Question: April recently retired, and has $480,000 in savings. She wants to invest as much as possible in a safe CD account paying 1.9% interest, and as little as possible in a riskier bond paying 5.1% interest. She needs to earn $14,240 a year in interest to cover expenses above SSI.
a) How much should she invest in each account? Enter your answers below.
1. **State the problem:** April has $480,000 to invest in two accounts: a safe CD at 1.9% interest and a riskier bond at 5.1% interest. She wants to maximize the amount in the CD and minimize the amount in the bond while earning $14,240 in total interest annually.
2. **Define variables:** Let $x$ be the amount invested in the CD, and $y$ be the amount invested in the bond.
3. **Write the equations:**
- Total investment: $$x + y = 480000$$
- Total interest needed: $$0.019x + 0.051y = 14240$$
4. **Express $y$ in terms of $x$ from the first equation:**
$$y = 480000 - x$$
5. **Substitute $y$ into the interest equation:**
$$0.019x + 0.051(480000 - x) = 14240$$
6. **Distribute and simplify:**
$$0.019x + 24480 - 0.051x = 14240$$
$$-0.032x + 24480 = 14240$$
7. **Isolate $x$:**
$$-0.032x = 14240 - 24480$$
$$-0.032x = -10240$$
8. **Divide both sides by $-0.032$ (showing cancellation):**
$$x = \frac{-10240}{-0.032} = \frac{\cancel{-10240}}{\cancel{-0.032}} = 320000$$
9. **Find $y$ using $y = 480000 - x$:**
$$y = 480000 - 320000 = 160000$$
10. **Interpretation:** April should invest $320000 in the safe CD and $160000 in the riskier bond to meet her interest goal while maximizing the safe investment.
**Final answer:**
- Amount in CD: $320000$
- Amount in bond: $160000$