1. **Problem statement:** Suppose two siblings currently have ages in the ratio 3:5. Years ago, the ratio of their ages was half the ratio of their ages years from now. Find their current ages. 2. **Define variables:** Let the current ages be $3x$ and $5x$. 3. **Set up expressions for ages years ago and years from now:** - Years ago, their ages were $3x - y$ and $5x - y$. - Years from now, their ages will be $3x + y$ and $5x + y$. 4. **Write the ratio condition:** The ratio years ago is half the ratio years from now: $$\frac{3x - y}{5x - y} = \frac{1}{2} \times \frac{3x + y}{5x + y}$$ 5. **Cross multiply to clear fractions:** $$2(3x - y)(5x + y) = (5x - y)(3x + y)$$ 6. **Expand both sides:** Left: $2[(3x)(5x) + (3x)(y) - y(5x) - y(y)] = 2(15x^2 + 3xy - 5xy - y^2) = 2(15x^2 - 2xy - y^2) = 30x^2 - 4xy - 2y^2$ Right: $(5x)(3x) + (5x)(y) - y(3x) - y(y) = 15x^2 + 5xy - 3xy - y^2 = 15x^2 + 2xy - y^2$ 7. **Set equation:** $$30x^2 - 4xy - 2y^2 = 15x^2 + 2xy - y^2$$ 8. **Bring all terms to one side:** $$30x^2 - 4xy - 2y^2 - 15x^2 - 2xy + y^2 = 0$$ Simplify: $$15x^2 - 6xy - y^2 = 0$$ 9. **Rewrite as:** $$15x^2 - 6xy - y^2 = 0$$ 10. **Solve for $y$ in terms of $x$:** Treat as quadratic in $y$: $$-y^2 - 6xy + 15x^2 = 0$$ Multiply both sides by $-1$: $$y^2 + 6xy - 15x^2 = 0$$ 11. **Use quadratic formula:** $$y = \frac{-6x \pm \sqrt{(6x)^2 - 4(1)(-15x^2)}}{2} = \frac{-6x \pm \sqrt{36x^2 + 60x^2}}{2} = \frac{-6x \pm \sqrt{96x^2}}{2}$$ 12. **Simplify square root:** $$\sqrt{96x^2} = x\sqrt{96} = x \times 4\sqrt{6}$$ 13. **So:** $$y = \frac{-6x \pm 4x\sqrt{6}}{2} = x \frac{-6 \pm 4\sqrt{6}}{2} = x(-3 \pm 2\sqrt{6})$$ 14. **Since $y$ is a positive number of years, choose positive root:** $$y = x(-3 + 2\sqrt{6})$$ 15. **Find current ages:** - First sibling: $3x$ - Second sibling: $5x$ 16. **Example:** Let $x=1$ for simplicity: - $y = -3 + 2\sqrt{6} \approx -3 + 4.89898 = 1.89898$ - Ages years ago: $3 - 1.89898 = 1.10102$, $5 - 1.89898 = 3.10102$ - Ages years from now: $3 + 1.89898 = 4.89898$, $5 + 1.89898 = 6.89898$ 17. **Check ratio condition:** - Ratio years ago: $\frac{1.10102}{3.10102} \approx 0.355 - Ratio years from now: $\frac{4.89898}{6.89898} \approx 0.710$ - Half of ratio years from now: $0.710 / 2 = 0.355$ 18. **Answer:** The current ages are $3x$ and $5x$ where $y = x(-3 + 2\sqrt{6})$ years satisfies the condition. For example, if $x=1$, ages are 3 and 5 years, and $y \approx 1.9$ years.