Suppose $\frac ab = \frac35$,$\frac bc=\frac{15}6$, and $\frac cd = 6$. What is the value of $\frac ad$? Express your answer in simplest form.
I notice that all the fractions relate two variables, so I can use cross-multiplication to get rid of the fractions and find the value of each variable in terms of another. For example, from the first equation, $\frac ab = \frac35$, I can multiply both sides by $5b$ and get $5a = 3b$. Similarly, from the second equation, $\frac bc=\frac{15}6$, I can multiply both sides by $6c$ and get $6b = 15c$. And from the third equation, $\frac cd = 6$, I can multiply both sides by $d$ and get $c = 6d$. Now I have three equations that I can use to substitute one variable for another and eventually find $\frac ad$. I see that the last equation gives me a value for $c$, so I can plug that into the second equation and get $6b = 15(6d)$. Simplifying this, I get $b = 15d$. Now I have a value for $b$, so I can plug that into the first equation and get $5a = 3(15d)$. Simplifying this, I get $a = 9d$. Now I have values for $a$ and $d$, so I can write $\frac ad$ as $\frac{9d}{d}$. Simplifying this, I get $\frac ad = 9$. 