Simple and Compound Interest

Simple Interest: Definition, Formula, and Calculations

In financial transactions involving borrowing or lending money, an additional sum is usually paid by the borrower to the lender as a charge for using the money for a specific period. This additional sum is called Interest.

The initial amount of money borrowed or lent is called the Principal (P).

The interest is typically calculated at a certain percentage of the principal amount per unit of time. This percentage is called the Rate of Interest (R). The rate is commonly quoted as 'per annum' (p.a.), meaning per year.

The duration for which the money is borrowed or lent is called the Time Period (T).

At the end of the time period, the borrower pays back the original principal amount along with the accumulated interest. The total sum paid back is called the Amount (A).

Simple Interest (SI)

Simple Interest (SI) is a type of interest calculation where the interest for each period is calculated solely on the original principal amount. The interest earned in previous periods is not added to the principal for calculating the interest in the current period. This results in the interest amount being the same for each unit of time (e.g., same interest every year if the rate is per annum).

Formula for Simple Interest

The Simple Interest is directly proportional to the Principal amount, the Rate of Interest, and the Time Period. The formula for calculating Simple Interest is:

Where:

• P = Principal amount
• R = Rate of Interest per annum (as a percentage, e.g., if rate is 8%, use 8)
• T = Time Period (in years)

Important Note on Units: For the formula $\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}$ to work correctly, the units of Rate (R) and Time (T) must be compatible. If the Rate is given 'per annum', the Time must be in 'years'. If the Time is given in months or days, convert it to years:

• Time in months: If time is $M$ months, $T = \frac{M}{12}$ years.
• Time in days: If time is $D$ days, $T = \frac{D}{365}$ years (use 366 for a leap year if specified, but typically 365 is used unless stated otherwise).

Formula for Amount (A) with Simple Interest

The total Amount (A) to be repaid is the sum of the Principal (P) and the Simple Interest (SI) earned. Using formula (i) and substituting the formula for SI (ii):

Factor out the Principal (P) from the right side:

Finding Principal, Rate, or Time from the SI Formula

The basic Simple Interest formula (ii) $\text{SI} = \frac{\text{P} \times \text{R} \times \text{T}}{100}$ can be rearranged to find any of the variables (P, R, or T) if the other three (SI, and the remaining two from P, R, T) are known.

• To find Principal (P):

  $\boldsymbol{\text{P} = \frac{\text{SI} \times 100}{\text{R} \times \text{T}}}$

  ... (iv)

• To find Rate (R) per annum:

  $\boldsymbol{\text{R} = \frac{\text{SI} \times 100}{\text{P} \times \text{T}}}$

  ... (v)

• To find Time (T) in years:

  $\boldsymbol{\text{T} = \frac{\text{SI} \times 100}{\text{P} \times \text{R}}}$

  ... (vi)

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Compound Interest: Definition and Comparison with Simple Interest

Compound Interest (CI) is a type of interest calculation where the interest earned in each period is added to the principal amount for the next period. This accumulated interest from previous periods then also starts earning interest in subsequent periods. This process of adding the interest back to the principal is called compounding, and it leads to interest earning 'interest on interest'.

Compound interest is the standard method used in most financial transactions today, including savings accounts, loans, and investments, because it reflects the true growth of money over time.

Comparison between Simple Interest and Compound Interest

The fundamental difference between Simple Interest (SI) and Compound Interest (CI) lies in the base on which the interest is calculated in subsequent periods:

Crucially, for a time period of exactly one year, the Simple Interest and Compound Interest calculated annually on the same principal and rate are equal. The difference between SI and CI becomes significant and increases as the time period extends beyond one year and the number of compounding periods increases.

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Compound Interest Formula and Calculation (Annual Compounding)

Calculating Compound Interest (CI) step-by-step for each period can be tedious, especially for longer time durations. Fortunately, there is a general formula that directly calculates the total Amount after compounding, from which the Compound Interest can be found.

Derivation of the Formula for Amount (Annual Compounding)

Let the Principal amount be P, the rate of interest per annum be R% (meaning $\frac{R}{100}$ as a fraction), and the number of years be $n$. We assume the interest is compounded annually (once a year).

At the end of the 1st year:

The interest for the first year is calculated on the original principal, P. Using the Simple Interest formula for 1 year:

The Amount at the end of the 1st year ($A_1$) is the sum of the principal and the interest earned in the first year:

At the end of the 2nd year:

For the second year, the principal is the amount at the end of the 1st year, which is $A_1 = \text{P} \left(1 + \frac{\text{R}}{100}\right)$. The interest for the second year is calculated on this new principal $A_1$ at the rate R% for 1 year.

The Amount at the end of the 2nd year ($A_2$) is the sum of the principal for the 2nd year ($A_1$) and the interest earned in the 2nd year:

Now, substitute the expression for $A_1$ from equation (i) into equation (ii):

At the end of the 3rd year:

The principal for the third year is the amount at the end of the 2nd year, $A_2 = \text{P} \left(1 + \frac{\text{R}}{100}\right)^2$. The interest for the third year is calculated on $A_2$ at rate R% for 1 year.

The Amount at the end of the 3rd year ($A_3$) is $A_2 + \text{Interest (Year 3)}$:

Substitute the expression for $A_2$ from equation (iii) into equation (iv):

Observing the pattern in equations (i), (iii), and (v), we can generalize the formula for the Amount (A) at the end of 'n' years, when compounded annually:

Where:

• P = Principal amount
• R = Rate of Interest per annum (as a percentage)
• n = Number of years (or the total number of compounding periods when compounded annually)

This formula is for situations where interest is compounded annually.

Formula for Compound Interest (CI)

The Compound Interest (CI) earned is the difference between the total Amount (A) at the end of the period and the original Principal (P). Using the definition of Amount from equation (i) in the previous section:

Substituting the formula for A (vi) into equation (vii):

We can factor out P:

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Compound Interest for Different Compounding Periods (Half-yearly, Quarterly, etc.)

Interest is not always compounded annually. The frequency of compounding can be different, such as half-yearly, quarterly, monthly, or even daily. When the interest is compounded more frequently than once a year, the standard annual compounding formula needs to be adjusted to reflect the shorter compounding intervals.

Adjustments to Rate and Time for Different Compounding Periods

Let the annual rate of interest be R% per annum, and the total time period be $n$ years.

If interest is compounded k times in a year (e.g., $k=2$ for half-yearly, $k=4$ for quarterly, $k=12$ for monthly):

• The annual rate R% is divided equally among the $k$ periods within a year. So, the rate of interest for each compounding period becomes $\frac{R}{k}\%$.
• The total time period of $n$ years will contain $n \times k$ compounding periods.

We use these adjusted values for the rate and number of periods in the compound interest formula derived for annual compounding (Amount $= \text{P} \left(1 + \frac{\text{Rate}}{100}\right)^{\text{Number of periods}}$).

Substituting the adjusted rate and number of periods:

This can also be written as:

Where:

• P = Principal amount
• R = Annual Rate of Interest (as a percentage)
• n = Total Time Period (in years)
• k = Number of times interest is compounded per year

The Compound Interest (CI) is still given by $\text{CI} = \text{A} - \text{P}$.

Specific Compounding Frequencies:

• Half-yearly Compounding: Interest is compounded every six months. This means there are $k=2$ compounding periods in a year.

  Rate per period = $\frac{\text{Annual Rate}}{2} = \frac{R}{2}\%$

  Number of periods = $2 \times (\text{Number of years}) = 2n$

  Formula for Amount: $\boldsymbol{\text{A} = \text{P} \left(1 + \frac{\text{R}/2}{100}\right)^{2n}}$

  Example: For a time of 1.5 years, $n=1.5$. Number of periods $= 2 \times 1.5 = 3$.
• Quarterly Compounding: Interest is compounded every three months. This means there are $k=4$ compounding periods in a year.

  Rate per period = $\frac{\text{Annual Rate}}{4} = \frac{R}{4}\%$

  Number of periods = $4 \times (\text{Number of years}) = 4n$

  Formula for Amount: $\boldsymbol{\text{A} = \text{P} \left(1 + \frac{\text{R}/4}{100}\right)^{4n}}$

  Example: For a time of 9 months ($0.75$ years), $n=0.75$. Number of periods $= 4 \times 0.75 = 3$.
• Monthly Compounding: Interest is compounded every month. This means there are $k=12$ compounding periods in a year.

  Rate per period = $\frac{\text{Annual Rate}}{12} = \frac{R}{12}\%$

  Number of periods = $12 \times (\text{Number of years}) = 12n$

  Formula for Amount: $\boldsymbol{\text{A} = \text{P} \left(1 + \frac{\text{R}/12}{100}\right)^{12n}}$

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When Time is a Fraction or Mixed Number (Annual Compounding)

If the total time period is not a whole number of years, say it is $n$ whole years and a fraction $\frac{a}{b}$ of a year (i.e., time $= n \frac{a}{b}$ years), and the interest is compounded annually, the formula needs a slight modification. The amount is compounded for the whole number of years $n$, and then simple interest is calculated on the amount accumulated at the end of $n$ years for the remaining fractional part of the year $\frac{a}{b}$.

Amount at the end of $n$ whole years $= \text{P} \left(1 + \frac{\text{R}}{100}\right)^n$. Let this amount be $A_n$.

Interest for the fractional part $\frac{a}{b}$ of a year is calculated as Simple Interest on $A_n$ for $\frac{a}{b}$ years at rate R% p.a.

The total Amount at the end of $n \frac{a}{b}$ years is $A_n + \text{Interest for fractional part}$.

Substitute the expression for $A_n$:

The Compound Interest (CI) is $A - P$.

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Applications of Compound Interest (Growth and Depreciation)

The compound interest concept is a specific instance of a broader mathematical model used to describe exponential growth or decay. This model is applicable in various real-world scenarios where a quantity changes by a fixed percentage over regular intervals. Common applications include population growth, the appreciation (increase in value) of assets like property, and the depreciation (decrease in value) of assets like machinery or vehicles.

Growth Applications (Compound Growth)

If a quantity increases at a constant percentage rate over a fixed period, its growth follows the pattern of compound interest. This is often referred to as compound growth.

Let $P_0$ be the initial quantity (or value) at the start, R be the rate of growth per period (expressed as a percentage), and $n$ be the number of periods over which the growth occurs.

The formula for the quantity after $n$ periods of growth is analogous to the Amount formula in compound interest:

Here, $P_0$ is the initial value (like the Principal), R is the growth rate per period (like the annual interest rate), and $n$ is the number of growth periods (like the number of years). The $(1 + \frac{R}{100})$ factor represents the multiplier for each period's growth.

Examples where this applies include:

• Population growth at a constant annual rate.
• Increase in the value of property or land at a fixed annual percentage rate.
• Growth of bacteria or other populations under ideal conditions.

Depreciation Applications (Exponential Decay)

Depreciation is the decrease in the value of an asset over time, usually due to usage, age, or obsolescence. If an asset depreciates at a fixed percentage rate per period, its value decreases exponentially. This is similar to negative compound interest or exponential decay.

Let $V_0$ be the initial value of the asset, R be the rate of depreciation per period (expressed as a percentage), and $n$ be the number of periods over which the depreciation occurs.

The formula for the value of the asset after $n$ periods of depreciation is:

The minus sign in the formula signifies that the value is decreasing. The $(1 - \frac{R}{100})$ factor represents the multiplier for each period's depreciation, resulting in a value less than the previous period's value.

Examples where this applies include:

• Depreciation in the value of machinery, vehicles, or electronic gadgets.
• Decrease in the value of certain investments or stocks.

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Solving Problems on Simple and Compound Interest

This section focuses on applying the formulas and concepts of Simple Interest (SI) and Compound Interest (CI) to solve various types of problems. It includes finding interest or amount, determining principal, rate, or time, and dealing with the difference between CI and SI.

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