Simple and compound interest · F04 · Sunflower Intelligence

Before you start

You will be able to

Calculate simple interest, and the interest accrued for part of a year under the actual/365, actual/360 and 30/360 day-count conventions.
Calculate the future value of an amount under compound interest, including over a fractional number of periods.
Find the compound rate that links two amounts over a given time, using an nth root.
Calculate the time needed to reach a target using logarithms, and compare the exact doubling time with the rule of 72.
Choose between simple and compound interest for a real arrangement, and explain why the choice changes the answer.

Review first

Interest and inflation — It introduced simple and compound interest, the formula FV = P(1 + r)ⁿ and the rule of 72. This lesson assumes them and makes you fluent with them.
Percentages as decimals (7% = 0.07) and the power key on a calculator (1.07 to the power 6).

Fictional case A fictional coffee business founded by Ana Lim. It starts as a catering and barista-training service, opens its own cafés and an online shop for beans, and plans a canned cold brew.

The idea

Fictional case

It is March 2027 and Lantern Coffee Co. is a year old. Interest now runs through its affairs in two different ways. Every quarter Lantern pays Coastline Bank $150, the interest on the $10,000 two-year note it signed on 1 March 2026 at 6% a year; the $10,000 itself is not due until 1 March 2028. And Ana is weighing up a $24,000 cold-brew system, which Lantern could pay for with a Coastline loan at 7.2% a year or with the supplier’s instalment plan.

Every one of these questions comes down to what money costs, or earns, over time. This course gives you the tools to answer them exactly, starting with the two ways interest is calculated.

Two ways to charge for time

Interest is the price of using money for a period. It is quoted as a rate, usually per year, on the principal, the amount lent or borrowed. F00’s lesson on interest and inflation introduced the two ways of calculating it: simple interest is charged on the principal only, while compound interest is also charged on interest already added to the balance. This lesson makes both precise enough to use on real contracts, where the dates, the day counts and the rounding all matter.

Simple interest for a time $t$ in years at an annual rate $r$ is $I = P \times r \times t$ . Lantern’s note is the textbook case. The interest is paid out every quarter, so the balance never grows and every quarter’s interest is the same: $10{,}000 \times 0.06 \times \tfrac{1}{4} = 150$ dollars. Over the note’s two years Lantern makes eight such payments, $1,200 in all, which is exactly $10{,}000 \times 0.06 \times 2$ .

Interest for part of a year

Interest builds up day by day, so between payment dates a loan carries accrued interest: interest earned but not yet paid. To measure it, a contract needs a rule for turning a number of days into a fraction of a year. That rule is called a day-count convention. Three are common:

Actual/365: Count the actual calendar days and divide by 365.
Actual/360: Count the actual calendar days and divide by 360. Because the “year” is shorter, the same quoted rate produces slightly more interest.
30/360: Treat every month as 30 days and the year as 360 days, so every quarter counts as exactly 90 days. Versions of this convention differ in how they treat the 31st of a month and the end of February.

Interest on $10,000 at 6% a year from 1 March to 1 June 2027 (92 calendar days)
Convention	Days counted	Fraction of a year	Interest
30/360	90	90 ÷ 360 = 0.25	$150.00
Actual/365	92	92 ÷ 365 ≈ 0.2521	$151.23
Actual/360	92	92 ÷ 360 ≈ 0.2556	$153.33

Interest = 10,000 × 0.06 × fraction of a year, rounded to the cent.

Coastline charges a quarter of a year’s interest every quarter, $150, whatever the number of days, which is what a 30/360 count gives. No convention is more correct than another: banks and markets use different ones, and the loan or deposit agreement says which applies. A dollar or two a quarter looks trivial, but on large balances it is real money, and “In practice” shows a treasurer choosing between two deposits on exactly this point.

Compound interest: interest on interest

When interest is added to the balance instead of being paid out, the next period’s interest is charged on the larger balance. Each period the balance is multiplied by $(1 + r)$ , where $r$ is the rate per period, so after $n$ periods an amount $\text{PV}$ today has grown to its future value, $\text{FV} = \text{PV}(1 + r)^n$ . The multiplier $(1 + r)^n$ is the growth factor.

Take $8,000 invested at 7% a year, compounded annually, for six years. The growth factor is $1.07^6 \approx 1.500730$ , so the future value is $8{,}000 \times 1.500730 \approx 12{,}005.84$ dollars. Simple interest would pay $8{,}000 \times 0.07 \times 6 = 3{,}360$ dollars, a balance of $11,360. The extra $645.84 is interest earned on earlier interest.

Two habits keep compound calculations right. First, the rate and the number of periods must use the same period: a rate per year with a number of years, a rate per month with a number of months. Second, keep the growth factor to at least six significant figures and round only the final answer. Rounding 1.500730 to 1.50 in the example above would cost $5.84.

One equation, four quantities

The equation $\text{FV} = \text{PV}(1 + r)^n$ links four quantities. Know any three and you can find the fourth, and each version answers a common question:

FV: what will this amount grow to? Multiply by the growth factor.
PV: what is a future amount worth today? Divide by the growth factor. That is discounting, the subject of the lesson on present and future value.
r: what rate of growth links two amounts? Take an nth root, as “In depth” shows.
n: how long until the money reaches a target? Use logarithms, explained from the beginning in “In depth”.

In depth

Simple interest, rearranged

Because $I = P \times r \times t$ is a product, it can be solved for any of its letters: the rate is $r = I \div (P \times t)$ and the time is $t = I \div (P \times r)$ . Suppose a supplier charges $54 for letting a café pay a $3,600 invoice three months late. The implied simple rate is $54 \div (3{,}600 \times 0.25) = 0.06$ , or 6% a year.

If simple interest is left to build up rather than paid out, the balance after $t$ years is $\text{FV} = P(1 + rt)$ , which grows by the same dollar amount every year: a straight line. Simple interest appears wherever interest is paid out as it falls due, as on Lantern’s note and on bonds that pay regular interest, and in many short-term loans and deposits, where interest is worked out on the principal for the days the money is out. OpenStax’s Contemporary Mathematics notes that banks have used, and still use, 360 days as the year in such calculations.

Counting days

Under actual/365 and actual/360, count the calendar days after the start date up to and including the end date. From 1 March to 16 April 2027 that is the 30 remaining days of March plus 16 days of April: 46 days. Under 30/360 every month has 30 days, and the count is

\text{Days} = 360\,(Y_2 - Y_1) + 30\,(M_2 - M_1) + (D_2 - D_1)

where $Y$ , $M$ and $D$ are the year, month and day of the start (1) and end (2) dates, with a day of 31 usually treated as 30. From 1 March to 16 April this gives $30 \times 1 + (16 - 1) = 45$ days. Leap years add a further wrinkle: there are 366 days from 1 March 2027 to 1 March 2028, because February 2028 has 29 days, and some conventions, often written actual/actual, divide by 366 in a leap year.

Converting a rate from one basis to another is a matter of proportion. A rate quoted on an actual/360 basis earns as much as a rate 365/360 times as large on an actual/365 basis, so 4.5% actual/360 is equivalent to $4.5\% \times 365 \div 360 = 4.5625\%$ actual/365. Compare short-term rates only after putting them on the same basis.

Compound growth over whole and fractional periods

The growth factor $(1 + r)^n$ makes sense for any $n$ , not only whole numbers: $1.06^{2.5} = 1.06^2 \times \sqrt{1.06} \approx 1.156817$ , so by the formula $10,000 left for two and a half years at 6% a year would grow to about $11,568.17. Many real accounts work differently. Interest is credited only on the compounding dates; for a part-period, some accounts pay simple interest on the balance when the money is withdrawn, while others pay nothing for interest not yet credited. The account terms say which. Two full years and then half a year of simple interest give $10{,}000 \times 1.06^2 \times (1 + 0.06 \times 0.5) = 11{,}573.08$ dollars, $4.91 more.

The difference comes from a fact worth remembering. Over less than one period, simple interest pays more than compound interest at the same rate; over exactly one period the two are equal; over more than one, compound interest pays more. The table shows it for $10,000 at 6% a year.

$10,000 at 6% a year: simple and compound balances
Years	Simple: 10,000(1 + 0.06t)	Compound: 10,000 × 1.06^t	Compound − simple
0.25	$10,150.00	$10,146.74	−$3.26
0.5	$10,300.00	$10,295.63	−$4.37
1	$10,600.00	$10,600.00	$0.00
2	$11,200.00	$11,236.00	$36.00
5	$13,000.00	$13,382.26	$382.26

Rounded to the cent. The two methods cross at exactly one period.

So for a fractional period, ask how the money actually earns interest. The formula with a fractional exponent is the right model for growth that compounds smoothly, such as an investment whose value changes every day, and for the comparisons in later lessons. The mixed method, compound to the last crediting date and simple after it, is the right model for an account that pays simple interest for a part-period; for one that pays nothing until interest is credited, use the balance at the last crediting date.

Solving for the rate: the nth root

If you know where an amount started, where it ended and how many periods passed, rearranging the future value equation gives the constant rate that links them:

r = \left(\frac{\text{FV}}{\text{PV}}\right)^{1/n} - 1

Raising a number to the power $1/n$ takes its nth root: the number that, multiplied by itself $n$ times, gives the original. On a calculator, use the power key (often marked $y^x$ or ^) with the exponent entered as $1 \div n$ . For example, an investment bought for $5,000 is sold for $8,000 seven years later. The ratio is $8{,}000 \div 5{,}000 = 1.6$ , and $1.6^{1/7} \approx 1.069449$ , so the rate is about 6.94% a year. Check it by going forwards: $1.069449^7 \approx 1.6$ .

Applied to a business measure such as revenue, this rate is called the compound annual growth rate (CAGR). It is not the total growth divided by the number of years: 60% ÷ 7 ≈ 8.57% overstates the rate, because it ignores the growth on growth. And it hides the path: a value that fell and then recovered can have the same CAGR as one that rose steadily.

Solving for time: logarithms from the beginning

To find the number of periods you must undo a power, and the tool for that is the logarithm. A logarithm answers one question: to what power must the base be raised to give this number? In base 10, the logarithm of 1,000 is 3, because 10³ = 1,000, and the logarithm of 100 is 2. The natural logarithm, written ln, uses as its base the number e ≈ 2.71828, which the next lesson meets in continuous compounding. Calculators have both keys (LOG and LN), and spreadsheets have the functions LOG10 and LN.

One property does all the work here: the logarithm of a power is the power times the logarithm, $\ln(a^n) = n \times \ln a$ . It turns an unknown exponent into an ordinary multiplication. Start from $(1 + r)^n = \text{FV} \div \text{PV}$ , take the logarithm of both sides, and divide:

n \ln(1 + r) = \ln\!\left(\frac{\text{FV}}{\text{PV}}\right) \quad\Longrightarrow\quad n = \frac{\ln(\text{FV}/\text{PV})}{\ln(1 + r)}

Any base works, as long as the same one is used above and below the line. How long does $10,000 take to grow to $15,000 at 6% a year? $n = \ln 1.5 \div \ln 1.06 = 0.405465 \div 0.058269 \approx 6.96$ years; base-10 logarithms give $0.176091 \div 0.025306 \approx 6.96$ too. If interest is credited once a year, though, the balance is $14,185.19 after six years and $15,036.30 after seven, so the target is first reached on the seventh crediting date.

Doubling time and the rule of 72

Setting $\text{FV} \div \text{PV} = 2$ gives the doubling time, $n = \ln 2 \div \ln(1 + r)$ , where $\ln 2 \approx 0.693147$ . The rule of 72, which F00 introduced, approximates it as 72 divided by the percentage rate. The table compares the two.

Periods needed to double at a constant compound rate
Rate per period	Rule of 72	Exact: ln 2 ÷ ln(1 + r)	Rule minus exact
1%	72.00	69.66	+2.34
2%	36.00	35.00	+1.00
4%	18.00	17.67	+0.33
6%	12.00	11.90	+0.10
8%	9.00	9.01	−0.01
12%	6.00	6.12	−0.12
24%	3.00	3.22	−0.22

Periods rounded to two decimal places.

The rule is exact at a rate of about 7.85% and drifts away on either side: it overstates the doubling time at low rates and understates it at high ones. Why 72? For small rates ln(1 + r) is close to r, so the doubling time is close to 69.3 divided by the percentage rate; because ln(1 + r) is a little smaller than r, the exact answer is a little longer, and at rates around 8% the correction brings 69.3 up to about 72.

The number 72 also divides neatly by 2, 3, 4, 6, 8, 9 and 12. Some people use 70 or 69.3, which suit low rates better. Use the rule for quick judgements and logarithms when the answer matters.

Formulas

Simple interest

I = P \times r \times t, \qquad \text{FV} = P(1 + rt)

$I$: Interest for the whole time$
$P$: Principal: the amount lent, borrowed or deposited (its present value)$
$r$: Annual interest rate, as a decimal (6% = 0.06)per year
$t$: Timeyears
$\text{FV}$: Balance at the end, if the interest is added rather than paid out$

Assumes. Interest is earned on the original principal only.

Rearranged: r = I ÷ (P × t) and t = I ÷ (P × r).

Interest for part of a year (day count)

I = P \times r \times \frac{D}{B}

$I$: Interest accrued over the days counted$
$P$: Principal$
$r$: Annual interest rate, as a decimalper year
$D$: Days counted under the convention: actual calendar days, or 30 a month under 30/360days
$B$: Days in the convention’s year: 365 (actual/365) or 360 (actual/360 and 30/360)days

Assumes. Simple interest within the period.

The contract sets the convention; conventions vary between countries, markets and products.

Future value under compound interest

\text{FV} = \text{PV}(1 + r)^n

$\text{FV}$: Future value: the balance after n periods$
$\text{PV}$: Present value: the amount today$
$r$: Interest rate per period, as a decimalper period
$n$: Number of periods (may be fractional)periods

Assumes. A constant rate; interest stays in the balance; nothing is added or withdrawn.

With a fractional n the formula assumes smooth compounding; an account that credits interest on fixed dates may pay simple interest for the part-period.

Rate linking two amounts (compound annual growth rate)

r = \left(\frac{\text{FV}}{\text{PV}}\right)^{1/n} - 1

$r$: Constant compound rate per periodper period
$\text{FV}$: Amount at the end$
$\text{PV}$: Amount at the start$
$n$: Number of periods between themperiods

With n in years this is the compound annual growth rate (CAGR).

Number of periods to reach a target

n = \frac{\ln(\text{FV}/\text{PV})}{\ln(1 + r)}

$n$: Number of periods neededperiods
$\text{FV}$: Target amount$
$\text{PV}$: Starting amount$
$r$: Interest rate per period, as a decimalper period

Assumes. Smooth compounding at a constant rate.

Any logarithm base works if the same one is used above and below. If interest is credited only at the end of each period, the target is reached at the next whole period.

Doubling time and the rule of 72

n_{\text{double}} = \frac{\ln 2}{\ln(1 + r)} \;\approx\; \frac{72}{100\,r}

$n_{\text{double}}$: Periods needed for an amount to doubleperiods
$r$: Interest rate per period, as a decimalper period

The rule of 72 is exact at about 7.85% and works well between about 2% and 15%.

Worked examples

A first example: What Lantern would owe to repay its note early

Fictional figures

Suppose Lantern decided to repay its Coastline note on 16 April 2027 and the bank allowed it without a penalty. The last interest payment was made on 1 March 2027. How much would Lantern pay? Use a 30/360 count, which is how the note charges interest, and compare the result with actual/365.

Count the days since the last payment. Under 30/360: 30 × (4 − 3) + (16 − 1) = 45 days. Counting calendar days: the 30 remaining days of March plus 16 days of April, 46 days.
The interest is simple, on the principal only, because every earlier quarter’s interest has been paid. Under 30/360: $10{,}000 \times 0.06 \times \tfrac{45}{360} = 75.00$ .
Under actual/365: $10{,}000 \times 0.06 \times \tfrac{46}{365} \approx 75.62$ .
Payoff = principal + accrued interest = 10,000 + 75.00 = $10,075.00 under the note’s own convention.

Answer. $10,075.00: the $10,000 principal plus $75.00 of interest for 45 days counted on a 30/360 basis. Counting actual days over 365 would give $75.62 of interest.

What it means. Repaying early stops the interest from that day, but interest already built up must still be paid. Whether repaying is wise is a separate question: it depends on what else Lantern could do with $10,075 and what it would cost to borrow again, which the lessons on discounting and loans take up.

A harder example: How long to save for a new oven?

Fictional figures

A fictional bakery has $18,000 in a deposit account paying 4.5% a year, compounded and credited once a year on the anniversary of the deposit. The owner needs $25,000 for a new oven. (a) By the formula, how long will the money take to reach $25,000? (b) If the owner withdraws between anniversaries, the account pays simple interest for the part-year. When can the owner first withdraw $25,000? (c) What annual rate would reach the target in five years?

(a) Use logarithms: $n = \ln(25{,}000 \div 18{,}000) \div \ln 1.045 = 0.328504 \div 0.044017 \approx 7.46$ years.
(b) Interest arrives only on the anniversaries. After seven years the balance is $18{,}000 \times 1.045^7 \approx 24{,}495.51$ , which is $25{,}000 - 24{,}495.51 = 504.49$ short.
During year 8 the balance earns simple interest of $24{,}495.51 \times 0.045 \approx 1{,}102.30$ a year, so the shortfall takes $504.49 \div 1{,}102.30 \approx 0.4577$ of a year, about 167 days.
(c) Use the nth root: $r = (25{,}000 \div 18{,}000)^{1/5} - 1 = 1.388889^{0.2} - 1 \approx 0.0679$ , or 6.79% a year.

Answer. (a) About 7.46 years. (b) Seven years and about 167 days. (c) About 6.79% a year.

What it means. Paying simple interest for the part-year makes the money available about two days earlier than the smooth-growth answer, because within a period simple interest pays more than compound. If the owner waits for interest to be credited, the balance passes $25,000 only on the eighth anniversary, when it reaches $25,597.81. To have the oven in five years the bakery would need about 6.8% a year, far above its account’s 4.5%: it must save more, borrow or wait.

Applied: a business decision: Is the growth plan realistic?

Fictional figures

A fictional battery-storage installer had revenue of $2.4 million in 2022 and $3.9 million in 2026. Its business plan, sent to a lender with a loan application, forecasts revenue of $6.0 million in 2029. Compare the growth the plan assumes with the growth the business has achieved.

Past growth over four years: $\text{CAGR} = (3.9 \div 2.4)^{1/4} - 1 = 1.625^{0.25} - 1 \approx 12.91\%$ a year. Dividing the total growth of 62.5% by four (15.63%) would overstate it.
Growth the plan needs over three years: $(6.0 \div 3.9)^{1/3} - 1 \approx 15.44\%$ a year.
At the past rate, 2029 revenue would be $3.9 \times 1.129050^3 \approx 5.61$ million dollars, and reaching $6.0 million would take $\ln(6.0 \div 3.9) \div \ln 1.129050 \approx 3.55$ years: not before the 2030 figures.

Answer. The plan assumes growth of about 15.44% a year, against 12.91% achieved from 2022 to 2026. At the past rate revenue would reach about $5.61 million in 2029; $6.0 million would take about 3.55 years.

What it means. The plan needs the business to grow faster, on average, than it ever has. That is not impossible, but the lender will ask what changes to justify it, such as new contracts, more installers or higher prices, and will test whether the loan can still be repaid at the slower, historical rate. Past growth is a reference point, not a forecast.

Explore

Common mistakes

Mixing periods
An annual rate must go with a number of years, a monthly rate with a number of months. Using 7% with 72 months, or 0.5% a month with 6 years, gives an answer that is wildly too large or too small. Write down the period before you write down the formula.
Dividing total growth by the number of years
Growth from $5,000 to $8,000 in seven years is 60% in total, but not 60% ÷ 7 ≈ 8.57% a year. That is simple-interest arithmetic. The compound rate is 1.6^(1/7) − 1 ≈ 6.94%. A brochure that quotes the larger figure as an “average return” overstates the growth.
Choosing the method by habit
Interest that is paid out as it falls due does not compound, so Lantern’s note costs $1,200 over two years, not 10,000 × (1.06² − 1) = $1,236. Interest left in the balance does compound. Ask where the interest goes before choosing the formula.
Assuming a day count
The same 6% on $10,000 for 1 March to 1 June is $150.00, $151.23 or $153.33 depending on the convention. Read the contract, and convert rates to the same basis before comparing them.
Rounding too early
Rounding the growth factor 1.500730 to 1.50 makes $8,000 at 7% for six years look like $12,000.00 instead of $12,005.84. Keep at least six significant figures in factors and round only the final answer.
Trusting the rule of 72 at extreme rates
At 1% the rule says 72 years to double; the exact answer is 69.66. At 24% it says 3 years; the exact answer is 3.22. Fine for a quick check, not for a contract or an exam answer.

Guided practice

Work through these with hints when you need them. Every hint and every worked solution is there to be used.

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On your own

No hints this time. Answer, check, and read the feedback; the solution appears once you have answered or if you ask for it. Your first answers count toward mastery.

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In practice

Interest arithmetic is routine in finance work, and the mistakes it catches are real.

Choosing a deposit. The treasurer of a fictional hotel group has $400,000 set aside for a tax payment due in 146 days, from 1 April to 25 August 2027. Bank A offers 4.50% on an actual/360 basis; Bank B offers 4.55% on an actual/365 basis. On the quoted rates Bank B looks better. Working out the interest says otherwise: Bank A pays 400,000 × 0.045 × 146 ÷ 360 = $7,300.00 and Bank B pays 400,000 × 0.0455 × 146 ÷ 365 = $7,280.00.

Bank A pays $20 more, because 4.50% on actual/360 is equivalent to 4.5625% on actual/365. Before deciding, the treasurer also checks what the rate leaves out: whether the money can be withdrawn early if the tax bill comes sooner, and how safe each bank is.

Testing a performance claim. An analyst reviewing a fictional fund reads in its brochure that it “doubled investors’ money in nine years, an average return of 11.1% a year”. The rule of 72 says doubling in nine years means about 8% a year, and the exact compound rate is 2^(1/9) − 1 ≈ 8.01%. The 11.1% is total growth (100%) divided by nine: simple-interest arithmetic applied to compound growth. The analyst reports the compound rate, which is the figure comparable with other investments.

Accruing and settling interest. Lantern’s bookkeeper accrues $50 of interest on the note at the end of each month, $10{,}000 \times 0.06 \times \tfrac{1}{12}$ , as in F01, so the accounts show the interest owed between payments. A lender preparing a payoff statement does the same calculation by days, using the day count in the contract, as the first worked example did.

Fictional case

For Lantern the next question is bigger. Coastline will lend the $24,000 for the cold-brew system at 7.2% a year, compounded monthly, over 48 months. What does that cost in a year once the monthly compounding is counted, and how does it compare with rates quoted on other bases? That is the next lesson, on effective annual rates. The full comparison with the supplier’s instalment plan needs annuities, which come in module 3.

Knowledge check

Six questions, one attempt each, no hints. Score 80% or more to complete the lesson; you can take it again at any time.

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Summary

Simple interest is charged on the principal only: I = P × r × t. It fits arrangements where interest is paid out as it falls due, such as Lantern’s note ($150 a quarter).
Interest for part of a year depends on the day-count convention (actual/365, actual/360 or 30/360). The contract decides; convert rates to the same basis before comparing.
Compound interest multiplies the balance by (1 + r) each period: FV = PV(1 + r)ⁿ. The rate and the number of periods must use the same period; round only at the end.
Within one period simple interest pays more than compound; after one period compound pays more. Accounts credit interest on fixed dates; for a part-period some pay simple interest and some pay nothing, as their terms say.
The rate linking two amounts is r = (FV ÷ PV)^(1/n) − 1, the compound annual growth rate when n is in years. Total growth divided by years overstates it.
The time to reach a target is n = ln(FV ÷ PV) ÷ ln(1 + r). Logarithms turn an unknown power into a multiplication.
Doubling time is ln 2 ÷ ln(1 + r). The rule of 72 approximates it and is exact near 7.85%.

Key formulas

Simple interest
Interest for part of a year (day count)
Future value under compound interest
Rate linking two amounts (compound annual growth rate)
Number of periods to reach a target
Doubling time and the rule of 72

Terms introduced

Accrued interest: Interest that has built up since the last payment or crediting date but has not yet been paid or added to the balance.
Day-count convention: The rule a contract or market uses to turn days into a fraction of a year for interest, such as actual/365, actual/360 or 30/360. Conventions vary between countries, markets and products.
Growth factor: The number (1 + r)ⁿ by which compound growth at a rate r per period multiplies an amount over n periods.
Compound annual growth rate: The constant yearly rate that takes a starting value to an ending value over a number of years: (end ÷ start)^(1/years) − 1. Often shortened to CAGR.
Logarithm: The power to which a base must be raised to give a number: the base-10 logarithm of 1,000 is 3 because 10³ = 1,000. The natural logarithm, ln, uses the base e ≈ 2.71828.
Doubling time: The time an amount takes to double at a constant compound rate r per period: ln 2 ÷ ln(1 + r) periods.
Rule of 72: A shortcut for doubling time: about 72 divided by the percentage rate per period. Close for rates between about 2% and 15%; exact near 7.85%.

Sources

Dahlquist, J. & Knight, R. (2022). Principles of Finance, section 7.2, “Time Value of Money (TVM) Basics”. OpenStax. openstax.org/books/principles-finance/pages/7-2-time-value-o…
Dahlquist, J. & Knight, R. (2022). Principles of Finance, section 7.3, “Methods for Solving Time Value of Money Problems” (solving for the number of periods and the rate). OpenStax. openstax.org/books/principles-finance/pages/7-3-methods-for-…
Kirk, D. (2023). Contemporary Mathematics, section 6.3, “Simple Interest”. OpenStax. openstax.org/books/contemporary-mathematics/pages/6-3-simple…
Abramson, J. (2021). College Algebra 2e, section 6.6, “Exponential and Logarithmic Equations”. OpenStax. openstax.org/books/college-algebra-2e/pages/6-6-exponential-…

Before you start

You will be able to

Review first

The idea

Two ways to charge for time

Interest for part of a year

Compound interest: interest on interest

One equation, four quantities

In depth

Simple interest, rearranged

Counting days

Compound growth over whole and fractional periods

Solving for the rate: the nth root

Solving for time: logarithms from the beginning

Doubling time and the rule of 72

Formulas

Simple interest

Interest for part of a year (day count)

Future value under compound interest

Rate linking two amounts (compound annual growth rate)

Number of periods to reach a target

Doubling time and the rule of 72

Worked examples

Explore

Common mistakes

Mixing periods

Dividing total growth by the number of years

Choosing the method by habit

Assuming a day count

Rounding too early

Trusting the rule of 72 at extreme rates

Guided practice

On your own

In practice

Knowledge check

Summary

Key formulas

Terms introduced

Sources