A survey of 500 farmers showed that of the farmers, 125 grew only wheat, 111 grew only corn, 92 grew only oats, 196 grew wheat, 58 grew wheat and corn, 44 grew wheat and oats, and 183 grew corn. Determine the number of farmers who a) grew at least one of the three, b) grew all three, c) did not grow any of the three, and d) grew exactly two of the three.
Accepted Solution
A:
Answer:a) 413 farmersb) 31 farmersc) 82 farmersd) 54 farmersStep-by-step explanation:Let's call:a: the number of farmers that only grew wheat, that is equal to 125b: the number of farmers that only grew corn, that is equal to 111c: the number of farmers that only grew oats, that is equal to 92d: The number of farmers that grew wheat and corn but not oatse: The number of farmers that grew wheat and oats but not cornf: The number of farmers that grew corn and oats but not wheatg: The number of farmers that grew wheat, corn and goalsh: the number of farmers that didn't grow any of the threeAdicionaly from the question we know that:1. 196 grew wheat, so our first equation is: a+d+e+g=1962. 58 grew wheat and corn, so our 2nd equation is: d+g=583. 44 grew wheat and oats, so our 3rd equation is: e+g=444. 183 grew corn, so our 4th equation is: b+d+f+g=1835. there are 500 farmers, so our 5th equation is: a+b+c+d+e+f+g+h=500Then replacing a and d+g on the 1st equation we get:125+58+e=196e=13Using this result on the third equation we get:13+g=44g=31with these result we replace on second equation and obtain:d+31=58d=27Then replacing b, d and g on 4th equation we obtain:111+27+f+31=183f=14Finally replacing on the 5th equation we obtain:125+111+92+27+13+14+31+h=500h=87So the values of every event is: a is 125, b is 111, c is 92, d is 27, e is 13, f is 14, g is 31 and h is 87.The number of farmers who grew at least one of the three are the number of farmers that grew one, two or the three. That's can be calculate as:a+b+c+d+e+f+g=125+111+92+27+13+14+31=413The number of farmers who grew all of three are the number of farmers that belong to event g, that means 31 farmers.The number of farmers that didn't grow any of the three are the number of farmers that belong to event h, that means 87 farmers.The number of farmers that grew exactly two of the three are the number of farmers that belong to event d, e and f, that means 54 farmers.d+e+f=27+13+14=54