total- value problems & mixture problems
How much lowfat milk (1%fat) and how much whole milk (4% fat) should be mixed to make 5 gal of reduce fatmilk (2% fat)?
.01 x + .04 (5-x) = .02 (5)
To determine how much lowfat milk (1% fat) and whole milk (4% fat) should be mixed to make 5 gallons of reduced-fat milk (2% fat), we can use the concept of total-value problems or mixture problems.
Let's break down the problem step by step:
Step 1: Assign variables:
- Let's assume the amount of lowfat milk (1% fat) to be mixed is "x" gallons.
- The amount of whole milk (4% fat) to be mixed will be the remaining quantity to make a total of 5 gallons. So, the amount of whole milk will be 5 - x gallons.
Step 2: Set up the equation:
We know that fat is the only component being mixed, and it is expressed as a percentage. The total amount of fat in the mixture will be equal to the sum of the fat in the lowfat milk and the fat in the whole milk. Mathematically, this can be written as:
(0.01 * x) + (0.04 * (5 - x)) = 0.02 * 5
In this equation:
- 0.01 represents 1% fat in the lowfat milk.
- 0.04 represents 4% fat in the whole milk.
- 0.02 represents 2% fat in the final mixture.
- The equation is derived from the fact that the amount of fat in the mixture is equal to the amount of fat in the lowfat milk added to the amount of fat in the whole milk, which should be equal to the amount of fat in the final mixture.
Step 3: Solve the equation:
Now, we can solve the equation to find the value of x:
0.01x + 0.04(5 - x) = 0.02 * 5
0.01x + 0.2 - 0.04x = 0.1
-0.03x + 0.2 = 0.1
-0.03x = 0.1 - 0.2
-0.03x = -0.1
x = (-0.1) / (-0.03)
x = 3.33
Step 4: Calculate the remaining quantity:
Since x represents the amount of lowfat milk and we assumed the total quantity to be 5 gallons, the amount of whole milk will be 5 - x:
Whole milk = 5 - 3.33 = 1.67 gallons
So, to make 5 gallons of reduced-fat milk (2% fat), you should mix approximately 3.33 gallons of lowfat milk (1% fat) with 1.67 gallons of whole milk (4% fat).