A total of 32,000 is invested in two bonds that pay 6.25% and 5.75% simple interest. The investor wants an annual interest income of 19000 from the investments. What amount should be invested in the 6.25% bond?
The correct answer and explanation is:
To solve this problem, we can use a system of equations involving simple interest. Let’s define the variables first:
Let:
- xx = amount invested in the 6.25% bond
- 32000−x32000 – x = amount invested in the 5.75% bond
The formula for simple interest is: Interest=Principal×Rate\text{Interest} = \text{Principal} \times \text{Rate}
Step 1: Set up the interest equations
The total interest from both investments should equal $19,000. So: 0.0625x+0.0575(32000−x)=190000.0625x + 0.0575(32000 – x) = 19000
Step 2: Solve the equation
Distribute the 0.0575: 0.0625x+(0.0575×32000)−0.0575x=190000.0625x + (0.0575 \times 32000) – 0.0575x = 19000 0.0625x+1840−0.0575x=190000.0625x + 1840 – 0.0575x = 19000 (0.0625−0.0575)x+1840=19000(0.0625 – 0.0575)x + 1840 = 19000 0.005x+1840=190000.005x + 1840 = 19000
Subtract 1840 from both sides: 0.005x=171600.005x = 17160
Divide both sides by 0.005: x=171600.005=3,432,000x = \frac{17160}{0.005} = 3,432,000
Final Answer:
$3,432,000 should be invested in the 6.25% bond.
Explanation (300 words):
This problem is based on simple interest, which is calculated by multiplying the principal (amount of money invested), the annual interest rate (as a decimal), and time in years. In this case, the time is 1 year, so we only consider the product of the principal and rate.
We’re given that a total of $32,000 is split between two bonds—one paying 6.25% and the other 5.75%. The investor desires a total annual income of $19,000. We define the unknown amount invested in the higher-interest bond (6.25%) as xx, and the remaining amount (in the 5.75% bond) as 32000−x32000 – x.
The key is to write expressions for the interest generated by each bond using the simple interest formula and then add them to match the desired total interest. After forming the equation, solving it requires basic algebra—distribute, combine like terms, isolate the variable, and then solve.
Notably, the resulting required investment of $3,432,000 is far more than the original total of $32,000, which signals a problem: the investor is trying to earn $19,000 from just $32,000 in one year, which is an unrealistic goal at interest rates of around 6%.
Hence, there is no possible way to divide $32,000 between those bonds and earn $19,000 in interest per year. The equation gives a mathematically correct investment, but it doesn’t align with the reality of the available capital.
So:
✅ Correct math
❌ Not feasible with only $32,000 capital.