Question

Pizza Palace has a small business loan for 30 months at 6% interest. The expression for the total loan amount to be paid is p (1+r)^t, where:
t is time in years,
r is interest rate as a decimal, and
p is the principal of the loan.

Find the principal of the loan, to the nearest dollar, when the total loan amount to be paid is $404,886 at 30 months.

A manager says, “If the interest rate was cut in half, the difference between the total loan amount and the principal would also be cut in half.”

The statement is not always true.

Provide a specific example to refute the manager’s statement.

Answer 1

Answer:

The principal of the loan is $350000

$26844 is not half 54886 so the statement is not true

Step-by-step explanation:

* Lets use the given rule to solve the question

- The total loan amount to be paid = p (1 + r)^t , where

# t is time in years,

# r is interest rate as a decimal

# p is the principal of the loan

- To find t divide the number of months by 12

∵ t = 30/12 = 2.5 years

∵ r = 6/100 = 0.06 ⇒ the interest rate in decimal

∵ The total loan amount to be paid = $404,886

∴ 404,886 = p (1 + 0.06)^2.5

∴ 404,886 = p (1.06)^2.5 ⇒ divide both sides by (1.06)^2.5

∴ p = 404,886 ÷ [(1.06)^2.5] ≅ $350,000

* The principal of the loan is $350,000

- To check the statement of the manager lets find the difference

  between the total loan amount and the principal

∵ The principal of the loan is $350,000

∵ the total loan amount to be paid is $404,886

∴ The difference = 404,886 - 350,000 = $54886

- Lets find the total loan amount to be paid when the interest rate

 was cut in half

∵ The total loan amount to be paid = p (1 + r)^t

∵ t = 30/12 = 2.5 years

∵ The half of 6% is 3%

∴ r = 3/100 = 0.03 ⇒ the interest rate in decimal

∵ p = $350,000

∴ The total loan amount to be paid = 350,000 (1 + 0.03)^2.5

∴ The total loan amount to be paid = 350,000 (1.03)^2.5

∴ The total loan amount to be paid = $376,844

- Lets find the difference between the total amount to be paid and

 the principal

∴ The difference = 376,844  - 350,000 = $26844

∵ $26844 is not half 54886

* The statement is not true