Grade 12 Mathematics
Grade 12 ¡ Term 1Mathematics

Finance, Growth & Decay

We extend finance to include loan repayment schedules, sinking funds, and the full analysis of present and future value of annuities in real-world contexts.

Week 5

3.1 Loan Schedules & Sinking Funds

  • Calculate outstanding balance on a loan at any point
  • Calculate sinking fund payments
  • Analyse and compare investment and debt scenarios
  • Apply time value of money to decision-making
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Real-World Connection

A sinking fund is a savings strategy used by municipalities and companies to set aside money regularly to pay for a large future expense — like replacing equipment or paying off a bond — so the full cost doesn't hit the budget all at once.

Outstanding Loan Balance

Pbalance=P(1+i)k−x⋅(1+i)k−1iP_{\text{balance}} = P(1+i)^k - x\cdot\frac{(1+i)^k-1}{i}

$P$ = original loan, $i$ = interest per period, $k$ = payments made, $x$ = payment per period

Sinking Fund

F=x⋅(1+i)n−1iF = x\cdot\frac{(1+i)^n-1}{i}

$F$ = future value needed; solve for $x$ to find required regular deposit

ℹī¸ Note

The outstanding balance formula is derived by computing the future value of the loan minus the future value of all payments made. It tells you exactly what you still owe after kk payments.

Worked Examples

Worked Example

Outstanding loan balance

Problem

R200 000 loan at 12% p.a. (monthly). Monthly payment R2 500. Find balance after 3 years (36 payments).

Worked Example

Sinking fund calculation

Problem

A machine worth R500 000 needs to be replaced in 6 years. Money earns 8% p.a. compounded monthly. Find the monthly deposit.
Activity — 8 Questions

CAPS Cognitive Level Distribution

L1 ¡ Knowledge2 Q
L2 ¡ Routine Procedures2 Q
L3 ¡ Complex Procedures2 Q
L4 ¡ Problem Solving2 Q
1
L1 ¡ Knowledge2 marks
State the formula for the future value of a sinking fund.
2
L1 ¡ Knowledge1 mark
A bond of R100 000 is taken at 9% p.a. compounded monthly. Find the monthly interest rate.
3
L2 ¡ Routine Procedures4 marks
Monthly payment R3 000 on a R250 000 loan at 10% p.a. (monthly). Find outstanding balance after 24 payments.
4
L2 ¡ Routine Procedures3 marks
How much must be invested quarterly for 4 years at 8% p.a. (quarterly) to accumulate R80 000?
5
L3 ¡ Complex Procedures5 marks
R120 000 loan at 14% p.a. compounded monthly, repaid over 5 years. Find: (a) the monthly payment; (b) total interest paid.
6
L3 ¡ Complex Procedures5 marks
After how many monthly payments of R2 000 on a R100 000 loan at 12% p.a. (monthly) is the loan fully repaid?
7
L4 ¡ Problem Solving6 marks
A company buys equipment for R1 000 000. It depreciates to R200 000 in 10 years (reducing balance). The replacement cost in 10 years is R1 500 000. Calculate the monthly sinking fund deposit at 8% p.a. (monthly) needed for the shortfall.
8
L4 ¡ Problem Solving5 marks
An investor makes the first of 12 annual payments of R5 000 NOW (annuity due). Find the present value at 10% p.a.

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Functions & Inverses

Finance, Growth & Decay Grade 12 Maths CAPS Notes & Examples | MathSciBuddy