1. Problem 15: How many ways can you make change for 25¢ using dimes, nickels, and pennies? 2. Define variables: - Let $d$ = number of dimes (10¢ each) - Let $n$ = number of nickels (5¢ each) - Let $p$ = number of pennies (1¢ each) 3. The equation for total cents is: $$10d + 5n + p = 25$$ 4. Since pennies can fill any leftover amount, for each combination of $d$ and $n$, $p$ is determined by: $$p = 25 - 10d - 5n$$ 5. Constraints: - $d, n, p \geq 0$ - $p$ must be an integer and non-negative 6. Find all integer pairs $(d,n)$ such that $p \geq 0$: - $d$ can be 0, 1, or 2 (since $3 \times 10 = 30 > 25$) 7. For each $d$: - $d=0$: $5n \leq 25 \Rightarrow n=0,1,2,3,4,5$ - $d=1$: $10 + 5n \leq 25 \Rightarrow 5n \leq 15 \Rightarrow n=0,1,2,3$ - $d=2$: $20 + 5n \leq 25 \Rightarrow 5n \leq 5 \Rightarrow n=0,1$ 8. Count total combinations: - For $d=0$: 6 values of $n$ - For $d=1$: 4 values of $n$ - For $d=2$: 2 values of $n$ Total ways = $6 + 4 + 2 = 12$ --- 9. Problem 16: How many 3-foot-by-3-foot squares of carpet are needed to cover a 12-foot-by-15-foot room? 10. Calculate the area of the room: $$12 \times 15 = 180 \text{ square feet}$$ 11. Calculate the area of one carpet square: $$3 \times 3 = 9 \text{ square feet}$$ 12. Number of carpet squares needed: $$\frac{180}{9} = 20$$ --- Final answers: - Number of ways to make 25¢: **12** - Number of carpet squares needed: **20**