4. There are \( x \) number of red and \( y \) number of blue balls all of the same size and weight in a box. The total number of balls in the box is 20 and \( (x+2):(2 y+1)=2: 5 \). (a) Find the values of \( x \) and \( y \).

To solve the given problem, we start by setting up our equations based on the information provided. We know: 1. The total number of balls is \( x + y = 20 \). 2. The ratio \( (x + 2):(2y + 1) = 2:5 \). From the ratio, we can express it as: \[ \frac{x + 2}{2y + 1} = \frac{2}{5} \] Cross-multiplying gives us: \[ 5(x + 2) = 2(2y + 1) \] Expanding both sides, we have: \[ 5x + 10 = 4y + 2 \] Rearranging this gives us: \[ 5x - 4y = -8 \quad (1) \] Now we have a system of equations: 1. \( x + y = 20 \) 2. \( 5x - 4y = -8 \) From equation (1), we can express \( y \) in terms of \( x \): \[ y = 20 - x \] Substituting this into equation \( 5x - 4y = -8 \): \[ 5x - 4(20 - x) = -8 \] This simplifies to: \[ 5x - 80 + 4x = -8 \] Combining like terms results in: \[ 9x - 80 = -8 \] Now add 80 to both sides: \[ 9x = 72 \] Dividing by 9 gives: \[ x = 8 \] We substitute \( x = 8 \) back into \( y = 20 - x \): \[ y = 20 - 8 = 12 \] Thus, the values of \( x \) and \( y \) are: \[ \boxed{8} \text{ (red balls)}, \quad \boxed{12} \text{ (blue balls)}. \]