The problem states that the ratio of rings to bracelets is $2:3$, and the ratio of bracelets to necklaces is $2:6$. We are given that Rosa has 10 rings and asked to find the number of necklaces she has.

1. Problem Description

The problem states that the ratio of rings to bracelets is $2:3$, and the ratio of bracelets to necklaces is $2:6$. We are given that Rosa has 10 rings and asked to find the number of necklaces she has.

2. Solution Steps

First, we need to find the number of bracelets. We know the ratio of rings to bracelets is $2:3$. Let $r$ be the number of rings and $b$ be the number of bracelets. Then we have

$\frac{r}{b} = \frac{2}{3}$

We are given that $r=10$, so we can substitute this into the equation:

$\frac{10}{b} = \frac{2}{3}$

Cross-multiplying gives $2b = 30$, so $b = \frac{30}{2} = 15$.

Now, we need to find the number of necklaces. We know the ratio of bracelets to necklaces is $2:6$. Let $n$ be the number of necklaces. Then we have

$\frac{b}{n} = \frac{2}{6}$

We know that $b=15$, so we can substitute this into the equation:

$\frac{15}{n} = \frac{2}{6}$

Cross-multiplying gives $2n = 15 \times 6 = 90$, so $n = \frac{90}{2} = 45$.

3. Final Answer

Rosa has 45 necklaces.