Finance 1 · Percentages & compound interest

Module 7 of 47Lesson 1 of 8 · Finance 1 · Percentages & compound interest

Financial mathematics: interest, depreciation, loans & annuities · Essential understanding 1

Percentages & compound interest

IB AI SL 1.4with Prof. Kitso

💻 Using the TI-Nspire CX Student Software on a laptop

The screens in these notes are drawn exactly as the Student Software shows them. The keypad on the left of the software works like the handheld — click its keys with the mouse.
menu: click menu on the on-screen keypad (or Documents Toolbox ▸ 🔧). The menu numbers and letters are the same as in these notes.
enter = Enter · tab = Tab (next box) · esc = Esc · del = Backspace.
Powers: type ^ (Shift+6), the power, then press → to come back down before typing more. x² = ^2 then →.
Fraction: type / · √: type sqrt( · π: type pi · eˣ: type exp( · store: type :=. For EE (×10ⁿ) and i, click them on the keypad.
New page: Ctrl+I (or the Insert menu) · change page: Ctrl+← / Ctrl+→ · graph entry line: Ctrl+G · var: Ctrl+L.
Radian / Degree: click Settings in the status bar at the bottom of the window. The mode (RAD or DEG) shows in the top bar.
IB exam mode (Press-to-Test): everything in these notes works on the non-CAS TI-Nspire with the IB exam settings. In exam mode there are no dot-product, cross-product, unit-vector or eigenvalue keys and no Diff Eq graphs — the notes show the allowed way (type the components, use polyRoots on the characteristic equation, do Euler steps on a Calculator page).
If you type nDeriv(, the software shows it as centralDiff( — it is the same command.

The big idea: With compound interest your interest also earns interest, so money grows faster and faster. Each year you multiply by the same number:

FV = PV \times \left(1 + \dfrac{r}{100}\right)^{n}

By the end of this lesson you can:

turn a percentage increase or decrease into a multiplier
explain the difference between simple and compound interest
use $FV = PV\left(1 + \dfrac{r}{100}\right)^{n}$ for interest compounded yearly
use the TI-Nspire Finance Solver to find a future value
find the interest earned and the percentage growth of an investment

🧑🏾 Imagine this…

You put P1 000 in a savings account that pays 10% a year.

Simple interest: the bank pays P100 every year, always on the original P1 000. After 3 years: P1 300.
Compound interest: year 1 gives P100, so you now have P1 100. Year 2 gives 10% of P1 100 = P110. Year 3 gives 10% of P1 210 = P121. After 3 years: P1 331.

The extra P31 is “interest on interest”. Over many years that small extra becomes big.

Principal / present value (PV)

the money you start with (the amount you invest or borrow).

Future value (FV)

how much the money is worth at the end.

Interest rate r%

the percentage added each period. In this lesson the period is one year.

Simple interest

interest is paid on the ORIGINAL amount only, so you get the same amount every year.

Compound interest

interest is paid on the amount you have NOW (original + interest already added).

Multiplier

the number you multiply by to apply a percentage change in one step.

★ The rules

Percentage increase of $r\%$ : multiply by $1 + \dfrac{r}{100}$ . Percentage decrease of $r\%$ : multiply by $1 - \dfrac{r}{100}$ .
Example: +5% → $\times 1.05$ ; −20% → $\times 0.8$ .
Compound interest, yearly: $FV = PV\left(1 + \dfrac{r}{100}\right)^{n}$ where $n$ = number of years. (in the IB formula booklet)
Simple interest (not needed in the exam, but good to compare): $FV = PV\left(1 + \dfrac{r \times n}{100}\right)$ .
Interest earned = FV − PV. Percentage growth $= \dfrac{\text{interest earned}}{PV} \times 100$ .
Compound growth is a geometric sequence: the common ratio is the multiplier $1 + \dfrac{r}{100}$ .

⚠️ Watch out

Do not multiply by r. For 5% you multiply by 1.05, not by 5 and not by 0.05.
The power is the number of years, not the rate. 1.05⁸ means “multiply by 1.05 eight times”.
Do not round the multiplier or any middle step. Keep the full calculator number and round only the final answer (money: 2 decimal places, or as the question says).

💡 Command terms

Calculate / find: a number with units. Write down: no working needed. Show that: your working must prove the printed answer. For compound interest, always show the line FV = PV(1 + $\dfrac{r}{100}$ )ⁿ with your numbers in it, even if you then use the GDC.

🧪 Try it yourselfTry the Finance Solver

These are the numbers from Example 3: P12 000 for 7 years at 4.5% a year. Press Solve next to FV. Then change N or I% and watch the future value change. (PpY and CpY stay at 1 for yearly interest.)

Worked examples

Example 1.1Percentage multipliers

(a) Write the multiplier for (i) an increase of 8%, (ii) an increase of 2.5%, (iii) a decrease of 15%, (iv) a decrease of 0.5%.
(b) A phone costs P3 200. Its price goes up by 8%. Find the new price.
(c) A jacket costing P640 is reduced by 12%. Find the sale price.

🧠 Think it through

Increase: add the percentage (as a decimal) to 1. Decrease: take it away from 1.
(i) $1 + 0.08$ , (ii) $1 + 0.025$ , (iii) $1 - 0.15$ , (iv) $1 - 0.005$ .
Then multiply the original amount by the multiplier.

✍️ By hand

(b)

3200 \times 1.08 = 3456

(c)

640 \times 0.88 = 563.20

🔢 On your TI-Nspire CX II

On a Calculator page type each product and press enter ↵💻 Enter.
Type 3200*1.08 enter ↵💻 Enter, then 640*0.88 enter ↵💻 Enter.

✅ Answer (a) (i) 1.08 (ii) 1.025 (iii) 0.85 (iv) 0.995. (b) P3 456. (c) P563.20.

Example 1.2Simple or compound? Year by year

Lesedi invests P5 000 for 3 years at 6% a year. Find the value at the end of 3 years if the interest is (a) simple, (b) compound.

🧠 Think it through

Simple: 6% of P5 000 = P300 every year, so 3 years gives P900 of interest.
Compound: multiply by 1.06 each year. Do it one year at a time first, so you can SEE what is happening.

✍️ By hand
(a) 
(b) Year 1: 
Year 2: 
Year 3: 
Same thing in one step: 

🔢 On your TI-Nspire CX II

Type 5000*1.06 enter ↵💻 Enter. Then type ans*1.06 enter ↵💻 Enter twice more (ans means “the last answer”).
Check with one line: 5000*1.06^3 enter ↵💻 Enter.

✅ Answer (a) P5 900. (b) P5 955.08. Compound gives P55.08 more interest than simple.

The first year is the same for both. The difference grows every year after that.

Example 1.3The compound interest formula and the Finance Solver

Thato invests P12 000 in an account that pays 4.5% interest per year, compounded yearly. Find the value of the investment after 7 years.

🧠 Think it through

Write the formula: $FV = PV\left(1 + \dfrac{r}{100}\right)^{n}$ with PV = 12 000, r = 4.5, n = 7.
The multiplier is 1.045 and we apply it 7 times.
On the Finance Solver: N = 7 (years), I% = 4.5, PV = −12 000 (money you pay in is negative), Pmt = 0 (no regular payments), PpY = CpY = 1 (yearly).

✍️ By hand

FV = 12000 \times \left(1 + \dfrac{4.5}{100}\right)^{7} = 12000 \times 1.045^{7} = 16\,330.34

🔢 On your TI-Nspire CX II

On a Calculator page, choose menu💻 click menu on keypad▸8: Finance▸1: Finance Solver.
Fill in N, I(%), PV, Pmt, PpY and CpY. Move between boxes with tab💻 Tab. Leave FV empty.
Everything filled in except FV
Move the cursor to the FV box and press enter ↵💻 Enter. The solver fills it in.
FV = 16330.3 → P16 330.34

✅ Answer After 7 years the investment is worth P16 330.34.

Why is PV negative? The solver thinks like a bank account: money that LEAVES your hand (you invest it) is negative, money you RECEIVE back is positive. FV comes out positive because you get it back.

Example 1.4Interest earned and percentage growth

Kabo invests P8 000 at 3.2% per year compounded yearly for 15 years.

(a) Find the value of the investment at the end.
(b) Find the total interest earned.
(c) Express the interest as a percentage of the amount Kabo invested.

🧠 Think it through

Part (a) is the same method as before: N = 15, I% = 3.2, PV = −8 000, PpY = CpY = 1.
Interest earned = FV − PV (final amount minus the amount you put in).
Percentage growth $= \dfrac{\text{interest}}{PV} \times 100$ .

✍️ By hand

(a)

8000 \times 1.032^{15} = 12\,831.74

(b)

12\,831.74 - 8000 = 4\,831.74

(c)

\dfrac{4\,831.74}{8000} \times 100 = 60.4\%

🔢 On your TI-Nspire CX II

Use menu💻 click menu on keypad▸8: Finance▸1: Finance Solver with N = 15, I(%) = 3.2, PV = −8000, Pmt = 0, PpY = 1, CpY = 1. Solve FV.
Then on the Calculator page: 12831.74-8000 and 4831.74/8000*100.

✅ Answer (a) P12 831.74. (b) P4 831.74. (c) 60.4% growth in total.

The total growth (60.4%) is much more than 15 × 3.2% = 48%. That extra is the “interest on interest”.

Example 1.5When does compound beat simple?

Account A pays simple interest of 6% per year on P10 000. Account B pays 4.5% per year compound interest on P10 000.

(a) Find the value of each account after 10 years and after 20 years.
(b) Find the first whole year in which account B is worth more than account A.

🧠 Think it through

Account A: P10 000 + P600 for every year, so value = 10 000 + 600n.
Account B: value = 10 000 × 1.045ⁿ.
For (b), try values of n close together until B overtakes A. Use the answer line trick: type the formula once, then change n.

✍️ By hand

A after 10 years:

10000 + 600 \times 10 = 16 000

B after 10 years:

10000 \times 1.045^{10} = 15\,529.69

A after 20 years:

22 000

; B after 20 years:

24\,117.14

🔢 On your TI-Nspire CX II

Compare year 13 and year 14 on the Calculator page.
Year 13: A is ahead. Year 14: B is ahead.

✅ Answer (a) After 10 years: A = P16 000, B = P15 529.69. After 20 years: A = P22 000, B = P24 117.14. (b) Year 14 (B = P18 519.45 > A = P18 400).

Compound interest starts slower here (lower rate) but its growth keeps speeding up. A straight line (simple interest) is always overtaken by an exponential curve in the end.

Challenge questions

⭐ Challenge 1.6A savings account for a tripPaper 1 style

Mpho invests P15 000 at 5.2% per year, compounded yearly, for 9 years.

(a) Find the value of the investment after 9 years, to the nearest pula.
(b) Find the interest earned.

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

Formula: FV = 15 000 × 1.052⁹, or Finance Solver with N = 9, I% = 5.2, PV = −15 000, PpY = CpY = 1.
Interest = FV − 15 000. Round only at the end.

🔢 On your TI-Nspire CX II

Finance Solver: solve for FV.
Then subtract the principal.

✅ Answer (a) P23 672 (nearest pula). (b) P8 672 (using the unrounded value: P8 671.89).

If the question says “nearest pula” for the final value, using the rounded value for the interest is fine. Keep full accuracy in all middle steps.

⭐ Challenge 1.7Simple versus compound over 12 yearsPaper 2 style

Onalenna has P40 000. Bank X offers 3.9% per year compound interest. Bank Y offers 5% per year simple interest. She plans to leave the money for 12 years.

(a) Find the value of her money in Bank X after 12 years.
(b) Find the interest earned in Bank X as a percentage of P40 000.
(c) Find the value of her money in Bank Y after 12 years.
(d) Which bank is better for 12 years? Her friend says “compound interest is always better than simple interest”. Comment.

✏️ Try it on paper and on your calculator first. Then check.

Show the full solution

🧠 Think it through

Bank X: 40 000 × 1.039¹². Bank Y: 40 000 + 12 × (5% of 40 000) = 40 000 + 12 × 2 000.
Compare the two final values.
Think about what the friend forgot: the two banks have different RATES, and the time is only 12 years.

🔢 On your TI-Nspire CX II

Bank X on the Finance Solver (N = 12, I% = 3.9, PV = −40 000, PpY = CpY = 1).
The percentage and Bank Y:

✅ Answer (a) P63 306.25. (b) 58.3%. (c) P64 000. (d) Bank Y is better after 12 years (P64 000 > P63 306.25). The friend is wrong in general: compound interest beats simple interest only when the rates are the same (or close enough) or the time is long enough. Here Y has a higher rate.

In exams, when you are asked to “comment”, give a number from your working and a reason.

Next: Lesson 2 · Compounding more often: half-yearly, quarterly, monthly

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL Finance 1 · Percentages & compound interest

IB DP Mathematics: Applications and Interpretation (AI) — SL and HL

Percentages & compound interest

Worked examples

Challenge questions