Compound Interest Formula

1. Formula
   Observation

   The boxed formula on the right side of the board reads $A = P(1 + r/n)$^{nt}.

2. Audio
   Observation

   At 00:00.000-00:07.360 the speaker says, "Formula here, A equals P times one plus R divided by N raised to the N times T power."

3. Audio
   Observation

   At 01:49.100-01:55.980 he identifies the final number as "the ending amount" including principal and interest.

A

Ending amount or future value after compounding; it includes the original investment and all accumulated interest.

Money amount in dollars in the worked example.

1. Formula
   Observation

   P appears as the multiplier outside the parentheses in $A = P(1 + r/n)$^{nt}.

2. Diagram
   Observation

   An arrow from the word PRINCIPAL points to P on the board.

3. Audio
   Observation

   At 00:08.000-00:10.240 the speaker says, "P is a principal, that's what you put into your investment."

P

Principal, the starting amount invested or deposited.

Money amount in dollars; in the example $P = 5000$.

1. Formula
   Observation

   r appears in the fraction $r/n$ inside the parentheses of $A = P(1 + r/n)$^{nt}.

2. Diagram
   Observation

   The board text below the formula states "r = interest rate as a decimal".

3. Audio
   Observation

   At 00:14.400-00:18.880 the speaker says, "R is the interest rate, but you want to convert that into a decimal."

r

Annual interest rate expressed as a decimal.

Decimal rate per year; in the example 8% is converted to 0.08.

1. Formula
   Observation

   n appears both in the denominator $r/n$ and in the exponent nt.

2. Diagram
   Observation

   The board text states "n = # of times you compound per year".

3. Audio
   Observation

   At 00:19.840-00:23.440 the speaker says, "N here and here represent the number of times you're compounding in a year."

4. Audio
   Observation

   At 00:24.000-00:28.880 he gives examples: monthly $n = 12$, weekly $n = 52$.

n

Number of compounding periods per year.

Positive integer count of periods per year; in the quarterly example $n = 4$.

1. Formula
   Observation

   t appears in the exponent nt.

2. Diagram
   Observation

   The board text states "t = time in years".

3. Audio
   Observation

   At 00:28.880-00:31.280 the speaker says, "And then T is a time in years, number of years."

t

Time measured in years.

Number of years; in the example $t = 2$.

1. Formula
   Observation

   The constant 1 appears inside the parentheses as $1 + r/n$.

2. Audio
   Observation

   At 00:11.040-00:13.600 the speaker says, "One represents like 100%, the original amount."

1

Represents 100% of the original amount before adding the periodic interest fraction.

1. Formula
   Observation

   The exponent in the boxed formula is written nt.

2. Audio
   Observation

   At 00:00.000-00:07.360 the speaker reads the exponent as "N times T power".

3. Audio
   Observation

   At 01:15.760-01:22.000 he interprets the example exponent as eight quarters in two years.

nt

Total number of compounding periods over the whole investment horizon.

1. Formula
   Observation

   The expression $r/n$ appears inside the parentheses of the general formula.

2. Audio
   Observation

   At 01:02.720-01:06.720 the speaker computes 0.08 divided by 4 as 0.02 and rewrites the base as 1.02.

3. Audio
   Observation

   At 01:08.000-01:15.120 he explains this means getting 2% every quarter instead of 8% annually.

$r/n$

Periodic interest rate per compounding period.

1. Formula
   Observation

   The right-hand boxed formula is $A = P(1 + r/n)$^{nt}.

2. Audio
   Observation

   The speaker reads the formula aloud at 00:00.000-00:07.360.

3. Diagram
   Observation

   Definitions are written beneath the formula: PRINCIPAL, n = # of times you compound per year, r = interest rate as a decimal, t = time in years.

The video presents the standard compound-interest formula for the ending amount after repeated periodic growth. The base $1 + r/n$ combines the original amount with one period's fractional interest, and the exponent $nt$ counts how many such periods occur.

1. Use r as a decimal annual rate.

2. Use n as the number of compounding periods per year.

3. Use t in years so that nt is the total number of periods.

1. Audio
   Observation

   At 00:08.000-00:10.240 the speaker says, "P is a principal, that's what you put into your investment."

2. Diagram
   Observation

   The word PRINCIPAL is written with an arrow pointing to P.

P denotes the initial amount invested or deposited before any interest is earned.

1. P is the starting amount used in the compound-interest formula.

1. Compound interest formula

1. Audio
   Observation

   At 00:11.040-00:13.600 the speaker says, "One represents like 100%, the original amount."

2. Formula
   Observation

   The 1 appears inside the factor ($1 + r/n$).

The constant 1 in $1 + r/n$ stands for 100% of the original amount, so adding $r/n$ expresses growth beyond the starting value.

1. This interpretation applies to the multiplicative growth factor in the compound-interest formula.

1. Compound interest formula

1. Audio
   Observation

   At 00:14.400-00:18.880 the speaker says, "R is the interest rate, but you want to convert that into a decimal."

2. Diagram
   Observation

   The board writes "r = interest rate as a decimal".

The annual interest rate must be converted from percent form to decimal form before substituting into the formula.

1. If the stated rate is a percent, divide by 100 before using it as r.

1. Compound interest formula

1. Audio
   Observation

   At 00:19.840-00:23.440 the speaker says, "N here and here represent the number of times you're compounding in a year."

2. Audio
   Observation

   At 00:24.000-00:28.880 he gives examples: monthly $n = 12$, weekly $n = 52$.

3. Diagram
   Observation

   The board writes "n = # of times you compound per year".

n is the number of compounding periods in one year. It appears both in the denominator of the periodic rate and in the total-period exponent.

1. Monthly compounding uses $n = 12$.

2. Weekly compounding uses $n = 52$.

3. Quarterly compounding uses $n = 4$, as shown in the example.

1. Compound interest formula

1. Audio
   Observation

   At 00:28.880-00:31.280 the speaker says, "And then T is a time in years, number of years."

2. Diagram
   Observation

   The board writes "t = time in years".

t is the investment duration measured in years, so that multiplying by n converts years into total compounding periods.

1. Use years for t when n is defined as periods per year.

1. Compound interest formula

1. Audio
   Observation

   At 00:43.440-00:58.320 the speaker substitutes the example values into the formula: $A = 5000$(1 + .08/4)^{$4*2$}.

2. Formula
   Observation

   The board shows $A = 5000$(1 + .08/4)^{$4*2$}.

To solve a compound-interest problem, identify P, r, n, and t from the wording, convert the percent rate to a decimal, and substitute directly into $A = P(1 + r/n)^{nt}$.

1. The rate must be converted to decimal before substitution.

2. n must match the compounding frequency.

3. t must be in years.

1. Compound interest formula
2. Interest rate r as a decimal
3. Compounding frequency n
4. Time t in years

1. Audio
   Observation

   At 00:59.040-01:06.720 the speaker simplifies .08 divided by 4 to .02 and rewrites the base as 1.02.

2. Formula
   Observation

   The board shows $5000(1.02)^8$.

The fraction $r/n$ is simplified first to obtain the interest rate per compounding period. Adding 1 gives the growth factor for one period.

1. This step is valid after r has been written as a decimal annual rate.

1. Substituting values into the compound-interest formula
2. Periodic interest rate meaning

1. Audio
   Observation

   At 01:08.000-01:15.120 the speaker explains that instead of 8% annually, the investment earns 2% every quarter.

2. Audio
   Observation

   At 01:15.760-01:22.000 he says there are eight quarters in two years, so the amount compounds eight times.

3. Formula
   Observation

   The exponent $4*2$ is simplified to 8 on the board.

The product $nt$ counts the total number of compounding intervals. In the example, 4 quarters per year times 2 years gives 8 total periods.

1. Use this interpretation when n is periods per year and t is years.

1. Compounding frequency n
2. Time t in years

1. Audio
   Observation

   At 01:26.860-01:31.340 the speaker says, "So if we simplify this down, let's see what we get. We'll do it on the calculator."

2. Audio
   Observation

   At 01:32.140-01:39.500 he reads the computation as 5000, 1.02 to the eighth power, 5000.

3. Formula
   Observation

   The board shows the final result $5858.30.

1. The exact calculator display is not shown clearly enough to verify intermediate digits beyond the final rounded amount.

After reducing the formula to $5000(1.02)^8$, the video evaluates the power and multiplies by the principal, then rounds the monetary answer to the nearest cent.

1. Round the final money amount to cents when reporting dollars and cents.

1. Simplifying the periodic growth factor
2. Interpreting the exponent as total compounding periods

1. Audio
   Observation

   At 01:49.100-01:55.980 the speaker says, "Okay, so that's the ending amount. That includes the original investment and the interest all together in this figure here, five thousand eight hundred and fifty-eight."

A is not just the interest earned; it is the total ending balance, combining the original principal and all accumulated interest.

1. This applies to the output of the compound-interest formula $A = P(1 + r/n)$^{nt}.

1. Compound interest formula

1. Audio
   Observation

   At 01:08.000-01:15.120 the speaker says, "So what this means is you're getting two percent. So instead of eight percent, you're getting two percent every quarter."

Dividing the annual decimal rate by the number of compounding periods per year gives the rate applied in each individual period.

1. This interpretation assumes equal compounding periods within a year.

1. Interest rate r as a decimal
2. Compounding frequency n

1. Audio
   Observation

   At 00:53.200-00:58.320 the speaker says, "divided by four since you're compounding quarterly, raised to the four times two."

2. Formula
   Observation

   The substituted expression uses 4 in the denominator and $4*2$ in the exponent.

If interest is compounded quarterly, then the number of compounding periods per year is $n = 4$.

1. The compounding frequency is quarterly.

2. n denotes the number of times interest is compounded per year.

For the given example with quarterly compounding.

1. Audio
   Observation

   At 01:15.760-01:22.000 the speaker says, "And there's eight quarters in two years, so this is going to be compounding eight times at two percent each time."

2. Formula
   Observation

   The exponent $4*2$ is simplified to 8 on the board.

For quarterly compounding over 2 years, the total number of compounding periods is 8.

1. $n = 4$ because compounding is quarterly.

2. $t = 2$ years.

For the specific example in the video.

1. Audio
   Observation

   At 01:49.100-01:55.980 the speaker says the ending amount includes the original investment and the interest all together.

The value computed by $A = P(1 + r/n)$^{nt} is the ending amount, which includes both the original principal and the accumulated interest.

1. A is computed from the compound-interest formula.

2. P is the principal.

For the compound-interest setting presented in the video.

1. Formula
   Observation

   The board progresses from $A = 5000$(1 + .08/4)^{$4*2$} to $5000(1.02)^8$.

2. Audio
   Observation

   At 00:43.440-00:58.320 the speaker substitutes the values.

3. Audio
   Observation

   At 00:59.040-01:06.720 he simplifies .08/4 to .02 and writes 1.02.

1. Expression

   $$A = 5000\left(1 + \frac{.08}{4}\right)^{4\cdot 2}$$
   Explanation

   Substitute the example values into the compound-interest formula: principal 5000, annual rate 0.08, quarterly compounding $n = 4$, and time $t = 2$ years.

   Justification

   Direct substitution into $A = P(1 + r/n)$^{nt}.

   Shown in the video
2. Expression

   $$\frac{.08}{4} = .02$$
   Explanation

   Compute the periodic rate by dividing the annual decimal rate by the number of periods per year.

   Justification

   Arithmetic simplification of $r/n$.

   Shown in the video
3. Expression

   $$1 + .02 = 1.02$$
   Explanation

   Add 1 to the periodic rate to form the one-period growth factor.

   Justification

   Definition of the base factor in the compound-interest formula.

   Shown in the video
4. Expression

   $$4\cdot 2 = 8$$
   Explanation

   Multiply the number of quarters per year by the number of years to get the total number of compounding periods.

   Justification

   The exponent nt counts total periods.

   Shown in the video
5. Expression

   $$A = 5000(1.02)^8$$
   Explanation

   Rewrite the substituted expression in simplified form ready for calculator evaluation.

   Justification

   Combining the previous simplifications.

   Shown in the video

The example reduces to $A = 5000(1.02)^8$ before numerical evaluation.

1. Audio
   Observation

   At 01:26.860-01:39.500 the speaker says he will use the calculator and reads out 5000, 1.02 to the eighth power, 5000.

2. Formula
   Observation

   The board shows the final amount $5858.30.

3. Audio
   Observation

   At 01:41.660-01:47.820 he states the answer as eight hundred fifty-eight dollars and thirty cents if you round to the pennies.

1. The calculator screen itself is not legible; only the spoken entry and the written final rounded result are clear.

1. Expression

   $$A = 5000(1.02)^8$$
   Explanation

   Start from the simplified expression obtained from the example data.

   Justification

   Previous derivation der-example-simplification.

   Shown in the video
2. Expression

   $$A \approx 5858.30$$
   Explanation

   Evaluate the power and multiply by 5000, then round the monetary result to the nearest cent.

   Justification

   Calculator evaluation followed by rounding to pennies as stated in the audio.

   Shown in the video

The ending amount is approximately $5858.30.

1. Audio
   Observation

   At 00:31.840-00:39.840 the speaker states the problem: investing $5,000 at 8% compounded quarterly for 2 years.

2. Diagram
   Observation

   The top-left board text reads $5000, 8%, QUARTERLY, 2 yrs.

3. Formula
   Observation

   The worked solution on the board is $A = 5000$(1 + .08/4)^{$4*2$}, then $5000(1.02)^8$, then $5858.30.

4. Audio
   Observation

   At 01:41.660-01:47.820 he reports the answer as $5858.30 after rounding to pennies.

Find the ending amount when $5,000 is invested at 8% compounded quarterly for 2 years.

1. $P = 5000$ dollars

2. Annual interest rate = 8%

3. Compounding frequency = quarterly

4. Time = 2 years

Compute the ending amount A.

1. Expression

   $$r = 0.08$$
   Explanation

   Convert the annual percentage rate to decimal form.

   Justification

   The video explicitly says to convert the interest rate into a decimal.

   Shown in the video
2. Expression

   $$n = 4$$
   Explanation

   Use quarterly compounding, so there are 4 periods per year.

   Justification

   The speaker states that quarterly compounding means dividing by 4.

   Shown in the video
3. Expression

   $$t = 2$$
   Explanation

   Use the investment duration in years.

   Justification

   The problem statement gives 2 years and the formula defines t in years.

   Shown in the video
4. Expression

   $$A = 5000\left(1 + \frac{0.08}{4}\right)^{4\cdot 2}$$
   Explanation

   Substitute the identified values into the compound-interest formula.

   Justification

   Direct application of $A = P(1 + r/n)$^{nt}.

   Shown in the video
5. Expression

   $$A = 5000(1.02)^8$$
   Explanation

   Simplify the periodic rate and total number of periods.

   Justification

   $0.08/4 = 0.02$, so $1 + 0.02 = 1.02$, and $4*2 = 8$.

   Shown in the video
6. Expression

   $$A \approx 5858.30$$
   Explanation

   Evaluate on a calculator and round to the nearest cent.

   Justification

   The speaker uses a calculator and states the rounded dollar-and-cent result.

   Shown in the video

$5858.30

The video verifies the result by calculator evaluation and by interpreting the answer as the ending amount including both principal and interest.

1. Diagram
   Observation

   The whiteboard is divided into a title/problem area on the left and a boxed formula/definitions area on the right.

2. Diagram
   Observation

   The title reads COMPOUND INTEREST, and the example data $5000, 8%, QUARTERLY, 2 yrs are written beneath it.

1. Title COMPOUND INTEREST

2. Example data $5000, 8%, QUARTERLY, 2 yrs

3. Boxed formula $A = P(1 + r/n)$^{nt}

4. Variable definitions under the formula

5. Worked solution lines below the example data

1. The instructor first points to the boxed formula and its variable definitions.

2. He then moves to the left side and writes the substituted example equation.

3. Next he writes the simplified expression $5000(1.02)^8$.

4. Finally he writes the numerical answer $5858.30.

1. The general formula remains visible on the right throughout the clip.

2. The example givens remain visible on the upper left throughout the clip.

The visual layout separates the general rule from its concrete application, making the substitution process easy to track.

1. Diagram
   Observation

   An arrow labeled PRINCIPAL points to P in the formula.

2. Audio
   Observation

   At 00:08.000-00:10.240 the speaker identifies P as the principal, what you put into your investment.

1. Word PRINCIPAL

2. Arrow

3. Symbol P in the formula

1. The instructor gestures toward the arrow and P while explaining the meaning of the principal.

1. The arrow remains fixed on the board once visible.

The diagram visually encodes that P is the initial invested amount.

1. Diagram
   Observation

   Arrows point from the definition text for n and t up toward the exponent nt in the formula.

2. Audio
   Observation

   At 00:19.840-00:31.280 the speaker explains n as compounding frequency and t as time in years.

1. Definition text for n

2. Definition text for t

3. Exponent nt

4. Connecting arrows

1. The instructor points between the definitions and the exponent while explaining their roles.

1. The exponent remains nt in the boxed formula.

The arrows show that the exponent is built from the compounding frequency and the number of years.

1. Animation
   Observation

   The instructor writes $A = 5000$(1 + .08/4)^{$4*2$} step by step on the left side of the board.

2. Audio
   Observation

   At 00:43.440-00:58.320 he narrates each substituted quantity as he writes it.

1. Handwritten equation $A = 5000$(1 + .08/4)^{$4*2$}

2. Marker

3. Board space under the example data

1. The equation is built from left to right as the speaker names each value.

2. The principal 5000 is written first, then the base with .08/4, then the exponent $4*2$.

1. The structure mirrors the general formula on the right.

The animation of writing makes the mapping from word problem to formula explicit.

1. Animation
   Observation

   The instructor writes the simplified line $5000(1.02)^8$ below the substituted expression.

2. Audio
   Observation

   At 00:59.040-01:06.720 he explains the simplification to 1.02 and the eighth power.

1. Handwritten equation $5000(1.02)^8$

1. The fractional rate .08/4 is replaced by .02 inside the base, producing 1.02.

2. The exponent $4*2$ is replaced by 8.

1. The principal 5000 remains outside the parentheses.

The visual simplification shows how the formula becomes a single power calculation.

1. Animation
   Observation

   The instructor writes the final amount $5858.30 on the board.

2. Audio
   Observation

   At 01:41.660-01:47.820 he states the answer as eight hundred fifty-eight dollars and thirty cents if you round to the pennies.

1. Handwritten result $5858.30

1. A new numeric line is added beneath the simplified expression.

1. The earlier formula lines remain visible above it.

The final written number is the computed ending balance rounded to cents.

1. Audio
   Observation

   At 00:14.400-00:18.880 the speaker says, "R is the interest rate, but you want to convert that into a decimal."

2. Diagram
   Observation

   The board explicitly writes "r = interest rate as a decimal".

A learner may substitute 8 for r when the rate is 8%.

The video states that r must be the interest rate as a decimal, so 8% should be entered as 0.08.

1. Audio
   Observation

   At 01:49.100-01:55.980 the speaker says the ending amount includes the original investment and the interest all together.

A learner may interpret the final number as interest alone.

The video explains that A is the ending amount, which includes both the principal and the accumulated interest.

1. Audio
   Observation

   At 01:08.000-01:15.120 the speaker says, "instead of eight percent, you're getting two percent every quarter."

A learner may think each quarter earns the full 8%.

The video shows that with quarterly compounding the periodic rate is $0.08/4 = 0.02$, or 2% per quarter.

1. Audio
   Observation

   The speaker defines P immediately after reading the formula.

2. Diagram
   Observation

   The arrow from PRINCIPAL points to P in the formula.

The definition of P is part of interpreting the compound-interest formula.

1. Audio
   Observation

   The speaker defines r as the interest rate converted to a decimal while pointing to the formula.

2. Diagram
   Observation

   The board text under the formula states r = interest rate as a decimal.

The meaning of r is one of the components required to use the formula correctly.

1. Audio
   Observation

   The speaker explains n while referencing both places it appears in the formula.

2. Diagram
   Observation

   The board definition n = # of times you compound per year sits under the formula with arrows to the exponent.

n is a built-in parameter of the compound-interest formula.

1. Audio
   Observation

   The speaker defines t as time in years after defining n.

2. Diagram
   Observation

   The board text t = time in years appears under the formula.

t is another required parameter of the compound-interest formula.

1. Audio
   Observation

   The speaker introduces the example and then substitutes its values into the formula.

2. Formula
   Observation

   The worked line $A = 5000$(1 + .08/4)^{$4*2$} directly instantiates the general formula.

The substitution method is applied to solve the worked example.

1. Audio
   Observation

   The speaker simplifies .08/4 to .02 and explains that this is the rate per quarter.

2. Formula
   Observation

   The base becomes 1.02 in the simplified expression.

Understanding the periodic rate is needed to justify replacing $r/n$ with 0.02 and the base with 1.02.

1. Audio
   Observation

   The speaker says there are eight quarters in two years and therefore eight compounding events.

2. Formula
   Observation

   The exponent $4*2$ is rewritten as 8.

The interpretation of nt as total periods justifies simplifying $4*2$ to 8 in the derivation.

1. Audio
   Observation

   The speaker identifies the final number as the ending amount including principal and interest.

2. Formula
   Observation

   The final written result is $5858.30.

The definition of A explains what the example's final numerical answer represents.

1. Audio
   Observation

   The speaker says the ending amount includes the original investment and the interest all together.

1. Compound interest formula
2. Meaning of the ending amount A

1. Audio
   Observation

   The speaker divides by 4 because the interest is compounded quarterly and explains that this gives 2% every quarter.

1. Compounding frequency n
2. Periodic interest rate meaning
3. Simplifying the periodic growth factor

1. Audio
   Observation

   The speaker says there are eight quarters in two years, so the amount compounds eight times.

1. Interpreting the exponent as total compounding periods
2. Simplifying the worked example before evaluation

1. Audio
   Observation

   The speaker says r is the interest rate but you want to convert that into a decimal.

1. Interest rate r as a decimal
2. Using the percent rate directly instead of converting to decimal

1. Audio
   Observation

   The speaker states the problem, substitutes values, simplifies, and computes the final amount.

2. Formula
   Observation

   The board shows the complete worked solution ending in $5858.30.

1. Worked example: $5000 at 8% compounded quarterly for 2 years
2. Substituting values into the compound-interest formula
3. Simplifying the worked example before evaluation
4. Evaluating $5000(1.02)^8$ and rounding to cents

1. Audio
   Observation

   The speaker says one represents like 100%, the original amount.

1. Meaning of the 1 in the base
2. Compound interest formula