Practical Peter was asked to cut a 99 foot rope into three smaller, equal length ropes.
However, as usual, Pete couldn't find his measuring tape so he guessed!
When he finally did find his tape (it was under his hat), he discovered that:
A) the second piece of rope was twice as long as the first piece, minus 35 feet (i.e. 2 x first, - 35).
B) the third piece of rope was half the length of the first, plus 15 feet (i.e. 0.5 x first, + 15)
Hint: Try breaking each cog size into prime factors.
Answer: 30 revolutions.
If we break each wheel into its prime factors, we get:
63 = 3 x 3 x 7
42 = 2 x 3 x 7
35 = 5 x 7
27 = 3 x 3 x 3
We now think of rotating the large wheel just once, and this is 3 x 3 x 7 teeth moved (3 x 21), and we can see that 42 tooth wheel also has a 3 x 7 (21 teeth) in it, with an extra 2. If we therefore rotate the 63 toothed wheel twice, the 42 will have rotated three times.
The answer involves cancelling any common factors from the large wheel. We can cancel 3, 3, 7 from any of the smaller ones to leave 2 (from the 42), 5 (from the 35) and 3 (from the 27). 2 x 5 x 3 = 30. QED.