Compound Interest, Reverse Percentage and Compound Proportional Change

Lessons

Senior 3 · Mathematics

Compound Interest, Reverse Percentage and Compound Proportional Change

4 lessons

Lesson 3 of 4

The compound interest formula

25 min

From a table to a formula

In the last lesson, 500,000 Frw at 8% grew by multiplying by 1.08 once a year: $500\,000 \times 1.08 \times 1.08 \times 1.08$ . For 3 years a table is fine. For 20 years, or for interest added every month, you need a formula that jumps straight to the answer.

Definition

The compound interest formula. A principal $P$ invested at a rate $r$ per period (as a decimal), compounded for $n$ periods, grows to the amount

A = P (1 + r)^{n} .

The compound interest is $A - P$ . With the rate written as a percentage $R %$ , the same formula is $A = P\left(1 + \frac{R}{100}\right)^n$ .

Example: 500,000 Frw at 8% for 3 years: $A = 500\,000 \times 1.08^3 = 629\,856$ Frw, as in the table of Lesson 2.

Important

Why the formula works. Each period multiplies the balance by $(1 + r)$ :

P \;\xrightarrow{\times(1 + r)}\; P(1 + r) \;\xrightarrow{\times(1 + r)}\; P(1 + r)^2 \;\xrightarrow{\times(1 + r)}\; \cdots \;\xrightarrow{\times(1 + r)}\; P(1 + r)^n.

Interest added $k$ times a year. If the yearly rate is $r$ and interest is added $k$ times a year for $t$ years, each period's rate is $\frac{r}{k}$ and there are $k t$ periods:

A = P\left(1 + \frac{r}{k}\right)^{kt}.

For half-yearly $k = 2$ , quarterly $k = 4$ , monthly $k = 12$ .

Worked example 1: five years at a SACCO

A member saves 800,000 Frw at 9% per year compounded annually (illustrative rate). Find the amount and the interest after 5 years.

Solution $P = 800\,000$ , $r = 0.09$ , $n = 5$ :

A = 800\,000 \times 1.09^5 = 800\,000 \times 1.538\,62\ldots = 1\,230\,899\ \text{Frw (to the nearest franc)}.

Interest: $1\,230\,899 - 800\,000 = 430\,899$ Frw. Simple interest would have given only $5 \times 72\,000 = 360\,000$ Frw.

Worked example 2: interest added every month

1,200,000 Frw is invested for 2 years at 12% per year compounded monthly. Find the amount, and compare it with annual compounding.

Solution Monthly: $\frac{r}{k} = \frac{0.12}{12} = 0.01$ and $kt = 12 \times 2 = 24$ periods.

A = 1\,200\,000 \times 1.01^{24} = 1\,523\,682\ \text{Frw}.

Annually: $1\,200\,000 \times 1.12^2 = 1\,505\,280$ Frw. Monthly compounding earns 18,402 Frw more, because each month's interest starts earning interest sooner.

Worked example 3: how much to save now?

Parents want 1,000,000 Frw in 3 years' time for secondary school fees. How much must they deposit now at 7% per year compounded annually?

Solution This is a reverse problem: $A$ is known and $P$ is not.

P(1.07)^3 = 1\,000\,000 \quad\Longrightarrow\quad P = \frac{1\,000\,000}{1.07^3} = 816\,298\ \text{Frw (to the nearest franc)}.

Round up in a real plan: 816,298 Frw is the least that is enough.

Worked example 4: how long to double?

After how many whole years will 500,000 Frw at 10% compounded annually first be worth more than 1,000,000 Frw?

Solution You need ${1.1}^{n} > 2$ . Make a table of powers with a calculator:

$n$	${1.1}^{n}$	$500\,000 \times 1.1^n$ (Frw)
1	1.1000	550,000
2	1.2100	605,000
3	1.3310	665,500
4	1.4641	732,050
5	1.6105	805,255
6	1.7716	885,781
7	1.9487	974,359
8	2.1436	1,071,794

After 7 years it is 974,359 Frw, still short; after 8 years it is 1,071,794 Frw. The answer is 8 years.

Worked example 4 as a chart: each bar is 1.1 times the one before, so the bars grow by more each year. The amount first passes 1,000,000 Frw after 8 years.

Worked example 5: finding the rate

200,000 Frw grows to 242,000 Frw in 2 years with interest compounded annually. Find the rate.

Solution $200\,000(1 + r)^2 = 242\,000$ , so $(1 + r)^{2} = 1.21$ . Take the positive square root: $1 + r = 1.1$ , so $r = 0.1$ , a rate of 10% per year.

Warning

Use the rate per period, as a decimal. 9% is $r = 0.09$ , not 9: $800\,000 \times 10^5$ is nonsense. And 12% per year compounded monthly is $0.01$ per month for $24$ months, not $0.12$ for 24 periods.

Power first, then multiply. $800\,000 \times 1.09^5$ means ${1.09}^{5}$ first; $(800\,000 \times 1.09)^5$ is a different, enormous number. Round only at the end.

Tip

On a scientific calculator: 800000 $\times$ 1.09 $x^{y}$ 5 $=$ . Estimate first: ${1.09}^{5}$ is a bit more than $1 + 5 \times 0.09 = 1.45$ , so the answer should be a little above $800\,000 \times 1.45 = 1\,160\,000$ .

Corbettmaths: Compound Interest. Watch how the percentage is turned into a multiplier and raised to the power of the number of years, with and without a calculator.

✦

Practice

Give money to the nearest franc. The last three are harder.

Find the amount when 250,000 Frw is invested for 4 years at 6% per year compounded annually.
Find the compound interest on 1,500,000 Frw at 8% per year for 10 years.
Find the amount when 400,000 Frw is invested for 2 years at 12% per year compounded quarterly.
How much must be deposited now at 8% compounded annually to have 2,000,000 Frw in 5 years?
50,000 Frw grows to 72,000 Frw in 2 years, compounded annually. Find the annual rate.
After how many whole years will 300,000 Frw at 15% compounded annually first be worth more than 600,000 Frw?
Bank A pays 10% per year compounded annually; Bank B pays 9.6% per year compounded monthly. Which gives more on 1,000,000 Frw after 1 year, and by how much?

Show answersHide answers

$250\,000 \times 1.06^4 = 315\,619$ Frw.
$1\,500\,000 \times 1.08^{10} = 3\,238\,387$ Frw, so the interest is $1\,738\,387$ Frw.
3% per quarter for 8 quarters: $400\,000 \times 1.03^8 = 506\,708$ Frw.
$\frac{2\,000\,000}{1.08^5} = 1\,361\,166$ Frw (round up to 1,361,167 Frw to be sure of reaching 2,000,000).
$(1 + r)^{2} = 1.44$ , so $1 + r = 1.2$ : 20% per year.
$1.15^4 \approx 1.749$ (not yet double), $1.15^5 \approx 2.011$ : 5 years.
A: 1,100,000 Frw. B: $1\,000\,000 \times 1.008^{12} = 1\,100\,339$ Frw. Bank B gives 339 Frw more, even though its yearly rate looks lower.

Sources and credits

Written by Ganzaa.
Video: Compound Interest, Corbettmaths, embedded from YouTube.

The worked examples, table, chart, practice items and quiz questions were written by Ganzaa. Interest rates are illustrative.

1From a table to a formula

From a table to a formula