Problem Solving

Three grades of milk: 1%, 2%, and 3% fat by volume. X gallons of 1%, y gallons
of 2%, and z gallons of 3% are mixed to give x + y + z gallons of 1.5%. What is x
in terms of y and z?
a. y + 3z
b. (y + z)/4
c. 2y + 3z
d. 3y + z
e. 3y + 4z

OA is A

Akansha wrote:Three grades of milk: 1%, 2%, and 3% fat by volume. X gallons of 1%, y gallons
of 2%, and z gallons of 3% are mixed to give x + y + z gallons of 1.5%. What is x
in terms of y and z?
a. y + 3z
b. (y + z)/4
c. 2y + 3z
d. 3y + z
e. 3y + 4.5z

OA is A

X=1% milkfat.
Y=2% milkfat.
If equal amounts of X and Y are combined, the percentage of milkfat in the resulting mixture will be the average of the two percentages above:
(1% + 2%)/2 = 1.5%.

To illustrate:
Let X = 200 liters and Y = 200 liters.
Milkfat in X = .01*200 = 2.
Milkfat in Y = .02*200 = 4.
Total milkfat = 2+4 = 6.
Total volume = 200+200 = 400.
Percentage of milkfat = 6/400 = (1.5)/100 = 1.5%.

Thus, the correct answer must work when equal amounts of X and Y are combined.

Plug in X=2, Y=2, Z=0.
This represents a mixture consisting of equal amounts of X and Y, with no Z.
Since equal amounts of X and Y are being combined, the resulting mixture will be 1.5% milkfat, as illustrated above.
The question asks for the value of X.
Given the values above, X=2. This is our target.

Now we plug Y=2 and Z=0 into the answers to see which yields our target of 2.

Only answer choice A works:
y+3z = 2+(3*0) = 2.

The correct answer is A.

Hi,
x gallons of milks contains x(1/100) fat by volume
y gallons of milks contains y(2/100) fat by volume
z gallons of milks contains z(3/100) fat by volume
So (x+y+z) gallons of milk contains (x+2y+3z)/100 fat ---eq(1)
But, we are given that the solution is 1.5% fat by volume. So, fat volume is (x+y+z)(3/200) --eq(2)
Equating eq(1) & eq(2); we get
(x+2y+3z)/100 = (x+y+z)(3/200)
=>2x+4y+6z = 3x+3y+3z
=>x = y+3z

Cheers!

I got neither of the explanations!
Any other way?

in the attached image the highlighted cell implies that,

1.5 % of mixtured solution = (.01X + .02Y + .03Z)
1.5/100(X+Y+Z) = (.01X + .02Y + .03Z)
1.5(X+Y+Z) = (1X + 2Y + 3Z)
0.5X=0.5Y + 1.5Z

therefore,
X = Y +3Z

If u're still having problem in understanding the above method, chk out tis URL


https://www.khanacademy.org/video/mixture-problems-2

has been discussed a lot.

1x+2y+3z = 1.5(x+y+z)

0.5x = 0.5y+1.5z
x=y+3z