Please help me with this question! Thank you SO much!!

Question

A cook needs to mix a regular ranch dressing containing 47 % fat with a light ranch dressing containing 37 % fat to result in 7 cups of a dressing containing 42 % fat. How many cups of each kind of dressing should be mixed?
The mixture should contain  how many cups of each:
 cups of regular dressing?

and
 cups of light dressing?

Answer 1

This is a "systems of equations" type problem.  The % make it look difficult, but it will get much easier when we deal with them.

 

First off, let's define our variables.

Let x be the # of cups of the (47% fat) dressing, and y is the # of cups for the (37% fat) dressing.

 

The first equation I want to write is about the number of cups of dressing.  If you are required to mix 7 cups of dressing, then Eq1: x+y=7

 

The second equation is going to be about the fat content.  The total fat in the final dressing is determined by the amount of fat provided by the first dressing plus the amount of fat provided by the 2nd dressing.  To determine the amount of fat, simply multiply the percent by volume.  I prefer to covert the percents to decimals.  We also need to note that the volume of the final dressing is the sum of both original dressings.

Fat in first dressing (x): 0.47x
Fat in second dressing (y): 0.37y
Eq2: 0.47x+0.37y=0.42(x+y)

 

Distribute the 0.42

0.47x+0.37y=0.42x+0.42y

 

Combine like terms (subtract 0.42x from both sides, subtract 0.37y from both sides).

0.05x=0.05y

 

Divide both sides by 0.05.

x=y

 

This shows us that the amount of x-dressing must be the same as the amount of y-dressing.

 

You can now substitute one variable for the other in Eq1.  

x+(x)=7

2x=7

x=3.5

 

Because x=y, then both are 3.5 cups.