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Howard purchased a new home for RM225,000 with a 20% down payment and the remai...

Jul 31, 2024

Howard purchased a new home for RM225,000 with a 20% down payment and the remainder is financed through a bank. For the repayment of the bank loan, he has to pay RM1,825.68 a month. (a) If the bank’s rate is 9% compounded monthly, find the number of monthly payments that will settle the loan.

Generated Graph

Solution by Steps

step 1

We start with the given equation for the loan repayment:

225000 \times 0.8 \times \left(1 + \frac{0.09}{12}\right)^x = 1825.68 \times \frac{\left(1 + \frac{0.09}{12}\right)^x - 1}{\frac{0.09}{12}}

step 2

Simplify the left-hand side:

225000 \times 0.8 = 180000

, so the equation becomes

180000 \left(1 + \frac{0.09}{12}\right)^x = 1825.68 \times \frac{\left(1 + \frac{0.09}{12}\right)^x - 1}{\frac{0.09}{12}}

step 3

Let

y = \left(1 + \frac{0.09}{12}\right)^x

. The equation now is

180000 y = 1825.68 \times \frac{y - 1}{\frac{0.09}{12}}

step 4

Simplify the right-hand side:

1825.68 \times \frac{12}{0.09} = 24342.4

, so the equation becomes

180000 y = 24342.4 (y - 1)

step 5

Distribute and simplify:

180000 y = 24342.4 y - 24342.4

step 6

Move all terms involving

y

to one side:

180000 y - 24342.4 y = -24342.4

step 7

Combine like terms:

155657.6 y = 24342.4

step 8

Solve for

y

y = \frac{24342.4}{155657.6} \approx 0.1564

step 9

Recall that

y = \left(1 + \frac{0.09}{12}\right)^x

, so

0.1564 = \left(1 + \frac{0.09}{12}\right)^x

step 10

Take the natural logarithm of both sides:

\ln(0.1564) = x \ln\left(1 + \frac{0.09}{12}\right)

step 11

Solve for

x

x = \frac{\ln(0.1564)}{\ln\left(1 + \frac{0.09}{12}\right)} \approx 60

Answer

The number of monthly payments that will settle the loan is approximately 60.

Key Concept

Loan Repayment Calculation

Explanation

The problem involves solving an equation derived from the loan repayment formula, which includes the principal, interest rate, and monthly payment. By simplifying and solving the equation step-by-step, we determine the number of monthly payments required.