\( \frac{5 x-25}{50}=2 x+50 \) b) The numerator of a fraction is greater than the denominator by 8 . If the numerator is decreased by 3 and the denominator is increased by 2 , the fraction is equivalent to \( \frac{4}{3} \). Find the fraction. Let the dente denominator ae \( x \)

To solve the equation \( \frac{5x - 25}{50} = 2x + 50 \), we first eliminate the fraction by multiplying both sides by 50: \[ 5x - 25 = 100x + 2500 \] Rearranging gives: \[ 5x - 100x = 2500 + 25 \] This simplifies to: \[ -95x = 2525 \] Dividing by -95: \[ x = -\frac{2525}{95} \] Now, for the fraction problem, you can let the denominator be \( x \). Therefore, the numerator would be \( x + 8 \). When the numerator is decreased by 3 and the denominator is increased by 2, the equation can be set up as: \[ \frac{x + 8 - 3}{x + 2} = \frac{4}{3} \] This simplifies to: \[ \frac{x + 5}{x + 2} = \frac{4}{3} \] Cross-multiplying gives: \[ 3(x + 5) = 4(x + 2) \] Expanding both sides leads to: \[ 3x + 15 = 4x + 8 \] Rearranging this gives: \[ 15 - 8 = 4x - 3x \] Which simplifies to: \[ x = 7 \] Substituting back to find the fraction, the numerator is \( 7 + 8 = 15 \). Thus, the fraction is \( \frac{15}{7} \). --- Throughout history, fractions have been vital in trade and commerce. Ancient civilizations, like the Egyptians, had a complex system to deal with fractions, often using unit fractions (like \( \frac{1}{2} \) or \( \frac{1}{3} \)). In everyday life, fractions appear everywhere! Whether it's measuring ingredients in cooking, splitting a bill at a restaurant, or understanding percentages in finance, knowing how to manipulate and understand fractions is crucial. The more you practice, the easier it becomes to see how useful they can be!