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Halmar the Great has boasted to his hordes of followers that many a notorious villain has fallen to his awesome sword: His total of 650 victims consists of evil sorcerers, trolls, and orcs. These he has slain with a total of 700 mighty thrusts of his sword; evil sorcerers and trolls each requiring two thrusts (to the chest) and orcs each requiring one thrust (to the neck). When asked about the number of trolls he has slain, he replies, "I, the mighty Halmar, despise trolls four times as much as I despise evil sorcerers. Accordingly, four times as many trolls as evil sorcerers have fallen to my sword!" How many of each type of villain has he slain?

*Let S, T, O be the number of sorcerers, trolls, and orcs respectively*

*The total number of victims is 650:*

*S + T + O = 650----- [1]*

*The total number he has slain is is 700, but there are two thrusts per kill for sorcerers and trolls and one for orcs:*

*2S + 2T + O = 700----- [2]*

*And there are 4 times as many trolls killed as sorcerers:*

*T = 4S----- [3]*

*Substitute [3] into [1] and [2]:*

*S + 4S + O = 650 -> 5S + O = 650*

*2S + 2(4S) + O = 700 -> 10S + O = 700*

*We have two equations and two unknowns:*

*Multiply the 1st by 2 and subtract from the 2nd:*

*10S + 2O = 1300*

*10S + O = 700*

*O = 600*

*Thus From above equation*

*5S + O = 650*

*Substituting value of O = 600*

*5S = 650 - 600*

*5S = 50*

*S = 10.*

*and*

*T = 4*10 = 40*

*Ans: 10 sorcerers, 40 trolls, and 600 orcs.*

*Answer: 10 sorcerers, 40 trolls, and 600 orcs.*

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