23 = efg69. We know a+b = 1 (mod 3) and a >= 4. Now there aren't many pairs left to try, so just guess and check until we find the smallest solution of the presumed form, which is 5823. · 4 years, 5 months ago