1. **State the problem:** We need to find a fraction $\frac{x}{y}$ such that: - When the numerator is multiplied by 4 and the denominator is decreased by 2, the fraction becomes $\frac{2}{2} = 1$. - When 15 is added to the numerator and 2 is subtracted from twice the denominator, the fraction becomes $\frac{9}{7}$. 2. **Write the equations from the problem:** From the first condition: $$\frac{4x}{y-2} = 1$$ From the second condition: $$\frac{x+15}{2y-2} = \frac{9}{7}$$ 3. **Simplify the first equation:** $$4x = y - 2$$ $$y = 4x + 2$$ 4. **Substitute $y$ into the second equation:** $$\frac{x+15}{2(4x+2) - 2} = \frac{9}{7}$$ Simplify the denominator: $$2(4x+2) - 2 = 8x + 4 - 2 = 8x + 2$$ So: $$\frac{x+15}{8x + 2} = \frac{9}{7}$$ 5. **Cross multiply to solve for $x$:** $$7(x + 15) = 9(8x + 2)$$ $$7x + 105 = 72x + 18$$ 6. **Bring all terms to one side:** $$105 - 18 = 72x - 7x$$ $$87 = 65x$$ 7. **Solve for $x$:** $$x = \frac{87}{65}$$ 8. **Find $y$ using $y = 4x + 2$:** $$y = 4 \times \frac{87}{65} + 2 = \frac{348}{65} + \frac{130}{65} = \frac{478}{65}$$ 9. **Final fraction:** $$\frac{x}{y} = \frac{\frac{87}{65}}{\frac{478}{65}} = \frac{87}{65} \times \frac{65}{478} = \frac{87}{478}$$ **Answer:** The fraction is $\frac{87}{478}$.