# 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?

can you show me how you got your answer please?

(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)

## Correct answers are:

A.) x= 90,000 & y= 60,000

B.) x= 130,000 & y= 20,000

## to think this was 14 years ago...

## And no one even answered until 8 year later!

## 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

## Damn, no one even showed how they got their answer. Which, I suppose, doesn't matter all that much anyway since the fastest reply to the question happened to be 8 years after the question had been asked. The questioner was probably already in a job by then lol.

## Yes, it's unfortunate that no one showed their work or provided an explanation for their answers. However, it's never too late to learn and understand the solution!

## 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.