This puzzle provides a long list of fractions and not much else. The title is probably the best place to begin. "Cancelled" heavily implies we want to cancel these fractions somehow, and they certainly all seem to be not yet in their most simplified forms. But converting them to simplest forms doesn't look particularly promising, and isn't really emphasising cancellation.
At this point it might seem prudent to turn to the smaller/simpler-looking fractions. The 16/64 and 19/95 examples simplify to 1/4 and 1/5 respectively, which is interesting as these simpler fractions could also have been achieved by deleting the common digits. In fact these pairs might be recognisable to some as the source of an age-old maths joke to the same effect, claiming you can cancel digits to simplify fractions.
So it seems possible this trick can be applied to all the provided fractions. Note for example that 154/451 = 14/41, and 712/4717 = 72/477 (= 8/53). In some cases we have to cancel two digits, e.g. 3514/26104 = 35/260. In other rare cases there is actually no way to cancel digits and keep the same fraction - the very first example doesn't even have common digits between the numerator and denominator (933/7464).
At any rate we can note that in the cases where two digits are removed, they always appear in the same order on the top and bottom, and the first digit is always a 1 or 2. In fact when concatenated, we can see there is no fraction whose removed digits make a number exceeding 26. So it seems likely we want to map the cancelled digits to their positions in the alphabet. Doing so, we can treat "uncancellable" fractions as spaces. For example, the first four fractions are 3514/26104 = 35/260, 1575/1953 = 75/93, 2346/24378 = 46/478, and 933/7464 cannot be cancelled. The cancelled digits (concatenated) in each case are 14, 15, 23, [none], which would map to the letters "NOW ".
Altogether this gives the message NOW ADD A HALF TO EACH UNUSED FRACTION. So let's follow this instruction. We must make the assumption here that when adding a half, the denominator remains the same even if cancellation could occur. Thus our first unused (space) fraction 933/7464 becomes 933/7464 + 1/2 = 4665/7464. This fraction now can be digit-cancelled, since 4665/7464 = 465/744.
Doing the same trick to the other unused fractions, we get all up that 4665/7464 = 465/744, 7936/8928 = 736/828, 4186/6188 = 46/68, 3757/6358 = 377/638, 8632/9628 = 832/928, 7172/9128 = 77/98, 12586/23548 = 186/348. This time the cancelled digits correspond to the letters FIREFLY, an infamously unfairly-cancelled television show.
|The answer is: firefly|