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.
My friend, Alex, struggles a little. I recently asked him to buy me some ribbon for my daughter's pretty pink bonnet.
He went to the haberdashery shop for the required length but accidentally interchanged the feet and inches.
When I measured the resulting ribbon I only had 5÷8 of the length I required.
I asked for 7 feet, 4 inches and my friend brought me 4 feet, 7 inches.
If we label what I asked for as A feet and B inches, which is (12 x A + B) inches, then I actually received B feet and A inches, which is (12 x B + A) inches. Which means that:
(5 ÷ 8) x (12 x A + B) = (12 x B + A)
A = (91 ÷ 52) x B
As B is the number of inches, it can only be between 1 and 12 and must give A as a whole number of feet. B = 4 is the only possible value, which means that A = 7. So I originally asked for 7 feet, 4 inches. QED.
Our local scrap merchant had to travel to the county recycling plant, 60 miles away.
On the way there he could only manage a steady 12mph, fully laden.
On the way back, he could manage a steady 16mph.
He placed 24 gallons of fuel in the tank ready for the round trip, thinking that this would be enough as the wagon managed to travel 8 miles per gallon. Unfortunately, a piece of metal pierced his fuel pipe at the start of the trip and the wagon started to lose 1 gallon of fuel per hour.
Luckily he only lost fuel when moving and he lost no fuel whilst unloading his scrap metal.
The trip was a total of 120 miles, which at 8 miles to the gallon, used 15 gallons.
The outward trip of 60 miles was at 12 mph, which took 5 hours. The return trip was at 16 mph, which took 3.75 hours. Which is a total journey time of 8.75 hours, which meant that 8.75 gallons were lost.
So, 15 gallons were used for the trip, 8.75 gallons were lost, which is a total of 23.75 gallons. So the 24 gallons was enough for the return trip.
I have three daughters and each has one brother. How many children do I have?