14. A student played a computer game 500 times and won 370 of these games. He then won the next \( x \) games and lost none. He has now won \( 75 \% \) of the games he has played. Find the value of \( x \).

To find the value of \( x \), we start by determining the total number of games the student has played after winning \( x \) additional games. Initially, the student played 500 games and won 370 of them. After winning \( x \) more games, the total number of games played becomes: \[ 500 + x \] The total number of games won now is: \[ 370 + x \] According to the problem, the student has won \( 75\% \) of the total games played. Therefore, we can set up the following equation: \[ \frac{370 + x}{500 + x} = 0.75 \] To eliminate the fraction, we can cross-multiply: \[ 370 + x = 0.75(500 + x) \] Expanding the right side: \[ 370 + x = 375 + 0.75x \] Next, we will isolate \( x \) by moving all terms involving \( x \) to one side and constant terms to the other side. First, subtract \( 0.75x \) from both sides: \[ 370 + x - 0.75x = 375 \] This simplifies to: \[ 370 + 0.25x = 375 \] Now, subtract 370 from both sides: \[ 0.25x = 375 - 370 \] This simplifies to: \[ 0.25x = 5 \] Now, divide both sides by 0.25 to solve for \( x \): \[ x = \frac{5}{0.25} = 20 \] Thus, the value of \( x \) is: \[ \boxed{20} \]