SOLUTION: Josh’s bicycle has 28-inch-diameter wheels and Julian’s has 14-inch-diameter wheels. Josh’s wheel rotates twice for every three rotations of the pedals; Julian’s wheel rotates thre

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Question 257443: Josh’s bicycle has 28-inch-diameter wheels and Julian’s has 14-inch-diameter wheels. Josh’s wheel rotates twice for every three rotations of the pedals; Julian’s wheel rotates three times for every two rotations of the pedals. Josh rotates the pedals twice as quickly as Julian (twice as many rotations per unit of time). Julian takes 8 minutes to bicycle one mile. How much time will Josh take?
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Josh’s bicycle has 28-inch-diameter wheels and Julian’s has 14-inch-diameter wheels. Josh’s wheel rotates twice for every three rotations of the pedals; Julian’s wheel rotates three times for every two rotations of the pedals. Josh rotates the pedals twice as quickly as Julian (twice as many rotations per unit of time). Julian takes 8 minutes to bicycle one mile. How much time will Josh take?
(60 minutes in an hour) X (N revolutions per minute) X ((D wheel diameter in inches) X pi per revolution) DIVIDED BY (5280 feet in a mile) times (12 inches in a foot) =
(speed in miles per hour)
60*(min/hr)*N*(rev/min)*(D*pi)*(in/rev) DIVIDED BY 5280*(ft/mi)* 12 * in/ft
60*N*D*pi*(in/hr) DIVIDED BY 5280*12*(in/mile)
(60*N*D*pi)/(5280*12) miles/hr
(5*N*D*pi)/5280 miles/hr (60/12=5)
(N*D*pi)/1056 miles/hr (1056*5=5280)
1 mile / 8 min * 60 minutes / 1 hr
1/8 * (miles/min) * 60 * (min/hr)
1/8 * 60 miles/hr
7.5 miles/hr or 1/8 mile/min
for Julian 7.5=(N*D*pi)/1056
7920=N*D*pi
7920=N*14*pi
180.0724498983=N in rev/min
1440.5795991861 revolutions in 8 minutes
10804.3469938955 rev/hr (60 min/hr)
Josh’s wheel rotates twice for every three rotations of the pedals; Julian's wheel rotates three times for every two rotations of the pedals. Josh rotates the pedals twice as quickly as Julian
2 rev / 3 pedal or 2/3 rev per pedal or 2/3*28*pi inches per pedal
58.643062867 inches per pedal
3 rev / 2 pedal or 3/2 rev per pedal or 3/2*14*pi inches per pedal
65.9734457254 inches per pedal
and Josh pedals twice as fast so Josh covers 117.286125734 inches in same time it takes Julian to go 65.9734457254 inches
a ratio of 1.7777777778 to 1
8 minutes / 1.7777777778 = 4.4999999999 = 4.5
Josh will take 4.5 minutes