# A retired couple needs \$12,000 per year in income to supplement their Social Security. They have \$150,000 to invest to obtain this income. They have decided on two investment options: AA bonds yielding 10% per annum and a bank CD yielding 5%.

(a) How much should be invested in each to realize exactly \$12,000 in interest income?
(b) If, after two years, the couple requires \$14,000 per year in income, how should they reallocate their investment to achieve the new amount?

(a) Let x be the amount invested in AA binds and y be the amount invested in CDs. You have two equations in two unknowns. They are
x + y = 150,000 (total principal available)
0.1 x + 0.05 y = 12,000 (desired income)
Now do the algebra.
x + 0.5 y = 120,000 (second equation x2)
0.5 y = 30,000. (subracting last equation from first)
y = 60,000. (doubling last equation)
x = 60,000.

(b) Use a similar procedure here, but the equations are:
x + y = 150,000 (total principal available)
0.1 x + 0.05 y = 14,000 (desired income)

A.) x= 90,000 & y= 60,000
B.) x= 130,000 & y= 20,000

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## sorry if I'm 14 years late but the answer is

A.) x= 90,000 & y= 60,000
B.) x= 130,000 & y= 20,000
and this was also made the year I was born

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## To solve this system of equations, we'll use the same approach as in part (a).

(a) Let x be the amount invested in AA bonds and y be the amount invested in CDs. The equations are:

x + y = 150,000 (total principal available)
0.1x + 0.05y = 12,000 (desired income)

To eliminate the decimals, we can multiply the second equation by 100:

10x + 5y = 120,000

Now we have the system of equations:

x + y = 150,000
10x + 5y = 120,000

To solve these equations, we can use the method of substitution or elimination. Let's use the substitution method.

From the first equation, we can express x in terms of y:

x = 150,000 - y

Substitute this value of x into the second equation:

10(150,000 - y) + 5y = 120,000
1,500,000 - 10y + 5y = 120,000
-5y = -1,380,000
y = 276,000

Now, substitute the value of y back into the first equation to find x:

x + 276,000 = 150,000
x = 150,000 - 276,000
x = -126,000

Since the amount invested cannot be negative, the solution (-126,000, 276,000) is not feasible.

Therefore, there is no feasible solution for part (a). This means that with the given options, there is no way to invest the \$150,000 to exactly obtain \$12,000 in interest income.

(b) Using a similar approach, we can solve the system of equations:

x + y = 150,000
10x + 5y = 14,000

Again, let's use the substitution method. From the first equation, we get:

x = 150,000 - y

Substitute this value of x into the second equation:

10(150,000 - y) + 5y = 14,000
1,500,000 - 10y + 5y = 14,000
-5y = -1,486,000
y = 297,200

Now substitute the value of y back into the first equation to find x:

x + 297,200 = 150,000
x = 150,000 - 297,200
x = -147,200

Again, the negative values are not feasible. So, there is no feasible solution for part (b) either. This means that with the given options, there is no way to invest the \$150,000 to exactly obtain \$14,000 in interest income.

In both cases, the retired couple will need to consider alternative investment options or adjust their income expectations to find a feasible solution.