Six numbers $a$, $b$, $c$, $d$, $e$, $f$ are such that $ab = 1$, $bc = \frac{1}{2}$, $cd = 6$, $de = 2$ and $ef = \frac{1}{2}$. What is the value of $(ad : be : cf)$?

### Concept & Algebraic Manipulation To find the terms $ad$, $be$, and $cf$, we can use the given paired products to isolate the desired combinations by multiplying and dividing adjacent pairs. $$ ad = \frac{(ab)(cd)}{bc} $$ $$ be = \frac{(bc)(de)}{cd} $$ $$ cf = \frac{(cd)(ef)}{de} $$ ### Step-by-Step Solution 1. Using the given values, we first calculate $ad$: $$ ad = \frac{ab \times cd}{bc} = \frac{1 \times 6}{\frac{1}{2}} = 6 \times 2 = 12 $$ 2. Next, calculate $be$: $$ be = \frac{bc \times de}{cd} = \frac{\frac{1}{2} \times 2}{6} = \frac{1}{6} $$ 3. Finally, calculate $cf$: $$ cf = \frac{cd \times ef}{de} = \frac{6 \times \frac{1}{2}}{2} = \frac{3}{2} $$ 4. Now, find the ratio $(ad : be : cf)$: $$ 12 : \frac{1}{6} : \frac{3}{2} $$ 5. To eliminate fractions, multiply the entire ratio by the Least Common Multiple (LCM) of the denominators, which is 6: $$ (12 \times 6) : \left(\frac{1}{6} \times 6\right) : \left(\frac{3}{2} \times 6\right) $$ $$ 72 : 1 : 9 $$ ### Exam Strategy & Shortcut Instead of solving for individual variables $a, b, c, d, e, f$, immediately look for algebraic groups. Seeing $ad$ alongside given products $ab, bc, cd$ suggests chaining products. Realizing $ad = \frac{abcd}{bc}$ saves immense time over substitution. ### Common Pitfall A common mistake is trying to solve for individual variable values (like setting $a = 1, b = 1$ to satisfy $ab = 1$), which might lead to contradictions later down the sequence or make calculations extremely complicated. ### Final Answer Therefore, the correct answer is **72 : 1 : 9**.