The strength of sugar solution is x% if 100 ml of solution contains ‘x’ grams of sugar. If three sugar solutions A, B, and C are mixed in the proportion 5 : 2 : 3, then the resulting solution has a strength of 40%. If instead the proportion is 3 : 5 : 2, then the resulting solution has a strength of 50%. A fourth solution, D, is produced by mixing A and C in the ratio 13 : 7. What is the ratio of the strength of D to that of B?

Let's denote the strength of the sugar solutions A, B, and C as a%, b%, and c% respectively.

From the first condition, we know that when A, B, and C are mixed in the ratio 5:2:3, the resulting solution has a strength of 40%. This can be written as:

5a + 2b + 3c = 40 * (5 + 2 + 3) = 400 ---- (1)

From the second condition, we know that when A, B, and C are mixed in the ratio 3:5:2, the resulting solution has a strength of 50%. This can be written as:

3a + 5b + 2c = 50 * (3 + 5 + 2) = 500 ---- (2)

We can solve these two equations to find the values of a, b, and c.

Subtracting equation (2) from equation (1) gives:

2a - 3b + c = -100

Solving this equation for c gives:

c = 100 + 3b - 2a ---- (3)

Substituting equation (3) into equation (1) gives:

5a + 2b + 3(100 + 3b - 2a) = 400

Solving this equation for a gives:

a = 20 - b ---- (4)

Substituting equation (4) into equation (3) gives:

c = 100 + 3b - 2(20 - b) = 140 - b ---- (5)

So, the strengths of the solutions A, B, and C are 20 - b%, b%, and 140 - b% respectively.

Now, let's find the strength of the solution D, which is produced by mixing A and C in the ratio 13:7. The strength of D can be calculated as:

d = (13a + 7c) / (13 + 7) = (13(20 - b) + 7(140 - b)) / 20 = 140 - b

So, the ratio of the strength of D to that of B is (140 - b) / b.

Since we don't know the value of b, we can't calculate the exact ratio. However, we can see that the ratio depends on the value of b. If b is small, the ratio will be large, and if b is large, the ratio will be small.