1. **State the problem:** Mr. Yong bought caps and T-shirts. The ratio of caps to T-shirts is 3:2. A T-shirt costs 5.50 more than a cap. He spent 259 more on caps than on T-shirts. Total spent is 2219. Find the cost of each cap. 2. **Define variables:** Let the cost of a cap be $x$. Then the cost of a T-shirt is $x + 5.50$. 3. **Number of items:** Let the number of caps be $3k$ and T-shirts be $2k$ (from ratio 3:2). 4. **Write equations for total cost:** Total cost caps = $3k \times x = 3kx$ Total cost T-shirts = $2k \times (x + 5.50) = 2k(x + 5.50)$ 5. **Use given conditions:** Total spent: $$3kx + 2k(x + 5.50) = 2219$$ Difference spent: $$3kx - 2k(x + 5.50) = 259$$ 6. **Simplify equations:** From total spent: $$3kx + 2kx + 11k = 2219$$ $$5kx + 11k = 2219$$ From difference: $$3kx - 2kx - 11k = 259$$ $$kx - 11k = 259$$ 7. **Factor out $k$ in difference equation:** $$k(x - 11) = 259$$ 8. **Express $k$ from difference equation:** $$k = \frac{259}{x - 11}$$ 9. **Substitute $k$ into total spent equation:** $$5k x + 11k = 2219$$ $$k(5x + 11) = 2219$$ Substitute $k$: $$\frac{259}{x - 11} (5x + 11) = 2219$$ 10. **Multiply both sides by $x - 11$:** $$259(5x + 11) = 2219(x - 11)$$ 11. **Expand both sides:** $$1295x + 2849 = 2219x - 24409$$ 12. **Bring all terms to one side:** $$1295x + 2849 - 2219x + 24409 = 0$$ $$-924x + 27258 = 0$$ 13. **Solve for $x$:** $$-924x = -27258$$ $$x = \frac{27258}{924}$$ 14. **Simplify fraction:** Divide numerator and denominator by 6: $$x = \frac{27258 \div 6}{924 \div 6} = \frac{4543}{154}$$ 15. **Calculate decimal:** $$x \approx 29.5$$ **Final answer:** The cost of each cap is approximately $29.50$.