1. **Problem statement:** Vincent was given a sum of money. He spent $\frac{3}{8}$ of it on 4 identical pens and 6 identical toys. Each pen costs thrice as much as a toy. Then he spent $\frac{4}{5}$ of the remaining money on toys. We need to find: (a) The total number of toys he bought. (b) The total amount he paid for all pens and toys if each toy costs 12. 2. **Formulas and rules:** - Let the cost of one toy be $t$ and one pen be $p = 3t$. - Total money = $M$. - Money spent on pens and toys initially = $\frac{3}{8}M$. - Remaining money after initial spending = $M - \frac{3}{8}M = \frac{5}{8}M$. - Money spent on toys later = $\frac{4}{5} \times \frac{5}{8}M = \frac{1}{2}M$. 3. **Step-by-step solution:** **Step 1:** Express initial spending on pens and toys. $$4p + 6t = \frac{3}{8}M$$ Substitute $p = 3t$: $$4(3t) + 6t = 12t + 6t = 18t = \frac{3}{8}M$$ **Step 2:** Solve for $t$ in terms of $M$: $$18t = \frac{3}{8}M \implies t = \frac{3}{8}M \times \frac{1}{18} = \frac{3}{8 \times 18}M = \frac{1}{48}M$$ **Step 3:** Calculate the money left after initial spending: $$M - \frac{3}{8}M = \frac{5}{8}M$$ **Step 4:** Money spent on toys later: $$\frac{4}{5} \times \frac{5}{8}M = \frac{4}{8}M = \frac{1}{2}M$$ **Step 5:** Number of toys bought later: Let number of toys bought later be $x$. Cost per toy is $t = \frac{1}{48}M$. So, $$x \times t = \frac{1}{2}M \implies x = \frac{\frac{1}{2}M}{t} = \frac{\frac{1}{2}M}{\frac{1}{48}M} = \frac{1}{2} \times 48 = 24$$ **Step 6:** Total number of toys bought: Initially 6 toys + 24 toys later = 30 toys. **Step 7:** If each toy costs 12, find $M$: From step 2, $$t = \frac{1}{48}M = 12 \implies M = 12 \times 48 = 576$$ **Step 8:** Calculate total cost of pens and toys: - Cost of pens: $$4p = 4 \times 3t = 12t = 12 \times 12 = 144$$ - Cost of toys: Initially 6 toys + 24 toys = 30 toys $$30 \times 12 = 360$$ - Total cost: $$144 + 360 = 504$$ **Final answers:** (a) Total toys bought = 30 (b) Total amount paid for pens and toys = 504