Kurt, Chris, and David paid a total of $100 to buy a pasture. They split up the cost according to the amount of grass their animals would eat. Kurt put in 9 horses; David put in 12 cows for twice the time; and Chris put in some sheep for 2.5 times as long as David’s cows. If Chris paid half the cost of the pasture, how many sheep did Chris have, and how much did Kurt and David each pay, if 6 cows eat as much as 4 horses and 10 sheep eat as much as 3 cows in a given amount of time?
First, note that “12 cows for twice the time” equals 24 cows for the same time, and “s sheep at 2½ cow time” equals 2.5s sheep at cow-time, or 5s sheep at horse-time. So, let’s make a chart; we’ll try to get everything converted to horses.
Kurt | David | Chris | |
x | 50 – x | 50 | money |
9 horses | 24 cows | 5s sheep | animals |
9 horses | 24 cows | 3s/2 cows | 10 sheep = 3 cows 5s sheep = 3s/2 cows |
9 horses | 16 horses | s horses = ½ the total = 25 horses = 125 sheep^{*} |
6 cows = 4 horses → 24 cows = 16 horses 3 cows = 2 horses 3/2 cows = 1 horse → 3s/2 cows = s horses |
9/25 of ½ = 9/50 | 16/25 of ½ = 16/50 | ½ = 25/50 | fraction of pasture used |
9/50 x 100 = $18 | 16/50 x 100 = $32 | 25/50 x 100 = $50 | Each pays |
Kurt paid $18; David paid $32.