A candy distributor needs to mix a 40% fat-content chocolate with a 60% fat content chocolate to create 200 kilograms of a 51% fat content chocolate. How many kilograms of each kind of chocolate must they use?
Let's assume the distributor needs to use x kilograms of the 40% fat-content chocolate.
Since the total amount of chocolate needed is 200 kilograms, the distributor will use (200 - x) kilograms of the 60% fat-content chocolate.
To find the total fat content in the mix, you need to calculate the sum of the fat content from each type of chocolate. The fat content of the 40% fat-content chocolate is 0.4x, and the fat content of the 60% fat-content chocolate is 0.6(200 - x).
Since the total mass of the mix is 200 kilograms and the desired fat content is 51%, the equation for the total fat content of the mix is: 0.4x + 0.6(200 - x) = 0.51 * 200.
Simplifying the equation:
0.4x + 0.6 * 200 - 0.6x = 0.51 * 200.
0.4x + 120 - 0.6x = 102.
0.4x - 0.6x = 102 - 120.
-0.2x = -18.
x = -18 / -0.2.
x = 90.
So, the distributor must use 90 kilograms of the 40% fat-content chocolate and (200 - 90) = 110 kilograms of the 60% fat-content chocolate.
To solve this problem, we can use a method called the "mixture" equation.
Let's assume the distributor needs to mix x kilograms of the 40% fat-content chocolate and (200 - x) kilograms of the 60% fat-content chocolate to get a total of 200 kilograms of the 51% fat-content chocolate.
Now, let's set up the equation based on the fat content:
0.40x + 0.60(200 - x) = 0.51(200)
Simplifying the equation:
0.40x + 120 - 0.60x = 102
Combining like terms:
-0.20x = -18
Dividing both sides by -0.20:
x = 90
Therefore, the distributor needs to mix 90 kilograms of the 40% fat-content chocolate and (200 - 90) = 110 kilograms of the 60% fat-content chocolate to obtain 200 kilograms of the 51% fat-content chocolate.