Betsy, a recent retiree, requires $6000 per year in extra income. She has $47,000 to invest and can invest in B-rated bonds paying 15% per year or in a Certificate of Deposit (CD) paying 6% per year. How much money should be invested in each to realize exactly $6000 in interest per year?
Solve this pair of equations, using algebra:
B + C = 47,000
0.15 B + 0.06 C = 6000
B is the Bond investment and C is the CD investment.
To determine how much money Betsy should invest in each option, we can set up a system of equations.
Let's denote the amount of money Betsy invests in B-rated bonds as "x" and the amount she invests in a Certificate of Deposit (CD) as "y".
According to the problem, Betsy needs to earn $6000 in interest per year.
The interest earned from investing in the bonds can be calculated as 0.15x (15% of the amount invested), while the interest earned from the CD can be calculated as 0.06y (6% of the amount invested).
Therefore, the first equation is:
0.15x + 0.06y = 6000
Additionally, Betsy has a total of $47,000 to invest, so the second equation is:
x + y = 47000
Now we have a system of equations:
0.15x + 0.06y = 6000
x + y = 47000
To solve this system of equations, we can use the substitution method or the elimination method.
Let's solve it using the elimination method. Multiply the second equation by 0.06 to make the coefficients of "y" in both equations the same:
0.06x + 0.06y = 2820
Now we can subtract this new equation from the first equation to eliminate "y":
(0.15x + 0.06y) - (0.06x + 0.06y) = 6000 - 2820
Simplifying the equation:
0.15x - 0.06x = 3180
Combining like terms:
0.09x = 3180
Dividing by 0.09:
x = 35,333.33
Now substitute the value of x back into the second equation to solve for y:
35,333.33 + y = 47,000
Subtract 35,333.33 from both sides:
y = 11,666.67
Therefore, Betsy should invest approximately $35,333.33 in B-rated bonds and $11,666.67 in a Certificate of Deposit (CD) to realize exactly $6000 in interest per year.