Minimum Loan Repayment and Number of Months of Loan Repayment

Minimum Loan Repayment and Number of Months of Loan Repayment

Jane borrows \( \$100 \ 000 \), which is to be repaid in equal monthly instalments. The interest rate is \( 6 \% \) per annum reducible, compounded monthly.

It can be shown that the amount, \( \$b_n \), owing after the \( n \) th repayment is given by the formula:

$$ b_n = 100 \ 000 r^n-x(1+r+r^2+ \cdots + r^{n-1}) $$

where \( r = 1.005 \) and \( \$ x \) is the monthly repayment.

Part 1

The minimum monthly repayment is the amount required to repay the loan in \( 200 \) instalments. Find the minimum monthly repayment.

\( \displaystyle \begin{align} 0 &= 100 \ 000 \times 1.005^{200}-x (1.005^0+1.005^1+1.005^2+ \cdots + 1.005^{199}) \\ 0 &= 100 \ 000 \times 1.005^{200}-x \times \frac{1.005^{200}-1}{1.005-1} \\ x \times \frac{1.005^{200}-1}{0.005} &= 100 \ 000 \times 1.005^{200} \\ x &= 100 \ 000 \times 1.005^{200} \times \frac{0.005}{1.005^{200}-1} \\ &= 792.1384 \cdots \require{AMSsymbols} \\ \therefore x &= \$ 792.14 \end{align} \)

Part 2

Jane wants to make repayments of \( \$ 1500 \) each month from the beginning of the loan. How many months will it take for her to repay the loan?

\( \displaystyle \begin{align} 100 \ 000 \times 1.005^n-1500 \times \frac{1.005^n-1}{1.005-1} &= 0 \\100 \ 000 \times 1.005^n-300 \ 000 \times (1.005^n-1) &= 0 \\100 \ 000 \times 1.005^n-300 \ 000 \times 1.005^n + 300 \ 000 &= 0 \\ -200 \ 000 \times 1.005^n &= -300 \ 000 \\ 1.005^n &= 1.5 \\ n &= \log_{1.005} 1.5 \\ &= 81.29 \cdots \\ \require{AMSsymbols} \therefore n &= 82 \text{ months} \end{align} \)

Unlock your full learning potential—download our expertly crafted slide files for free and transform your self-study sessions!

Discover more enlightening videos by visiting our YouTube channel!

 

Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume




Related Articles

Responses

Your email address will not be published. Required fields are marked *