I want to know how to solve this, not just the answer. Thanks.

The ratio of rubies to emeralds was 3 to 1, and the ratio of emeralds to diamonds was 2 to 1. If there were 18 rubies, emeralds, and diamonds in all, how many of each were there?

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- Feb 28th 2011, 04:04 PMwizkid94Proportion Problem Help
I want to know how to solve this, not just the answer. Thanks.

The ratio of rubies to emeralds was 3 to 1, and the ratio of emeralds to diamonds was 2 to 1. If there were 18 rubies, emeralds, and diamonds in all, how many of each were there? - Feb 28th 2011, 04:13 PMQuacky
Ruby:Emerald are in the following ratio:

$\displaystyle 3:1$

Can you appreciate that this is equivalent to:

$\displaystyle 6:2$?

So If I have R:E:D, I could write it all in one ratio as:

$\displaystyle 6:2:1$

**Working backwards, for every one diamond, there are two emeralds. And for every emerald, there are three times as many rubies**. This is the core statement.

Now, in the ratio above there are $\displaystyle 9$ gems. We could say that for every $\displaystyle 9$ gems, $\displaystyle \frac{6}{9}$ are rubies. So, if that is true, how many rubies would there be in a collection of $\displaystyle 18$ gems? That is to say, for every $\displaystyle 18$ gems, $\displaystyle \frac{?}{18}$ would be rubies? Use equivalent fractions to work it out. - Feb 28th 2011, 11:22 PMlanierms
You can also use a constant $\displaystyle k$ to figure it out quite easily.

Solution:

Ruby:Emerald = $\displaystyle 3:1$.

So, Ruby = $\displaystyle 3k$, Emerald = $\displaystyle k$.

Emerald: Diamond = $\displaystyle 2:1$ = $\displaystyle k:$ Diamond.

$\displaystyle \therefore Diamond=\frac{1}{2}k$

So, there are 3k rubies, k emeralds and half k diamonds.

Now, if we add all of them,

$\displaystyle 3k+k+\frac{1}{2}k=18$

$\displaystyle \therefore k=4$.

So, there are 12 rubies, 4 emeralds and 2 diamonds.

The solution posted by**Quacky**is great, but I don't think problems will always be simple like this one. - Mar 1st 2011, 07:23 PMwizkid94