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%.