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Pythagorean Theorem for Shortest Distance

Topic: Logical ReasoningChapter: Chapter 10
c=a2+b2c = \sqrt{a^2 + b^2}
This theorem is used to calculate the shortest distance (displacement) between the starting and ending points when movements create a right-angled triangle. 'a' and 'b' represent the net displacements along two perpendicular axes (e.g., North-South and East-West), and 'c' is the shortest distance or hypotenuse.

Net Displacement (Component Method)

Topic: Logical ReasoningChapter: Chapter 10
Net East-West Displacement (x)=∑East movements−∑West movementsNet North-South Displacement (y)=∑North movements−∑South movements\text{Net East-West Displacement (x)} = \sum \text{East movements} - \sum \text{West movements} \\ \text{Net North-South Displacement (y)} = \sum \text{North movements} - \sum \text{South movements}
This method involves breaking down all individual movements into their respective East-West (x-axis) and North-South (y-axis) components. The net displacement in each direction is calculated by summing movements in one direction and subtracting movements in the opposite direction. These net components (x and y) can then be used with the Pythagorean theorem to find the shortest distance from the origin.

Pythagorean Theorem (for shortest distance/displacement)

Topic: Direction SenseChapter: Chapter 10
c2=a2+b2orc=a2+b2c^2 = a^2 + b^2 \quad \text{or} \quad c = \sqrt{a^2 + b^2}
This formula is used to calculate the shortest distance (displacement) between the starting and ending points when movements are made at right angles (e.g., moving East then North). Here, 'c' represents the hypotenuse (the shortest distance), and 'a' and 'b' represent the lengths of the other two sides of the right-angled triangle formed by the movements.

Pythagorean Theorem for Shortest Distance

Topic: Direction SenseChapter: Chapter 10

c=a2+b2c = \sqrt{a^2 + b^2}

Calculates the shortest direct distance (hypotenuse, cc) between the starting and ending points when movements are made along two perpendicular directions (aa and bb). Here, aa represents the net displacement along the North-South axis, and bb represents the net displacement along the East-West axis.

Final Orientation Angle

Topic: Direction SenseChapter: Chapter 10

Final Angle=(Initial Angle+∑Clockwise Turns−∑Anti-clockwise Turns)(mod360)\text{Final Angle} = (\text{Initial Angle} + \sum \text{Clockwise Turns} - \sum \text{Anti-clockwise Turns}) \pmod{360}

Determines the final direction (orientation) in degrees relative to North (0 or 360 degrees) after a series of clockwise and anti-clockwise turns. Standard angle mappings are: North = 0/360 degrees, East = 90 degrees, South = 180 degrees, West = 270 degrees. A right turn typically adds 90 degrees, and a left turn subtracts 90 degrees.

Simple Interest (SI)

Topic: Time Value of MoneyChapter: Chapter 10
SI=P×R×TSI = P \times R \times T
This formula calculates the interest earned only on the principal amount. 'P' is the Principal amount, 'R' is the annual interest rate (as a decimal), and 'T' is the time period in years. It's used for short-term loans or deposits where interest is not compounded.

Future Value (FV) with Simple Interest

Topic: Time Value of MoneyChapter: Chapter 10
FV=P×(1+R×T)FV = P \times (1 + R \times T)
This calculates the total amount (principal + 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

Topic: Time Value of MoneyChapter: Chapter 10
FV=P×(1+i)nFV = P \times (1 + i)^n
This formula determines the value of an investment at a future date, considering that interest is earned on both the principal and previously accumulated interest. 'P' is the Principal amount, 'i' is the interest rate per compounding period (annual rate 'R' divided by number of compounding periods per year 'm', i.e., i = R/m), and 'n' is the total number of compounding periods (time in years 'T' multiplied by 'm', i.e., n = T * m).

Present Value (PV) with Compound Interest

Topic: Time Value of MoneyChapter: Chapter 10
PV=FV(1+i)norPV=FV×(1+i)−nPV = \frac{FV}{(1 + i)^n} \quad \text{or} \quad PV = FV \times (1 + i)^{-n}
This formula calculates the current value of a future sum of money, discounted at a specific interest rate. 'FV' is the Future Value, 'i' is the interest rate per compounding period, and 'n' is the total number of compounding periods. It's used to determine how much needs to be invested today to achieve a certain future sum.

Effective Annual Rate (EAR)

Topic: Time Value of MoneyChapter: Chapter 10
EAR=(1+Rm)m−1EAR = \left(1 + \frac{R}{m}\right)^m - 1
This formula converts a nominal annual interest rate (R) compounded 'm' times per year into an equivalent annual rate that accounts for the effect of compounding. It helps in comparing different investment options with varying compounding frequencies.

Future Value of an Ordinary Annuity (FVOA)

Topic: Time Value of MoneyChapter: Chapter 10
FVOA=A×[(1+i)n−1i]FVOA = A \times \left[\frac{(1 + i)^n - 1}{i}\right]
This calculates the future value of a series of equal payments ('A') made at the end of each period for 'n' periods, compounded at an interest rate 'i' per period. An ordinary annuity assumes payments occur at the end of each period.

Future Value of an Annuity Due (FVAD)

Topic: Time Value of MoneyChapter: Chapter 10
FVAD=A×[(1+i)n−1i]×(1+i)FVAD = A \times \left[\frac{(1 + i)^n - 1}{i}\right] \times (1 + i)
This calculates the future value of a series of equal payments ('A') made at the beginning of each period for 'n' periods, compounded at an interest rate 'i' per period. An annuity due assumes payments occur at the beginning of each period, hence the extra compounding factor (1+i).

Present Value of an Ordinary Annuity (PVOA)

Topic: Time Value of MoneyChapter: Chapter 10
PVOA=A×[1−(1+i)−ni]PVOA = A \times \left[\frac{1 - (1 + i)^{-n}}{i}\right]
This formula determines the current value of a series of equal payments ('A') to be received at the end of each period for 'n' periods, discounted at an interest rate 'i' per period. It's used to value streams of future payments.

Present Value of an Annuity Due (PVAD)

Topic: Time Value of MoneyChapter: Chapter 10
PVAD=A×[1−(1+i)−ni]×(1+i)PVAD = A \times \left[\frac{1 - (1 + i)^{-n}}{i}\right] \times (1 + i)
This calculates the current value of a series of equal payments ('A') to be received at the beginning of each period for 'n' periods, discounted at an interest rate 'i' per period. The additional (1+i) factor accounts for the earlier receipt of payments compared to an ordinary annuity.

Present Value of a Perpetuity (PVP)

Topic: Time Value of MoneyChapter: Chapter 10
PVP=AiPVP = \frac{A}{i}
This formula calculates the present value of an infinite series of equal payments ('A') received at the end of each period, discounted at an interest rate 'i' per period. It's used for valuing assets that are expected to generate cash flows indefinitely.

Amount (Simple Interest)

Topic: Time Value of MoneyChapter: Chapter 10
A=P+SIorA=P(1+R×T100)A = P + SI \quad \text{or} \quad A = P \left(1 + \frac{R \times T}{100}\right)
Calculates the total amount (Principal + Simple Interest) after a given period. A = Amount, P = Principal, R = Annual Rate of Interest (in percentage), T = Time period (in years).

Compound Interest (CI)

Topic: Time Value of MoneyChapter: Chapter 10
CI=A−PCI = A - P
Calculates the compound interest earned by subtracting the principal from the total compound amount. A = Compound Amount, P = Principal.

Compound Amount (A) - Annually Compounded

Topic: Time Value of MoneyChapter: Chapter 10
A=P(1+R100)TA = P \left(1 + \frac{R}{100}\right)^T
Calculates the total amount (Principal + Compound Interest) when interest is compounded annually. P = Principal, R = Annual Rate of Interest (in percentage), T = Time period (in years).

Compound Amount (A) - Compounded 'n' times a year

Topic: Time Value of MoneyChapter: Chapter 10
A=P(1+R100n)nTA = P \left(1 + \frac{R}{100n}\right)^{nT}
Calculates the total amount when interest is compounded 'n' times in a year. P = Principal, R = Annual Rate of Interest (in percentage), n = Number of compounding periods per year, T = Time period (in years).

Perpetuity (Present Value)

Topic: Time Value of MoneyChapter: Chapter 10
PV=PiPV = \frac{P}{i}
Calculates the present value of an infinite series of equal payments (P) starting from the end of the first period. P = Payment per period, i = Interest rate per period (as a decimal).
Chapter 10 Formulas — CA Foundation | Kartavvya