### Instruction: If $\displaystyle\frac{q}{r} = 9$, $\displaystyle\frac{s}{r} = 6$, and $\displaystyle \frac{s}{t} = \frac{1}{2}$, then what is $\displaystyle\frac{t}{q}$?### Response: I see that this problem involves several fractions that are equal to each other, so I wonder if I can use cross-multiplication to simplify them.
For example, if I cross-multiply the first fraction, I get $q \cdot 1 = r \cdot 9$, or $q = 9r$.
Similarly, I can cross-multiply the second fraction and get $s = 6r$.
And for the third fraction, I get $2s = t$.
Now I have expressions for $q$, $s$, and $t$ in terms of $r$, so I can use them to rewrite the fraction I want to find, $\displaystyle\frac{t}{q}$.
I replace $t$ with $2s$ and $q$ with $9r$, and get $\displaystyle\frac{2s}{9r}$.
Then I replace $s$ with $6r$, and get $\displaystyle\frac{12r}{9r}$.
I see that I can simplify this fraction by canceling out a common factor of $r$, and get $\displaystyle\frac{12}{9}$.
I can also reduce this fraction by dividing both the numerator and denominator by 3, and get $\displaystyle\frac{4}{3}$.