19. One number is 16 more than another. If the smaller number is subtracted trom two thirds of the larger number, the result is one fourth of the sum of the two numbers. Find the numbers.

Let's denote the smaller number as \( x \) and the larger number as \( x + 16 \). According to the problem, if we subtract \( x \) from two-thirds of the larger number, we should get one-fourth of the sum of the two numbers. We can set up the equation as follows: \[ \frac{2}{3}(x + 16) - x = \frac{1}{4}(x + (x + 16)) \] Now, let’s simplify this equation step-by-step! First, simplify the left side: \[ \frac{2}{3}(x + 16) - x = \frac{2x + 32}{3} - x = \frac{2x + 32 - 3x}{3} = \frac{-x + 32}{3} \] Now, simplify the right side: \[ \frac{1}{4}(x + (x + 16)) = \frac{1}{4}(2x + 16) = \frac{2x + 16}{4} = \frac{x + 8}{2} \] Now we have: \[ \frac{-x + 32}{3} = \frac{x + 8}{2} \] Cross-multiplying to eliminate the fractions gives: \[ -2(-x + 32) = 3(x + 8) \] Expanding both sides results in: \[ 2x - 64 = 3x + 24 \] Now, isolate \( x \): \[ 2x - 3x = 24 + 64 \implies -x = 88 \implies x = -88 \] So the smaller number \( x = -88 \) and the larger number is: \[ x + 16 = -88 + 16 = -72 \] Thus, the two numbers are \( -88 \) and \( -72 \).