Productive use of machinery on a farm made it possible to decrease the number of tractors by 12. how many tractors does the farm have now if it used to have 1.5 times as many?

team one had one-fourth as many people as team two. after 6 quit team two and 12 people were transferred from team two to team one, both teams became equal. how many people were originally on team one?

1. the answer is 24. think of x as the original amount, and y as the new amount. y times 1.5 is x, and y+12 is x. reverse that to figure out y, which is what we need, and you have x/1.5 = y as well as x-12 = y. Use the equal values method and make an equation x/1.5=x-12. solve this equation to get x, which is 36. to figure out the new amount, y, you need to subtract 12, which would help you get 24 as your final answer. 

2. once again, create an equation. let's call team 1 x and team 2 y. team one has 1/4th as many as team 2, so that would be x=1/4y. An easier way to write that is 4x=y. after 6 people quit team two, that would be y-6. after the transfer, that would be y-6-12, and x+12 for the teams. they are equal after these, so y-6-12=x+12. solve this equation to get y-18= x+12. if you recall earlier, y was 4 times x, so substitute that into y to get 4x-18=x+12. Solve the equation to get 10 people on team one originally. your final answer is 10 people. 

Logical equivalent of the give question is its contrapositive