Caps Cost Bf119D

Let's solve this step by step! 🎉 **Step 1:** Understand the problem 👕🧢 Imagine you have some caps and T-shirts. The ratio of caps to T-shirts is 3 : 2. That means for every 3 caps, there are 2 T-shirts. **Step 2:** Use letters to help 🧮 Let the number of caps be $3x$. Let the number of T-shirts be $2x$. **Step 3:** Cost difference 💰 A T-shirt costs $5.50$ more than a cap. If the cost of a cap is $c$, then the cost of a T-shirt is $c + 5.50$. **Step 4:** Money spent on caps and T-shirts 🧢 + 👕 Mr. Yong spent $259$ more on caps than T-shirts. Money on caps = number of caps × cost per cap = $3x \times c = 3xc$. Money on T-shirts = number of T-shirts × cost per T-shirt = $2x \times (c + 5.50) = 2x(c+5.50)$. **Step 5:** Write the equation for the difference Money on caps is $259$ more than money on T-shirts: $$3xc = 2x(c+5.50) + 259$$ **Step 6:** Total money spent Total spent is $2219$: $$3xc + 2x(c + 5.50) = 2219$$ **Step 7:** Solve the equations From Step 5: $$3xc = 2xc + 11x + 259$$ Subtract $2xc$ from both sides: $$3xc - 2xc = 11x + 259$$ $$xc = 11x + 259$$ Divide both sides by $x$ (assuming $x \neq 0$): $$c = 11 + \frac{259}{x}$$ **Step 8:** Use total money equation from Step 6: $$3xc + 2x(c + 5.50) = 2219$$ Substitute $c$: $$3x \times c + 2x \times (c + 5.50) = 2219$$ $$3x c + 2x c + 11x = 2219$$ Combine like terms: $$5x c + 11x = 2219$$ Substitute $c$ from Step 7: $$5x \times \left(11 + \frac{259}{x}\right) + 11x = 2219$$ Simplify: $$5x \times 11 + 5x \times \frac{259}{x} + 11x = 2219$$ $$55x + 1295 + 11x = 2219$$ $$66x + 1295 = 2219$$ Subtract 1295: $$66x = 924$$ Divide: $$x = 14$$ **Step 9:** Find the cost of a cap Use $c = 11 + \frac{259}{x}$: $$c = 11 + \frac{259}{14} = 11 + 18.5 = 29.5$$ 🎉 Great! The cost of each cap is $29.50$. **Visual groups:** + =