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Chapter 2

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Simple Interest (SI)

Topic: Time Value of MoneyChapter: Chapter 2
SI=P×R×TSI = P \times R \times T
This formula calculates the interest earned on the principal amount only. P is the Principal amount, R is the Annual Interest Rate (as a decimal), and T is the Time period (in years).

Future Value (FV) with Simple Interest

Topic: Time Value of MoneyChapter: Chapter 2
FV=P×(1+R×T)FV = P \times (1 + R \times T)
This calculates the total amount (principal plus simple interest) at the end of the investment period. P is the Principal, R is the Annual Interest Rate (as a decimal), and T is the Time period (in years).

Future Value (FV) with Compound Interest (Annual Compounding)

Topic: Time Value of MoneyChapter: Chapter 2
FV=P×(1+R)NFV = P \times (1 + R)^N
This formula determines the future value of a single sum when interest is compounded annually. P is the Principal, R is the Annual Interest Rate (as a decimal), and N is the Number of Compounding Periods (in years).

Future Value (FV) with Compound Interest (Multiple Compounding Periods per Year)

Topic: Time Value of MoneyChapter: Chapter 2
FV=P×(1+Rm)N×mFV = P \times \left(1 + \frac{R}{m}\right)^{N \times m}
This calculates the future value when interest is compounded 'm' times per year. P is the Principal, R is the Annual Nominal Interest Rate (as a decimal), N is the Number of Years, and m is the Number of Compounding Periods per year.

Present Value (PV) with Compound Interest

Topic: Time Value of MoneyChapter: Chapter 2
PV=FV(1+R)NorPV=FV×(1+R)−NPV = \frac{FV}{(1 + R)^N} \quad \text{or} \quad PV = FV \times (1 + R)^{-N}
This formula calculates the current value of a future sum of money, discounted at a specific rate. FV is the Future Value, R is the Discount Rate (per period, as a decimal), and N is the Number of Periods.

Future Value of an Ordinary Annuity (FVOA)

Topic: Time Value of MoneyChapter: Chapter 2
FVOA=A×[(1+R)N−1R]FVOA = A \times \left[\frac{(1 + R)^N - 1}{R}\right]
This calculates the future value of a series of equal payments made at the end of each period. A is the Annuity Payment per period, R is the Interest Rate per period (as a decimal), and N is the Total Number of Payments.

Present Value of an Ordinary Annuity (PVOA)

Topic: Time Value of MoneyChapter: Chapter 2
PVOA=A×[1−(1+R)−NR]PVOA = A \times \left[\frac{1 - (1 + R)^{-N}}{R}\right]
This calculates the present value of a series of equal payments made at the end of each period. A is the Annuity Payment per period, R is the Discount Rate per period (as a decimal), and N is the Total Number of Payments.

Future Value of an Annuity Due (FVAD)

Topic: Time Value of MoneyChapter: Chapter 2
FVAD=A×[(1+R)N−1R]×(1+R)FVAD = A \times \left[\frac{(1 + R)^N - 1}{R}\right] \times (1 + R)
This calculates the future value of a series of equal payments made at the beginning of each period. A is the Annuity Payment per period, R is the Interest Rate per period (as a decimal), and N is the Total Number of Payments.

Present Value of an Annuity Due (PVAD)

Topic: Time Value of MoneyChapter: Chapter 2
PVAD=A×[1−(1+R)−NR]×(1+R)PVAD = A \times \left[\frac{1 - (1 + R)^{-N}}{R}\right] \times (1 + R)
This calculates the present value of a series of equal payments made at the beginning of each period. A is the Annuity Payment per period, R is the Discount Rate per period (as a decimal), and N is the Total Number of Payments.

Present Value of a Perpetuity (PVP)

Topic: Time Value of MoneyChapter: Chapter 2
PVP=ARPVP = \frac{A}{R}
This calculates the present value of an infinite series of equal payments. A is the Periodic Payment, and R is the Discount Rate per period (as a decimal).

Effective Annual Rate (EAR)

Topic: Time Value of MoneyChapter: Chapter 2
EAR=(1+Rnominalm)m−1EAR = \left(1 + \frac{R_{\text{nominal}}}{m}\right)^m - 1
This formula converts a nominal interest rate with 'm' compounding periods per year into an equivalent annual rate. R_nominal is the Annual Nominal Interest Rate (as a decimal), and m is the Number of Compounding Periods per year.

Standard Form of a Linear Equation (One Variable)

Topic: Linear and Quadratic EquationsChapter: Chapter 2
ax+b=0ax + b = 0
This is the most basic form of a linear equation with one variable 'x'. Here, 'a' is the coefficient of 'x' (where 'a' must not be zero), and 'b' is a constant. The solution 'x' represents the value that satisfies the equation.

Standard Form of a Linear Equation (Two Variables)

Topic: Linear and Quadratic EquationsChapter: Chapter 2
ax+by+c=0ax + by + c = 0
This form represents a straight line in a two-dimensional coordinate system. 'x' and 'y' are variables, while 'a', 'b', and 'c' are constants. 'a' and 'b' cannot both be zero. This is used for problems involving two unknown quantities with a linear relationship.

Slope-Intercept Form of a Linear Equation

Topic: Linear and Quadratic EquationsChapter: Chapter 2
y=mx+cy = mx + c
This form is particularly useful for understanding the characteristics of a straight line. 'm' represents the slope (gradient) of the line, indicating its steepness and direction, and 'c' represents the y-intercept, which is the point where the line crosses the y-axis (i.e., the value of y when x=0).

Standard Form of a Quadratic Equation

Topic: Linear and Quadratic EquationsChapter: Chapter 2
ax2+bx+c=0ax^2 + bx + c = 0
This is the general form of a quadratic equation, where 'x' is the variable, and 'a', 'b', and 'c' are constants. A critical condition is that 'a' must not be zero (a
eq 0), otherwise, it reduces to a linear equation. Quadratic equations are used to model parabolic relationships.

Quadratic Formula (Sridharacharya's Formula)

Topic: Linear and Quadratic EquationsChapter: Chapter 2
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
This formula provides the solutions (roots) for any quadratic equation in its standard form. The ' \pm ' sign indicates that there can be two distinct roots, one repeated root, or two complex conjugate roots, depending on the discriminant. 'a', 'b', and 'c' are the coefficients from the standard form.

Discriminant of a Quadratic Equation

Topic: Linear and Quadratic EquationsChapter: Chapter 2
Δ=b2−4ac\Delta = b^2 - 4ac
The discriminant (often denoted by \Delta) is a crucial part of the quadratic formula that determines the nature of the roots of a quadratic equation. If \Delta > 0, there are two distinct real roots. If \Delta = 0, there is one real and repeated root. If \Delta < 0, there are two distinct complex (imaginary) roots.

Sum of Roots of a Quadratic Equation

Topic: Linear and Quadratic EquationsChapter: Chapter 2
α+β=−ba\alpha + \beta = -\frac{b}{a}
For a quadratic equation ax^2 + bx + c = 0, if \alpha and \beta are its roots, their sum is given by this formula. This relationship is useful for quickly checking roots or forming equations without explicitly finding the roots.

Product of Roots of a Quadratic Equation

Topic: Linear and Quadratic EquationsChapter: Chapter 2
αβ=ca\alpha \beta = \frac{c}{a}
For a quadratic equation ax^2 + bx + c = 0, if \alpha and \beta are its roots, their product is given by this formula. This, along with the sum of roots, is instrumental in constructing quadratic equations from given roots.

Formation of a Quadratic Equation from its Roots

Topic: Linear and Quadratic EquationsChapter: Chapter 2
x2−(α+β)x+αβ=0x^2 - (\alpha + \beta)x + \alpha \beta = 0
If the roots (\alpha and \beta) of a quadratic equation are known, this formula allows for the direct construction of the quadratic equation. It is derived from the relationships between roots and coefficients.

General Linear Equation in One Variable

Topic: Logical ReasoningChapter: Chapter 2
ax+b=0ax + b = 0
This is the standard form of a linear equation where 'x' is the variable, 'a' is the coefficient of x (a \u2260 0), and 'b' is a constant. The goal is to find the value of 'x' that satisfies the equation.

General Quadratic Equation

Topic: Logical ReasoningChapter: Chapter 2
ax2+bx+c=0ax^2 + bx + c = 0
This is the standard form of a quadratic equation where 'x' is the variable, 'a', 'b', and 'c' are coefficients and constants respectively (a \u2260 0). Quadratic equations typically have two roots or solutions.

Quadratic Formula (Shreedharacharya's Formula)

Topic: Logical ReasoningChapter: Chapter 2
x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
This formula is used to find the roots (solutions) of any quadratic equation in the form ax^2 + bx + c = 0. The ' \u00b1 ' sign indicates that there are two possible roots.

Discriminant of a Quadratic Equation

Topic: Logical ReasoningChapter: Chapter 2
D=b2−4acD = b^2 - 4ac
The discriminant 'D' determines the nature of the roots of a quadratic equation. If D > 0, roots are real and distinct. If D = 0, roots are real and equal. If D < 0, roots are imaginary (not real).

Sum of Roots of a Quadratic Equation

Topic: Logical ReasoningChapter: Chapter 2
α+β=−ba\alpha + \beta = -\frac{b}{a}
For a quadratic equation ax^2 + bx + c = 0, if \alpha and \beta are its roots, their sum is given by this formula. This is useful for checking solutions or forming equations.

Product of Roots of a Quadratic Equation

Topic: Logical ReasoningChapter: Chapter 2
αβ=ca\alpha \beta = \frac{c}{a}
For a quadratic equation ax^2 + bx + c = 0, if \alpha and \beta are its roots, their product is given by this formula. This is also useful for checking solutions or forming equations.

Formation of Quadratic Equation from Roots

Topic: Logical ReasoningChapter: Chapter 2
x2−(α+β)x+αβ=0x^2 - (\alpha + \beta)x + \alpha \beta = 0
If the roots (\alpha and \beta) of a quadratic equation are known, this formula can be used to construct the quadratic equation directly.
Chapter 2 Formulas — CA Foundation | Kartavvya