place the rational numbers 5/6, 9/8, 1/2, 3/10, 3/4, 1/2, 5/4, 1/3 and 15/8 in one of nine circles so that the product of every three connected circles is 5/16

Question

Any help is appreciated.

Answer 1

Let’s take each fraction in turn, divide it into 5/16 and find out what pair of other fractions further down the line make this quotient.

1. 5/6 into 5/16=5/16×6/5=3/8, so we need a pair of fractions whose product is 3/8: 9/8×1/3, 1/2×3/4 (two circles contain 1/2), 3/10×5/4
2. 9/8 into 5/16=5/16×8/9=5/18. No pairs because 18=2×9=3×6 and there are no remaining fractions with 9 or 6 as the denominator
3. 1/2 into 5/16=5/8. We have 1/2×5/4, 1/3×15/8. See also (6)
4. 3/10 into 5/16=25/24. No pairs.
5. 3/4 into 5/16=5/12: 5/4×1/3.
6. 1/2 into 5/16=5/8. No more pairs (see (3))
7. 5/4 into 5/16=1/4. No more pairs. See (1), (3) and (5)
8. 1/3 into 5/16=15/16. No more pairs. See (1), (3) and (5)
9. 15/8. See (3)

Now we have all combinations.

If we give labels to the fractions using letters A to I for the fractions in order: A=5/6, B=9/8, ..., I=15/8, we can represent the products that give 5/16 as:

ABH (red) ACE (blue) AEF (blue) ADG (green) CFG (pink) CHI yellow) EGH (cyan) FHI (yellow)

Because C and F are both 1/2 they spawn triangles of the same colour (hence the pair of blue and yellow triangles). It may be helpful to think of the triangles in different planes in three dimensions.

The pictures are representations of a 3-dimensional structure, resembling a roof where the gable ends are triangles ACE and HFI and the ridge is CF. The base of the roof is the rectangle AEIH. DAH is a straight line, as is BAE. Triangle CGF sits on the ridge, where G, the apex, is the highest vertical circle. The black dashed line EI doesn’t form a meaningful link, but the coloured triangles show which circles are connected by the rule that the products of their associated fractions are equal to 5/16.