A photographer decides to decrease a picture she took in sequence to fit it within a certain frame. She requires the picture to be one-third of the area of the original. If the original picture was 4 inches through 6 inches, how many inches is the smaller dimension of the decreased picture if each dimension changes the same amount?
Let x = the amount of reduction. Then 4 - x = the width of the decreased picture and 6 - x = the length of the decreased picture. Since area is length times width, and one-third of the old area of 24 is 8, the equation for the area of the decreased picture would be (4 - x)(6 - x) = 8. Multiply the binomials by using the distributive property: 24 - 4x - 6x + x^{2} = 8; combine like terms: 24 - 10x + x^{2} = 8. Subtract 8 from both sides: 24 - 8 - 10x + x^{2} = 8 - 8. Simplify and place in standard form: x^{2} - 10x + 16 = 0. Factor the trinomial into 2 binomials: (x - 2)(x - 8) = 0. Set each factor equal to zero and solve: x - 2 = 0 or x - 8 = 0; x = 2 or x = 8. The solution of 8 is not reasonable because it is greater than the original dimensions of the picture. Accept the solution of x = 2 and the smaller dimension of the reduced picture would be 4 - 2 = 2 inches.