How many gallons of milk containing 5% butterfat must be mixed with 90 gallons of 1% milk to obtain 2% milk?
gallons of 5% milk --- x
.05x + .01(90) = .02(x+90)
times 100
5x + 1(90) = 2(x+90)
5x + 90 = 2x + 180
3x = 90
x = 30
Well, it seems you're looking for a milk-mixing adventure! Let's see if we can figure this out together.
To start off, we have 90 gallons of 1% milk and we want to end up with 2% milk. That means we need to add some milk with a higher butterfat content than 1% to achieve the desired result.
Let's assume we need x gallons of the 5% butterfat milk.
Now, let's do some math. We want to find the total amount of butterfat in the final mixture.
The total butterfat in the 1% milk is (90 gallons) * (0.01 butterfat content) = 0.9 gallons of butterfat.
For the 5% butterfat milk, we have (x gallons) * (0.05 butterfat content) = 0.05x gallons of butterfat.
In the final mixture, the total butterfat should be (90 + x) gallons * (0.02 butterfat content) = 0.02(90 + x) gallons of butterfat.
According to the law of conservation of butterfat (or something like that), the total amount of butterfat before and after mixing should be the same. So, we can set up the equation:
0.9 gallons + 0.05x gallons = 0.02(90 + x) gallons
Now, let's solve this equation step by step:
0.9 + 0.05x = 1.8 + 0.02x
0.05x - 0.02x = 1.8 - 0.9
0.03x = 0.9
x = 0.9 / 0.03
x = 30
So, you will need to mix 30 gallons of the 5% butterfat milk with 90 gallons of 1% milk to obtain 2% milk.
Well, that was udderly fascinating, wasn't it? Happy milk mixing!
To find the number of gallons of 5% butterfat milk needed, let's define the variables:
Let x be the number of gallons of 5% butterfat milk we need.
We know that the total volume of milk after mixing the two types is 90 + x gallons.
The equation for the butterfat concentration can be expressed as:
(0.05 * x + 0.01 * 90) / (90 + x) = 0.02
Simplifying the equation, we have:
(0.05x + 0.009) / (90 + x) = 0.02
Now, let's solve for x:
0.05x + 0.009 = 0.02 * (90 + x)
0.05x + 0.009 = 1.8 + 0.02x
0.05x - 0.02x = 1.8 - 0.009
0.03x = 1.791
x = 1.791 / 0.03
x ≈ 59.7
Therefore, approximately 59.7 gallons of 5% butterfat milk must be mixed with 90 gallons of 1% milk to obtain 2% milk.
To solve this problem, we need to determine the amount of milk with 5% butterfat that needs to be mixed with the 90 gallons of 1% milk to obtain a desired mixture with 2% butterfat.
Let's break down the problem step by step:
Step 1: Understand the given information.
- We have two types of milk: one with 5% butterfat and the other with 1% butterfat.
- We want to mix them to get a final milk with 2% butterfat.
- We are given the amount of the 1% milk, which is 90 gallons.
Step 2: Assign variables to the unknowns.
- Let's assume the number of gallons of milk with 5% butterfat that needs to be mixed is "x".
Step 3: Set up the equation using the butterfat content.
- The butterfat content in the 5% milk is 0.05 (5% written as a decimal).
- The butterfat content in the 1% milk is 0.01 (1% written as a decimal).
- The butterfat content in the 2% milk is 0.02 (2% written as a decimal).
- The total amount of butterfat in the 5% milk is 0.05x.
- The total amount of butterfat in the 1% milk is 0.01 * 90 = 0.9.
Since we want to obtain a final milk mixture with 2% butterfat, we can set up the equation:
0.05x + 0.01 * 90 = 0.02 * (90 + x)
Step 4: Solve the equation for "x".
Let's solve the equation to find the value of "x" which represents the number of gallons of 5% milk needed.
0.05x + 0.9 = 1.8 + 0.02x
0.05x - 0.02x = 1.8 - 0.9
0.03x = 0.9
x = 0.9 / 0.03
x = 30
Therefore, you need to mix 30 gallons of milk with 5% butterfat with the 90 gallons of 1% milk to obtain 2% milk.