A butcher has supplies of lean beef containing 15% fat, and fat trim containing 100% fat. How many kg of lean beef and fat trim are needed to make 50 kg of hamburger containing 25% fat? (round your answers to 1 decimal place.)
To solve this problem, we'll use a simple algebraic approach. Let's assume that the amount of lean beef used is x kg, and the amount of fat trim used is y kg.
Given that the lean beef contains 15% fat, and we want the hamburger to contain 25% fat, we can set up the following equation:
0.15x + 1.00y = 0.25 * 50
Let's simplify this equation:
0.15x + y = 12.5
To find the values of x and y, we need another equation to relate the amounts of lean beef and fat trim. Since the total weight of the hamburger is 50 kg, we have:
x + y = 50
Now we have a system of equations:
0.15x + y = 12.5
x + y = 50
We can solve this system of equations using substitution or elimination method. Let's use the elimination method in this case:
Multiply the second equation by -1:
-1(x + y) = -1 * 50
-x - y = -50
Now add the two equations together:
0.15x - x + y - y = 12.5 - 50
-0.85x - 2y = -37.5
Simplify the equation:
-0.85x - 2y = -37.5
Now we have one equation in terms of x and y. Let's solve for x or y:
y = (-37.5 + 0.85x) / 2
We can substitute this value of y back into the second equation:
x + (-37.5 + 0.85x) / 2 = 50
Simplify the equation:
2x + (-37.5 + 0.85x) = 100
Now solve for x:
2x - 37.5 + 0.85x = 100
2.85x = 137.5
x ≈ 48.2
To find y, substitute the value of x back into the equation:
y = (-37.5 + 0.85(48.2)) / 2
y ≈ 1.8
So, approximately 48.2 kg of lean beef and 1.8 kg of fat trim are needed to make 50 kg of hamburger containing 25% fat.
This is just like the interest problem.
If there are x kg of 15% fat, then the rest (50-x) is 100% fat. So, just add up all the fat amounts:
.15x + 1.00(50-x) = .25*50