Question

in what ratio must two sorts of sugar costing $390 and $315 be mixed in order to produce a mixture worth $69 per kg?

Answer 1

no way can you mix two expensive items to produce a cheap mixture.
Also, you have not specified the cost per kg of either of the two sugars.

Answer 2

let the highest prime number that can go into 390 and 315 represent the number of sugar for each set.
the highest prime is 5
390/5=#78 per kg for this set
also, 315/5=#63 per kg for this set
for each sale of #78 per kg, loss = (78-69)=#9
for each sale of #63 per kg, gain= (69-63)#6
hence, ratio will be 9:6 = 3:2
check
6×78=468
9×63=567
567+468=1035
similarly; 69×15=1035.