Answer:
Explanation:
There may be other ways to do this. Here's the approach I took. Let the integer part of x be represented by "a", and the integer part of y be represented by "b". Now, we have ...
(a +2/3)(b +3/4) = 10
Multiplying by 12, this becomes ...
(3a +2)(4b +3) = 120
There are only 16 divisors of 120, so they can be easily enough checked one at a time.
120 = 1·120 = 2·60 = 3·40 = 4·30 = 5·24 = 6·20 = 8·15 = 10·12
Of these divisors, the ones that are of the form 3a+2 are ...
2, 5, 8, 20 . . . . a=0, 1, 2, 6
The ones that are of the form 4b+3 are ...
3, 15 . . . . b = 0, 3
The factor pair that consists of a number on the first list and a number on the second list is ...
120 = 8·15
corresponding to a=2 and b=3. So, the mixed numbers you want are ...
x = 2 2/3
y = 3 3/4
_____
Check
(2 2/3)·(3 3/4) =2·3 +2·3/4 +2/3·3 +2/3·3/4 = 6 +6/4 +2 +2/4 = 8+8/4 = 10