Solution:

Given:
- The rate of gross profit is same in all departments.
- The ratio of sales in these departments is 5:3:2.

To find:
- The ratio of cost price to sales of these departments.

Explanation:

Let us assume that the rate of gross profit in each department is x%.

Therefore, the cost price of each department will be (100 - x)% of the selling price.

Let the sales in the three departments be 5x, 3x, and 2x respectively.

Therefore, the total sales of the business = 5x + 3x + 2x = 10x.

Now, let us calculate the gross profit in each department.

Gross profit in 1st department = 5x * (x/100) = 5x^2/100
Gross profit in 2nd department = 3x * (x/100) = 3x^2/100
Gross profit in 3rd department = 2x * (x/100) = 2x^2/100

Therefore, the total gross profit = (5x^2/100) + (3x^2/100) + (2x^2/100) = 10x^2/100 = x^2/10.

Now, let us calculate the cost price of each department.

Cost price of 1st department = (100 - x)% of 5x = (100 - x)*5x/100 = 5x - (x/100)*5x = (95/100)*5x = 4.75x.
Cost price of 2nd department = (100 - x)% of 3x = (100 - x)*3x/100 = 3x - (x/100)*3x = (97/100)*3x = 2.91x.
Cost price of 3rd department = (100 - x)% of 2x = (100 - x)*2x/100 = 2x - (x/100)*2x = (98/100)*2x = 1.96x.

Therefore, the total cost price = 4.75x + 2.91x + 1.96x = 9.62x.

Now, let us calculate the ratio of cost price to sales of these departments.

Ratio of cost price to sales of 1st department = 4.75x/5x = 0.95.
Ratio of cost price to sales of 2nd department = 2.91x/3x = 0.97.
Ratio of cost price to sales of 3rd department = 1.96x/2x = 0.98.

Therefore, the ratio of cost price to sales of these departments is 0.95 : 0.97 : 0.98, which can be simplified as 95 : 97 : 98.

Final Answer:
The ratio of cost price to sales of these departments is 95 : 97 : 98.