Complete the multiplication given below:
3 3 ##
#--- * 2--- = 14---
7 # 14
where each hash(#) represents a digit, whether same or different.
Note: Numerator is strictly less than the denominator. Accordingly, fractions like: (7+7/3) or (8+11/11) are NOT permissible.
Since all fractions are proper fractions and the #'s represent non-degenerate digits, we know
2 3/9 <= 2 3/# < 2 3/4 and 14 10/14 <= 14 ##/14 < 14 13/14.
Then the smallest the first term # 3/7 can be is
(14 10/14) / (2 3/4) = 412/77 = 5.3506
And the largest the first term can be is
(14 13/14) / (2 3/9) = 627/98 = 6.3980
3/7 = 0.4286, so then # 3/7 = 5 3/7 = 5.4286 must be the first term.
Now multiply both sides by 1/(5 3/7) = 7/38 to get
2 3/# = (196 + ##)/76
The right side needs to simplify so as to not have 19 as a factor in the denominator and ## is one of 10,11,12,13. ##=13 is the only option to do that.
Then 2 3/# = (196 + 13)/76 = 11/4. Then the last missing digit is 4.
The original equation then must be (5 3/7) * (2 3/4) = (14 13/14)
Solution
