If 𝑚 is nonzero and 𝑎 over 𝑏 equals 𝑐 over 𝑑 which equals 𝑒 over 𝑓 which equals 𝑚, which of the following is equivalent to 𝑎𝑐𝑒 over 𝑏𝑑𝑓? Is it (A) three 𝑚 cubed, (B) 𝑚 cubed, (C) 𝑚, or three 𝑚?
We’re told a relationship between 𝑎, 𝑏; 𝑐, 𝑑; and 𝑒, 𝑓. We’re told that 𝑎 and 𝑏 are in the same proportion as 𝑐 and 𝑑 as are 𝑒 and 𝑓 and this proportion is equal to 𝑚. And so, let’s remind ourselves how to multiply fractions. When we multiply a pair of fractions for instance, we multiply their numerator and then separately multiply their denominators. By thinking about the reverse of that process and extending it to more than two fractions, we can say that 𝑎 over 𝑏 times 𝑐 over 𝑑 times 𝑒 over 𝑓 must be equal to 𝑎𝑐𝑒 over 𝑏𝑑𝑓. And that’s great because we know that each of these fractions individually is equal to 𝑚. So, we can write 𝑎𝑐𝑒 over 𝑏𝑑𝑓 as 𝑚 times 𝑚 times 𝑚.
Now, of course, 𝑚 times 𝑚 times 𝑚 is 𝑚 cubed. So, given that 𝑚 is a nonzero and 𝑎 over 𝑏 equals 𝑐 over 𝑑 which equals 𝑒 over 𝑓 which equals 𝑚, then we know the expression that’s equal to 𝑎𝑐𝑒 over 𝑏𝑑𝑓 is 𝑚 cubed. In this case, that’s option (B).