The First Three Problems of AMC 12: Explained

Problem Level: 3~4
Problem Source: American Math Competitions 12


Problem 1:
Two is 10% of x and 20% of y . What is x-y?


Problem 2:
The equations 2x+7=3 and bx-10=-2  have the same solution. What is the value of b?


Problem 3:
A rectangle with diagonal length x is twice as long as it is wide. What is the area of the rectangle in terms of x?


(answers)



Solutions:
Problem 1:
It is important to get the variable straight. 2 = 0.1 x and 2 = 0.2y.
Once the equations are there, everything becomes simpler. x =20 and y=10.
x-y=20-10=10
Answer = 10

Problem 2:

They have the same solution.  So lets solve 2x+7=3 first.  Taking 7 from both sides and we know that x= -2. Plugging it in the second equation (because the have the same solution, we get : -2b-10=-2.

Solving that we get b =-4 which becomes our answer.

Problem 3:

Let w be the width of the rectangle and 2w be the length. By the Pythagorean Theorem,(2w)^2+w^2=x^2 so 5w^2=x^2 and w = x/  The area would become 2w*w=2w^2.

Substituting w for x/ we get 2/5 x^2 (our answer)