Suppose that $a$ is inversely proportional to $b$.  Let $a_1,a_2$ be two nonzero values of $a$ such that $\frac{a_1}{a_2}=\frac{2}{3}$.  Let the corresponding $b$ values be $b_1,b_2$.  If $b_1,b_2$ are nonzero, find the value of $\frac{b_1}{b_2}$.
If $a$ is inversely proportional to $b$, then the product $ab$ is a constant.  For $a_1$ and $a_2$ this implies: $$a_1\cdot b_1=a_2\cdot b_2$$We can divide both sides of this equation by $b_1\cdot a_2$ to find: \begin{align*}
\frac{a_1\cdot b_1}{b_1\cdot a_2}&=\frac{a_2\cdot b_2}{b_1\cdot a_2}\\
\Rightarrow\qquad \frac{2}{3}=\frac{a_1}{a_2}&=\frac{b_2}{b_1}\\
\Rightarrow\qquad \boxed{\frac{3}{2}}&=\frac{b_1}{b_2}
\end{align*}
\frac{3}{2}
Level 3
Algebra
Problem: Suppose that $a$ is inversely proportional to $b$.  Let $a_1,a_2$ be two nonzero values of $a$ such that $\frac{a_1}{a_2}=\frac{2}{3}$.  Let the corresponding $b$ values be $b_1,b_2$.  If $b_1,b_2$ are nonzero, find the value of $\frac{b_1}{b_2}$.
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>Since $a$ is inversely proportional to $b$, we can express this relationship mathematically as $a = \frac{k}{b}$ for some constant $k$. This implies that $ab = k$, meaning that the product of $a$ and $b$ remains constant for any pair of corresponding values.

Given the two values of $a$, $a_1$ and $a_2$, we can write the relationships for their corresponding $b$ values as follows:
1. For $a_1$: $a_1 b_1 = k$
2. For $a_2$: $a_2 b_2 = k$

Since both expressions equal $k$, we can set them equal to each other:
$a_1 b_1 = a_2 b_2$.

From the problem, we know that $\frac{a_1}{a_2} = \frac{2}{3}$. We can express $a_1$ in terms of $a_2$:
$a_1 = \frac{2}{3} a_2$.

Substituting this into the equation $a_1 b_1 = a_2 b_2$ gives:
\(\frac{2}{3} a_2 b_1 = a_2 b_2\).

Assuming $a_2 \neq 0$, we can divide both sides by $a_2$:
\(\frac{2}{3} b_1 = b_2\).

To find the ratio $\frac{b_1}{b_2}$, we rearrange the equation:
\(b_1 = \frac{3}{2} b_2\).

Now, taking the ratio $\frac{b_1}{b_2}$, we have:
\(\frac{b_1}{b_2} = \frac{3}{2}\).

Thus, the final answer is $\frac{3}{2}$.<|END_OF_LATENT|> The final answer is $\frac{3}{2}$.