We are asked to find all ratios that are equivalent to $12:18$ from the following options: $15:21$, $6:9$, $24:36$, $13:19$, and $2:3$.

1. Problem Description

We are asked to find all ratios that are equivalent to $12:18$ from the following options: $15:21$, $6:9$, $24:36$, $13:19$, and $2:3$.

2. Solution Steps

First, simplify the given ratio $12:18$. We can divide both numbers by their greatest common divisor, which is

6. $12/6 = 2$

$18/6 = 3$

So, $12:18$ simplifies to $2:3$.

Now, we check each option:

* $15:21$

Divide both by 3: $15/3 = 5$, $21/3 = 7$. So, $15:21$ simplifies to $5:7$. This is not equal to $2:3$.

* $6:9$

Divide both by 3: $6/3 = 2$, $9/3 = 3$. So, $6:9$ simplifies to $2:3$. This is equivalent to $12:18$.

* $24:36$

Divide both by 12: $24/12 = 2$, $36/12 = 3$. So, $24:36$ simplifies to $2:3$. This is equivalent to $12:18$.

* $13:19$

This cannot be simplified. $13:19$ is not equal to $2:3$.

* $2:3$

This is already in simplest form, and is the same as the simplified form of $12:18$.

3. Final Answer

The equivalent ratios are $6:9$, $24:36$, and $2:3$.