If $\displaystyle\frac{q}{r} = 9$, $\displaystyle\frac{s}{r} = 6$, and $\displaystyle \frac{s}{t} = \frac{1}{2}$, then what is $\displaystyle\frac{t}{q}$?
If we multiply the second equation, the reciprocal of the first equation, and the reciprocal of the third equation, we get \[\frac{s}{r}\cdot\frac{r}{q}\cdot \frac{t}{s} = 6\cdot \frac{1}{9}\cdot2\Rightarrow \frac{t}{q}= \boxed{\frac{4}{3}}.\]
\frac{4}{3}
Level 3
Algebra
Problem: If $\displaystyle\frac{q}{r} = 9$, $\displaystyle\frac{s}{r} = 6$, and $\displaystyle \frac{s}{t} = \frac{1}{2}$, then what is $\displaystyle\frac{t}{q}$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To find the value of $\frac{t}{q}$, we start by expressing $q$, $s$, and $t$ in terms of $r$. From the first equation, $\frac{q}{r} = 9$, we can derive that $q = 9r$. From the second equation, $\frac{s}{r} = 6$, we find that $s = 6r$. The third equation, $\frac{s}{t} = \frac{1}{2}$, allows us to express $t$ in terms of $s$: $s = \frac{1}{2}t$, which rearranges to $t = 2s$. 

Substituting the expression for $s$ into the equation for $t$, we have $t = 2(6r) = 12r$. Now we have $q = 9r$ and $t = 12r$. 

Next, we compute $\frac{t}{q}$: 

\[
\frac{t}{q} = \frac{12r}{9r} = \frac{12}{9} = \frac{4}{3}.
\]

Thus, the final answer is $\frac{4}{3}$.<|END_OF_LATENT|> The final answer is $\frac{4}{3}$.