Question

Madison Finance has a total of $22 million earmarked for homeowner loans and auto loans, where x is homeowner loans in millions of dollars and y is auto loans in millions of dollars. On the average, homeowner loans have a 8% annual rate of return, whereas auto loans yield a 13% annual rate of return. Management has also stipulated that the total amount of homeowner loans should be greater than or equal to 4 times the total amount of automobile loans. Determine the total amount of loans of each type Madison should extend to each category to maximize its returns P in millions of dollars.Maxmize P=total loansdistributtion of loans

Answer 1

Answer:

Maximum returns = $1,980,000

x = Homeowner loans = $17,600,000

y = Auto loans = $4,400,000

Madison should extend $17,600,000 in home owners loans and $4,400,000 in auto loans in order to have a maximum return of $1,980,000

Step-by-step explanation:

From the question we are told that:

Madison Finance has a total of $22 million earmarked for homeowner loans and auto loans, where x is homeowner loans in millions of dollars and y is auto loans in millions of dollars.

On the average, homeowner loans have a 8% annual rate of return, whereas auto loans yield a 13% annual rate of return.

We have the equation:

Maximum Return = 8% × x + 13% × y

= 0.08x + 0.13y

We are told in the question that

Management has also stipulated that the total amount of homeowner loans should be greater than or equal to 4 times the total amount of automobile loans.

Hence,

x ≥ 4y

So:

x + y = $22,000,000

4y + y = $22,000,000

5y = $22,000,000

y = $22,000,000/5

y = 4,400,000

x = 4y

x = 4(4,400,000)

x = 17,600,000

Maximum Return = 0.08x + 0.13y

= 0.08 × 17,600,000 + 0.13 × 4,400,000

= $1,980,000

Therefore, Madison should extend $17,600,000 in home owners loans and $4,400,000 in auto loans in order to have a maximum return of $1,980,000