Question

A confectioner has 300 pounds of a chocolate that is 1 part cocoa butter to 7 parts caramel. How much chocolate that is 1 part cocoa butter to 15 parts caramel should be combined with the first in order to create a new chocolate that is 1 part cocoa butter to 9 parts caramel?

Answer 1

Let's call our unknown quantity $n$. We'll need to add $n$ lbs of chocolate with a 1/15 ratio of cocoa butter to caramel to 300 lbs of chocolate with a 1/7 ratio of cocoa butter to caramel to yield $300+n$ lbs of chocolate with a 1/9 ratio of cocoa butter to caramel. We can set this up as an equation, but it's important to note something first: the ratio of the caramel to the cocoa butter is provided, but this is not the same thing as the fraction of the chocolate the cocoa butter takes up.

The 300 pounds of chocolate have 1 + 7 = 8 parts of cocoa butter and caramel total, which means that cocoa butter takes up 1/8 of those total parts; the $n$ pounds we're adding on has 1 + 15 = 16 total parts, which means the cocoa butter takes up 1/16 of those; and the $300+n$ pounds being produced have 1 + 9 = 10 total parts, so the cocoa butter takes up 1/10 of those parts. With this in mind, we can set up the following equation:

$300\big( \frac{1}{8}\big)+n\big( \frac{1}{16}\big)=(300+n)\big( \frac{1}{10}\big)$

which we can rewrite as

$\frac{300}{8}+ \frac{n}{16}= \frac{300+n}{10}$

From here, it would be helpful to combine the fractions on the left side of the equation. To do this, we'll convert the denominator of $\frac{300}{8}$ to 16, multiplying it by $\frac{2}{2}$ to obtain the fraction $\frac{600}{16}$. Combining that with $\frac{n}{16}$, we have:

$\frac{600+n}{16}= \frac{300+n}{10}$

To get rid of the denominator on the left, we'll multiply both sides of the equation by 16, and to eliminate the one on the right, we'll multiply both sides by 10:

$(10)(16)\big( \frac{600+n}{16}\big)=\big( \frac{300+n}{10}\big)(10)(16)$

Simplifying:

$10(600+n)=(300+n)16\\ 6000+10n=4800+16n$

And finally, we solve for $n$:

$6000-4800=16n-10n\\ 1200=6n\\ 200=n$

So, the confectioner needs 200 lbs of chocolate that's 1 part cocoa butter and 15 parts caramel.