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?
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?
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.
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.