COMPOUND INTEREST

1. Definition

Compound Interest is the interest calculated on both the original principal and the accumulated interest from previous periods. Unlike Simple Interest, the interest increases after every compounding period because each new interest is added to the principal before calculating the next interest.

Compound Interest is commonly used in banks, savings accounts, investments, loans, mortgages and business finance because it allows money to grow faster over time.

Principal (P)

The Principal is the original amount of money borrowed or invested before any compound interest is added.

Example

If Grace deposits $8,000 into a savings account, then:

Principal (P) = $8,000

Amount (A)

The Amount is the total money after compound interest has been added to the principal.

Example

If an investment grows from $5,000 to $6,250, then:

Amount (A) = $6,250

Compound Interest (CI)

Compound Interest is the extra money earned or paid after interest is repeatedly added to the principal during each compounding period.

Example

If an investment of $10,000 grows to $12,500, then:

Compound Interest = $12,500 − $10,000 = $2,500

Rate (r)

The Annual Interest Rate is the percentage of interest charged or earned on the principal per year.

Annual Interest Rate (r): The nominal interest rate per year, usually expressed as a percentage per annum (% p.a.), but must be converted to a decimal when used in this formula:

A = P ( 1 + r n ) ^nt

Example

If a bank pays 5% compound interest every year, then:

Percentage Rate: 5% per annum
Formula Value (r): 0.05 (calculated as 5 ÷ 100)

Periodic Interest Rate (r/n): The actual interest rate applied during each individual compounding period (for example, if the rate is 5% compounded monthly, the periodic rate used inside the brackets is 0.05 ÷ 12).

Time (t)

Time is the total period or duration for which the money is borrowed or invested.

It is usually measured in years unless stated otherwise.

Example

If money is invested for 36 months, then:

Time (t) = 36 ÷ 12 = 3 years

2. Formula

The Compound Interest Formula is:

Compound Interest Formula

A = P ( 1 + r n ) ^nt

Where:

A = Total Accrued Amount (Final Balance)
P = Principal Amount (Initial Investment/Loan)
r = Annual Interest Rate (as a decimal, e.g., 0.05 for 5%)
n = Number of compounding periods per year (e.g., 12 for monthly)
t = Number of years the money is invested for

After finding the Amount, the Compound Interest is calculated using the formula:

CI = A − P

where :

CI = Compound Interest

A = Amount

P = Principal

Relationship Between Principal, Amount and Compound Interest

The Amount is obtained by adding the Compound Interest to the Principal.

Example

Principal = $8,000

Compound Interest = $1,724

Amount = Principal + Compound Interest

Amount = $8,000 + $1,724

Amount = $9,724

Worked Examples

Example 1

Calculate the Compound Interest and the Final Amount on a principal of $8,000 invested at 5% per annum, compounded annually for 3 years.

Solution

Step 1: Identify the given values.

P = $8,000

r = 5% = 0.05

n = 1 (Compounded annually)

t = 3 years

These are the quantities needed for the calculation.

Step 2: Write the Compound Interest formula.

A = P ( 1 + r n ) ^nt

This formula calculates the final amount.

Step 3: Substitute the values.

A = 8,000 ( 1 + 0.05 1 ) ^{1 × 3}

The known values are substituted correctly.

Step 4: Simplify.

A = 8000(1.05)³

A = 8000 × 1.157625

A = $9,261

This is the total accumulated amount.

Step 5: Calculate the Compound Interest.

CI = A − P

CI = 9261 − 8000

CI = $1,261

This is the compound interest earned.

Final Answers

Amount = $9,261

Compound Interest = $1,261

Example 2

A customer deposits $12,000 into a savings account paying 6% per annum, compounded annually. Calculate the Final Amount and Compound Interest after 4 years.

Solution

Step 1: Identify the given values.

P = $12,000

r = 6% = 0.06

n = 1

t = 4 years

These are the quantities needed for the calculation.

Step 2: Write the Compound Interest formula.

A = P ( 1 + r n ) ^nt

This formula calculates the total amount.

Step 3: Substitute the values into the formula..

A = 12,000 ( 1 + 0.06 1 ) ^{1 × 4}

Step 4: Simplify.

A = 12000(1.262477)

A = $15,150

This is the accumulated amount.

Step 5: Find the Compound Interest.

CI = 15150 − 12000

CI = $3,150

This is the compound interest earned.

Final Answers

Amount = $15,150

Compound Interest = $3,150

Example 3

An investment of $15,000 grows to $18,232 after 4 years, compounded annually. Calculate the Compound Interest earned.

Solution

Step 1: Identify the given values.

P = $15,000

A = $18,232

We are required to find the Compound Interest.

Step 2: Write the formula.

CI = A − P

This formula calculates the compound interest earned.

Step 3: Substitute the values.

CI = 18,232 − 15,000

Step 4: Simplify.

CI = $3,232

This is the compound interest earned.

Final Answer

Compound Interest = $3,232

Example 4

A principal of $20,000 is invested at 8% per annum, compounded quarterly for 2 years. Calculate the Final Amount and the Compound Interest.

Solution

Step 1: Identify the given values.

P = $20,000

r = 8% = 0.08

n = 4 (Compounded quarterly)

Note: A year has 4 quarters (Jan–Mar, Apr–Jun, Jul–Sep, Oct–Dec); So, the interest is compounded 4 times per year.

t = 2 years

These are the quantities needed for the calculation.

Step 2: Write the Compound Interest formula.

A = P ( 1 + r n ) ^nt

Step 3: Substitute the values.

A = 20,000 ( 1 + 0.08 4 ) ^{4 × 2}

The known values are substituted correctly.

Step 4: Simplify.

A = 20,000(1.02)⁸

A = 20,000 × 1.171659

A = $23,433.18

This is the total accumulated amount.

Step 5: Calculate the Compound Interest.

CI = A − P

CI = 23,433.18 − 20,000

CI = $3,433.18

This is the compound interest earned.

Final Answers

Amount = $23,433.18

Compound Interest = $3,433.18

Example 5

A principal of $10,000 grows to $12,155.06 after 4 years, compounded annually. Determine the annual rate of interest.

Solution

Step 1: Identify the given values.

P = $10,000

A = $12,155.06

n = 1 (Compounded annually)

t = 4 years

We are required to find the annual interest rate.

Step 2: Write the Compound Interest formula.

A = P ( 1 + r n ) ^nt

This formula relates the amount to the principal, rate and time.

Step 3: Make r the subject of the formula.

Step-by-Step Derivation to Make r the Subject

Step 1: Start with the standard compound interest formula.

A = P ( 1 + r n ) ^nt

Step 2: Divide both sides by the Principal (P) to isolate the bracket.

A P = ( 1 + r n ) ^nt

Step 3: Eliminate the exponent (nt) by raising both sides to the power of 1/nt.

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

Step 4: Subtract 1 from both sides to isolate the fraction term containing r.

( A P ) ^{1
nt} − 1 = r n

Step 5: Multiply the entire opposite side by n to isolate r completely.

r = n [ ( A P ) ^{1
nt} − 1 ]

This rearranged formula allows us to calculate the annual rate.

Step 4: Substitute the values.

r = [ ( 12,155.06 10,000 ) ^{1
4} − 1 ]

The known values are substituted correctly.

Step 5: Simplify.

r = ( 1.215506 ) ^{1
4} − 1

r = 1.05 − 1

r = 0.05

r = 0.05 × 100% = 5%

This is the annual compound interest rate.

Final Answer

Rate = 5% per annum

Sample Questions

Attempt the following Questions:

(i) A local contractor deposits a sum of money into a business asset portfolio yielding an annual interest rate of 6% compounded semi-annually. If the total accumulated balance grows to exactly $11,255.09 at the end of 2 years, determine the original principal amount that was invested.

(ii) Mary secures an enterprise development lease of $15,000 from a commercial financial provider to upgrade her production line machinery. If the institution charges interest compounded quarterly over a duration of 3 years, causing her total payoff amount to be $19,023.60, calculate the annual rate of interest applied.

(iii) John places a premium principal investment of $8,000 into a high-yield agricultural project certificate that promises a fixed interest rate of 12% per annum compounded monthly. Calculate how long, in total years, John must lock away his money for the total value of the asset portfolio to reach $11,497.60.

(iv) An entrepreneur takes out an infrastructure development loan of $24,000 from a cooperative bank to construct an irrigation supply system. If the financial contract establishes a compound interest rate of 8% per annum compounded quarterly, calculate the standalone compound interest amount the borrower will accumulate over a timeframe of 18 months.

(v) A student allocates a legacy saving gift of $3,500 into a specialized educational trust fund. If the capital account yields an interest framework running at an annual rate of 4.5% compounded semi-annually, find the total compound interest earned by the account over a period of 4 years.

Check The Answers Below:

TAKE NOTE:

Always write down the correct standard formula and explicitly identify your known variables before performing any algebraic substitutions.
Ensure the annual nominal interest rate (r) is converted into a standard operational decimal form before entry into the math blocks (e.g., 5% becomes 0.05).
The Time variable (t) must always be formatted in structural years. If timeline data is presented in months, convert it by dividing by 12 (e.g., 18 months = 1.5 years).
The Principal (P) represents the initial capital foundation baseline borrowed or invested before any successive interest operations accumulate.
The value of n represents how many times interest is compounded each year.
- Compounded Annually: n = 1
- Compounded Semi-Annually (Half-Yearly): n = 2
- Compounded Quarterly: n = 4
- Compounded Monthly: n = 12
The compound interest formula directly resolves the comprehensive integrated Total Amount (A) first. To isolate the standalone interest earnings, you must subtract the original principal balance: CI = A − P.
Unlike Simple Interest setups where returns accumulate uniformly on the baseline principal only, Compound Interest updates the active balance base sequentially across every compounding interval.
Remember the standard compound interest formulas:
A = P ( 1 + r n ) ^nt
r = n [ ( A P ) ^{1
nt} − 1 ]
Always retain correct currency sign markers (e.g., $, £, €) next to your numeric metrics inside final result responses to maintain proper accounting notation standards.
Examine contextual instruction goals meticulously to verify whether the problem requests the total global accrued value amount (A) or strictly the absolute interest premium addition (CI).

COMPOUND INTEREST

1. Definition

Principal (P)

Example

Amount (A)

Example

Compound Interest (CI)

Example

Rate (r)

Time (t)

Example

2. Formula

Compound Interest Formula

Relationship Between Principal, Amount and Compound Interest

Example

Worked Examples

Example 1

Solution

Final Answers

Example 2

Solution

Final Answers

Example 3

Solution

Final Answer

Example 4

Solution

Final Answers

Example 5

Solution

Step-by-Step Derivation to Make r the Subject

Final Answer

Sample Questions

Check The Answers Below:

Answers:

TAKE NOTE:

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