1. **Identify the variables for problem 12:** - Let $x$ be the amount invested at 5.5% interest. - Let $y$ be the amount invested at 3.75% interest. 2. **List the other information:** - Total investment: $x + y = 20000000$ - Total interest earned after one year: $0.055x + 0.0375y = 1008125$ 3. **Write two equations:** $$\begin{cases} x + y = 20000000 \\ 0.055x + 0.0375y = 1008125 \end{cases}$$ 4. **Work problem:** - From the first equation, express $y$ in terms of $x$: $$y = 20000000 - x$$ - Substitute into the second equation: $$0.055x + 0.0375(20000000 - x) = 1008125$$ - Simplify: $$0.055x + 750000 - 0.0375x = 1008125$$ $$0.0175x + 750000 = 1008125$$ - Subtract 750000 from both sides: $$0.0175x = 258125$$ - Solve for $x$: $$x = \frac{258125}{0.0175} = 14778571.43$$ - Find $y$: $$y = 20000000 - 14778571.43 = 5221428.57$$ **Answer:** - Invested approximately $14778571.43$ at 5.5% interest. - Invested approximately $5221428.57$ at 3.75% interest. --- 1. **Identify the variables for problem 14:** - Let $c$ be the number of cookies sold. - Let $p$ be the number of cupcakes sold. 2. **List the other information:** - Total items sold: $c + p = 180$ - Total money collected: $0.25c + 0.50p = 66$ 3. **Write two equations:** $$\begin{cases} c + p = 180 \\ 0.25c + 0.50p = 66 \end{cases}$$ 4. **Work problem:** - From the first equation, express $c$ in terms of $p$: $$c = 180 - p$$ - Substitute into the second equation: $$0.25(180 - p) + 0.50p = 66$$ - Simplify: $$45 - 0.25p + 0.50p = 66$$ $$45 + 0.25p = 66$$ - Subtract 45 from both sides: $$0.25p = 21$$ - Solve for $p$: $$p = \frac{21}{0.25} = 84$$ - Find $c$: $$c = 180 - 84 = 96$$ **Answer:** - Sold 96 cookies. - Sold 84 cupcakes.