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)
to think this was 14 years ago...
Correct answers are:
A.) x= 90,000 & y= 60,000
B.) x= 130,000 & y= 20,000
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.
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.