Betsy, a recent retiree, requires $6,000 per year in extra income. She has $60,000 to invest and can invest in B-rated bonds paying 13% per year or in a certificate of deposit (CD) paying 3% per year. How much money should be invested in each to realize exactly $6,000 in interest per year?
The amount of money invested at 13%?
The amount of money invested at 3%?
To determine how much money should be invested at each rate, we can set up a system of equations.
Let's assume Betsy invests x amount of dollars in B-rated bonds at 13% interest per year, and y amount of dollars in a CD at 3% interest per year.
The interest earned from the bonds would be 0.13x, and the interest earned from the CD would be 0.03y.
According to the problem, Betsy requires $6,000 per year in extra income. So, we can set up the following equation:
0.13x + 0.03y = 6,000
We also know that Betsy has $60,000 to invest, which means x + y = 60,000.
We now have a system of equations:
0.13x + 0.03y = 6,000
x + y = 60,000
To solve this system, we can use the method of substitution:
1. Rearrange the second equation to express y in terms of x:
y = 60,000 - x
2. Substitute this expression for y in the first equation:
0.13x + 0.03(60,000 - x) = 6,000
3. Simplify and solve for x:
0.13x + 1,800 - 0.03x = 6,000
0.10x = 4,200
x = 42,000
Now, we can find the amount of money invested at 3% by substituting the value of x back into the second equation:
y = 60,000 - x
y = 60,000 - 42,000
y = 18,000
Therefore, Betsy should invest $42,000 at 13% and $18,000 at 3% to realize exactly $6,000 in interest per year.
To determine the amount of money to be invested at 13% and 3% respectively, we can set up a system of equations:
Let x be the amount invested at 13% and y be the amount invested at 3%.
The interest earned from the investment at 13% is given by 0.13x, and the interest earned from the investment at 3% is given by 0.03y.
According to the problem, the total interest earned is $6,000 per year, so we have the equation:
0.13x + 0.03y = 6,000. (equation 1)
We also know that the total amount invested is $60,000, so we have the equation:
x + y = 60,000. (equation 2)
To solve this system of equations, we can use substitution or elimination method. Let's use substitution method:
Solving equation 2 for x, we get:
x = 60,000 - y.
Substituting x in equation 1, we have:
0.13(60,000 - y) + 0.03y = 6,000.
Distributing and simplifying:
7,800 - 0.13y + 0.03y = 6,000.
Combining like terms:
0.03y - 0.13y = 6,000 - 7,800.
-0.10y = -1,800.
Dividing both sides by -0.10:
y = -1,800 / -0.10.
y = 18,000.
Now, substitute the value of y back into equation 2 to solve for x:
x + 18,000 = 60,000.
Subtracting 18,000 from both sides:
x = 60,000 - 18,000.
x = 42,000.
Therefore, Betsy should invest $42,000 at 13% and $18,000 at 3%.
Investments:
x dollars @ 13%,
y dollars @ 3%,
x + y = 60000,
0.13x + 0.03y = 6000,
Multiply the 1st Eq by -0.13:
-0.13x - 0.13y = -7800,
0.13x + 0.03y = 6000,
Add the 2 Eqs:
-0.1y = -1800,
y = 18000,
Substitute 18000 for y in the 1st Eq:
x + 18000 = 60000,
x = 60000 - 18000 = 42000.
She invested:
$42000 @ 13%,
$18000 @ 3%.