Nick can brush his teeth 20 seconds faster than his girlfriend. If they work together they can brush his teeth in 30 seconds. How long does it take by himself?
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The girlfriend can brush Nick's mouth in G seconds., rate = M/G seconds
Nick's can brush his teeth in G-20 seconds, rate = M/(G-20) seconds
Working together they can brush nick's mouth in 30 seconds
1 Mouth = 30(1/G + 1/(G-20))
1 = 30/G + 30/(G-20) (multiply thru by G)
G = 30 + 30G/(G-20) (multiply thru by (G-20)
G^2 - 20G = 30G -600 + 30G = 60G - 600
G^2 -20G -60G + 600 = 0
G^2 - 80G + 600 = 0
G = ((80 +/- sqrt((6400-2400))/2
G = (80 +/- sqrt(4000))/2 = (80 +/- 63.25)/2 = 71.6 and 8 seconds
It takes the girlfriend 71.6 seconds to brush Nick's mouth, and Nick takes 51,6 seconds to brush his mouth.
Nick's can brush his teeth in G-20 seconds, rate = M/(G-20) seconds
Working together they can brush nick's mouth in 30 seconds
1 Mouth = 30(1/G + 1/(G-20))
1 = 30/G + 30/(G-20) (multiply thru by G)
G = 30 + 30G/(G-20) (multiply thru by (G-20)
G^2 - 20G = 30G -600 + 30G = 60G - 600
G^2 -20G -60G + 600 = 0
G^2 - 80G + 600 = 0
G = ((80 +/- sqrt((6400-2400))/2
G = (80 +/- sqrt(4000))/2 = (80 +/- 63.25)/2 = 71.6 and 8 seconds
It takes the girlfriend 71.6 seconds to brush Nick's mouth, and Nick takes 51,6 seconds to brush his mouth.