A mixture of two kinds of beef has a total fat content of 8%; find the ratio of the mixture

Question

I am struggling a bit on weighted average problems on the GMAT. I'm a bit confused as to how they work and the best way to solve them. I'm kind of relying on intuition, but I'd like a more formal way to solve the problem.

1. A mixture of "lean" ground beef (10% fat) and "super-lean" ground beef (3% fat) has a total fat content of 8%. What is the ratio of "lean" ground beef to "super lean" ground beef?

So the way I solve this is by thinking first that the ratio has to be weighted towards the "lean" ground beef because 8% is above the average of the two numbers (6.5%). But how weighted towards "lean" is it?

Well, what I do is I take the difference absolute value between the actual average (8%) and subtract 3% from it to get 5%. Then I take the absolute value between the average (8%) and the high number (10%). Now I know the ratio is 5:2.

But what's a more formal way to do this?

Answer 1

I'd approach it in this way:

Let $L$ be the total amount of lean beef, $L_f$ its net fat content. Same for $S$ (super lean) and $S_f$. And let $M$ be the mixture (and $M_f$ its fat).

Then you know that $$L_f =0.1\, L$$ $$S_f=0.03 \, S$$ $$M_f=0.08 \,M$$ Also: $M=L+S$ and $M_f = L_f+S_f$

Then

$$ 0.1 L + 0.03 S = 0.08 \, (L+S)$$

$$0.02\, L = 0.05 \, S \implies L/S = 5/2$$

Answer 2

Suppose you have $1$ pound of super-lean beef. You then have $0.03$ pounds of fat in there. If you add $x$ pounds of lean beef, you then have $1+x$ pounds of beef, of which $0.03+0.1x$ pounds are fat. The condition you want to satisfy is that $0.03+0.1x$ is $8$ percent—that is, $0.08$—of $1+x$. Symbolically:

$$ 0.03+0.1x = 0.08(1+x) = 0.08+0.08x $$

Subtracting $0.03+0.08x$ from both sides gives us

$$ 0.02x = 0.05$ $$ $$ 2x = 5 $$ $$ x = \frac{5}{2} $$

In plain English, you need to mix $2.5$ pounds of lean beef to every pound of super-lean beef to get a mixture that has a fat content of $8$ percent.