2150.62 – Lowest Terms?

For certain values of the whole number $x$ , the fraction $\frac{9x+7}{6x + 9}$ is not in lowest terms. Find some examples of this and explain what is going on.

Solution

Using a spread sheet perhaps, you'll find that some of these x-values are 5, 18, 31, 44, 57, and so on. Notice that these are in the form $x = 13k+5$ . Plugging in, we can see what is happening:

\begin{aligned} \frac{9x+7}{6x + 9} &=& \frac{9(13k+5) + 7}{6(13k+5) + 9} \\ &=& \frac{117k+ 52}{78k+ 39} \\ &=& \frac{13 \cdot 9k+ 13 \cdot 4}{13 \cdot 6k+ 13 \cdot 3} \\ &=& \frac{13 (9k+4)}{13 (6k+3)}. \end{aligned}

My, my!