Timelines and Changing Interest Rates | IEB Grade 11 Maths

Real investments rarely stay the same for years: rates change, and people add or take out money. A timeline keeps track of every change, and you grow each amount forward stage by stage. Annuity formulae are Grade 12 work, so here each payment is handled separately.

Drawing timelines

Draw a horizontal line marked in years: $T_{0}$ (now), $T_{1}$ , $T_{2}$ and so on.

Write each deposit above the line and each withdrawal below it, at the time it happens.
Mark each interest rate and its compounding period between the times it applies.

A timeline takes 30 seconds and prevents most errors in multi-step questions.

Changing interest rates

Grow the amount to the time the rate changes, then continue with the new rate.

Example. R12 000 at 6% p.a. compounded annually for 2 years, then 8% p.a. compounded annually for 3 years:

$A = 12000 (1, 06)^{2} (1, 08)^{3} \approx R 16984, 95$

Change both $i$ and $n$ when the compounding period changes.

Additional deposits and withdrawals

Method 1: grow each amount separately to the end date, then add deposits and subtract withdrawals.

Example. R20 000 at 9% p.a. compounded monthly, with R5 000 added after 1 year. Value after 3 years:

$20000 (1, 0075)^{36} + 5000 (1, 0075)^{24} \approx R 32154, 98$

Method 2: stage by stage. Grow to the date of the transaction, add or subtract, then continue.

Example. R30 000 at 7,5% p.a. compounded quarterly; R8 000 withdrawn after 2 years. Value after 4 years:

$[30000 (1, 01875)^{8} - 8000] (1, 01875)^{8} \approx R 31101, 66$

Finding a deposit. Write each deposit's value at the end date in terms of $x$ , set the total equal to the target and solve.

Key terms

Timeline: A diagram showing amounts, dates and rates in order.
Future value: What an amount will be worth at a later date.
Withdrawal: Money taken out of an investment.

Common mistakes

Growing a later deposit for the full investment period.
Forgetting to change $i$ and $n$ when the compounding period changes.
Subtracting a withdrawal from the final amount without growing it.
Rounding between stages.

Practice questions

Reveal steps one at a time, then tick the marks you earned.

Question 1

4 marks total

routine

R12 000 is invested at 6% p.a. compounded annually for 2 years. The rate then changes to 8% p.a. compounded annually for a further 3 years. Calculate the final value.

Worked solution

Hidden - try the question first

Solution

After 2 years: $12000 (1, 06)^{2}$
Next 3 years: $\times (1, 08)^{3}$
$A = 12000 (1, 06)^{2} (1, 08)^{3}$
$\approx$ R16 984,95

Question 2

5 marks total

routine

R20 000 is invested at 9% p.a. compounded monthly. After 1 year, a further R5 000 is deposited. Calculate the value of the investment 3 years after the first deposit.

Worked solution

Hidden - try the question first

Solution

$i = 0, 0075$ per month
First deposit grows for 36 months: $20000 (1, 0075)^{36}$
Second deposit grows for 24 months: $5000 (1, 0075)^{24}$
Total $= 20000 (1, 0075)^{36} + 5000 (1, 0075)^{24}$
$\approx$ R32 154,98

Question 3

6 marks total

complex

R30 000 is invested at 7,5% p.a. compounded quarterly. After 2 years, R8 000 is withdrawn.

(a) Draw a timeline. (2)

(b) Calculate the value of the investment after 4 years. (4)

Worked solution

Hidden - try the question first

Solution

(a) R30 000 at $12000 (1, 06)^{2}$ ; R8 000 out at $T_{2}$ ; end at $T_{4}$
7,5% p.a. compounded quarterly marked throughout
(b) At $T_{2}$ : $30000 (1, 01875)^{8}$
$- 8000$
$\times (1, 01875)^{8}$
$\approx$ R31 101,66

Question 4

9 marks total

problem-solving

Sipho wants R50 000 in 5 years. He deposits $x$ now and $2 x$ after 2 years. Interest is 8% p.a. compounded monthly for the first 3 years and 9% p.a. compounded quarterly after that.

(a) Draw a timeline. (2)

(b) Calculate $x$ . (7)

Worked solution

Hidden - try the question first

Solution

(a) $12000 (1, 06)^{2}$ at $T_{0}$ ; $2 x$ at $T_{2}$ ; target R50 000 at $T_{5}$
8% monthly from $T_{0}$ to $T_{3}$ ; 9% quarterly from $T_{3}$ to $T_{5}$
(b) First deposit: $x (1 + \frac{0 , 08}{12})^{36} (1, 0225)^{8}$
Second deposit: $2 x (1 + \frac{0 , 08}{12})^{12} (1, 0225)^{8}$
Sum $= 50000$
$x [(1, 2702 \dots) (1, 1948 \dots) + 2 (1, 0830 \dots) (1, 1948 \dots)] = 4, 10572 \dots x = 50000$ (keep the factors unrounded in the calculator)
$x \approx$ R12 178,13

FAQs

Why draw a timeline?

It shows exactly how long each amount grows and at which rate, which prevents most mistakes.

How do I handle a withdrawal?

Either subtract it at the time it happens and keep growing the balance, or grow it separately to the end and subtract.

Are annuities part of Grade 11?

No. Annuity formulae are Grade 12 work. In Grade 11 each deposit is grown separately.

Timelines, changing rates and multiple transactions

Drawing timelines

Changing interest rates

Additional deposits and withdrawals

Key terms

Common mistakes

Practice questions

FAQs

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