Final answer:
To prepare 100 kg of a 22% sucrose solution, we must mix 70 kg of the 10% sucrose solution with 30 kg of the 50% sucrose solution using the concept of mass percent.
Step-by-step explanation:
To solve this problem, we use the concept of mass percent (w/w) which is defined as the mass of the solute divided by the mass of the solution, multiplied by 100%. We are aiming to mix a 10% sucrose solution with a 50% sucrose solution to create 100 kg of a 22% sucrose solution.
Let's define:
x = mass of the 10% sucrose solution (in kg)
(100-x) = mass of the 50% sucrose solution (in kg).
We can then set up an equation based on the total mass of sucrose from both solutions being equal to the mass of sucrose in the final 22% solution.
10% of x (from the 10% solution) + 50% of (100 - x) (from the 50% solution) = 22% of 100 kg (final solution).
Solving for x gives us the mass of each solution needed:
0.10x + 0.50(100 - x) = 22
0.10x + 50 - 0.50x = 22
-0.40x = -28
x = 70 kg
Therefore, to obtain 100 kg of a 22% sucrose solution, we should mix 70 kg of the 10% sucrose solution with 30 kg of the 50% sucrose solution.