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

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.

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.

To find the ratio in which the two types of sugar should be mixed, we can use the concept of weighted averages.

Let's assume we need x kg of sugar from the 390-dollar per kg sugar and y kg of sugar from the 315-dollar per kg sugar to produce a mixture worth $69 per kg.

The total cost of the 390-dollar per kg sugar will be 390x dollars, and the total cost of the 315-dollar per kg sugar will be 315y dollars.

According to the problem, the total mixture is worth $69 per kg.

So, we can set up the equation:

(390x + 315y) / (x + y) = 69

To simplify the equation, let's multiply both sides by (x + y):

390x + 315y = 69(x + y)

Expanding the equation:

390x + 315y = 69x + 69y

Rearranging the equation:

390x - 69x = 69y - 315y

321x = -246y

Dividing both sides by -246:

x / y = -246 / 321

Since we are dealing with a ratio, we can ignore the negative sign:

x / y = 246 / 321

The ratio of the two types of sugar that should be mixed is 246:321.

To determine the ratio in which the two sugars should be mixed to produce a mixture worth $69 per kg, we can use a weighted average formula. Let's assume that x is the ratio of the more expensive sugar and y is the ratio of the cheaper sugar.

The cost per kg of the mixture is equivalent to the weighted average of the costs of the two sugars. So we can set up the following equation:

(390x + 315y) / (x + y) = 69

To solve for the ratio x:y, we can use algebraic manipulation. First, we multiply both sides of the equation by (x + y) to get rid of the denominator:

390x + 315y = 69(x + y)

Next, we can distribute on the right side:

390x + 315y = 69x + 69y

Combining like terms:

390x - 69x = 69y - 315y

Simplifying further:

321x = -246y

Now, we can divide both sides of the equation by -246:

(321x) / (-246) = (y)

This gives us the ratio of x:y. However, since we want a specific ratio, we can choose one variable as a constant to find the other. Let's assume y = 1, then x = (321/(-246)).

Therefore, the ratio in which the two sugars must be mixed to produce a mixture worth $69 per kg is approximately 1 : (-321/246).