Standard XI Mathematics & Statistics Commerce Part 2

Partition Values

• Definition and Concept

  • Partition values are values that divide a total set of observations, when arranged in ascending order of magnitude, into a specified number of equal parts.
  • The procedure of dividing the data into equal parts is called partitioning.

• Median

  • Median is a special type of partition value that divides an ordered array of data into two equal parts, such that 50% of the observations lie below it and 50% lie above it.
  • Ungrouped Data Formulae:
  • For an odd number of observations ($n$):       $\text{Median} = \left(\frac{n+1}{2}\right)^{\text{th}} \text{ observation}$
  • For an even number of observations ($n$):       $\text{Median} = \frac{\left(\frac{n}{2}\right)^{\text{th}} \text{ observation} + \left(\frac{n+2}{2}\right)^{\text{th}} \text{ observation}}{2}$
  • Grouped Continuous Data Formula:     $\text{Median} = L + \frac{h}{f} \left(\frac{N}{2} - \text{c.f.}\right)$
  • $L$: Lower boundary of the median class
  • $h$: Class width/length of the median class
  • $f$: Frequency of the median class
  • $\text{c.f.}$: Less-than cumulative frequency of the class preceding the median class
  • $N$: Total frequency ($N = \sum f$)
  • Median class is the class containing the $\left(\frac{N}{2}\right)^{\text{th}}$ observation (first class whose cumulative frequency $\ge \frac{N}{2}$).

• Quartiles ($Q_1, Q_2, Q_3$)

  • Quartiles divide the data into 4 equal parts, each containing 25% of the total observations.
  • $Q_1$ (First or Lower Quartile): 25% of observations are below $Q_1$ and 75% are above $Q_1$.
  • $Q_2$ (Second Quartile / Median): 50% of observations are below $Q_2$ and 50% are above $Q_2$.
  • $Q_3$ (Third or Upper Quartile): 75% of observations are below $Q_3$ and 25% are above $Q_3$.
  • Ungrouped Data Formula:     $Q_i = \text{value of } i\left(\frac{n+1}{4}\right)^{\text{th}} \text{ observation}, \quad i \in \{1, 2, 3\}$
  • Fractional Position Interpolation Rule: If $i\left(\frac{n+1}{4}\right) = k + m$ (where $k$ is an integer and $0 < m < 1$), then:       $Q_i = \text{value of } k^{\text{th}} \text{ observation} + m \times (\text{value of } (k+1)^{\text{th}} \text{ observation} - \text{value of } k^{\text{th}} \text{ observation})$
  • Grouped Data Formula:     $Q_i = L + \frac{h}{f} \left(\frac{iN}{4} - \text{c.f.}\right), \quad i \in \{1, 2, 3\}$
  • $Q_i$ class is the class containing the $\left(\frac{iN}{4}\right)^{\text{th}}$ observation.

• Deciles ($D_1, D_2, \dots, D_9$)

  • Deciles divide the data into 10 equal parts, each containing 10% of the observations.
  • Ungrouped Data Formula:     $D_i = \text{value of } i\left(\frac{n+1}{10}\right)^{\text{th}} \text{ observation}, \quad i \in \{1, 2, \dots, 9\}$
  • Grouped Data Formula:     $D_i = L + \frac{h}{f} \left(\frac{iN}{10} - \text{c.f.}\right), \quad i \in \{1, 2, \dots, 9\}$
  • $D_i$ class is the class containing the $\left(\frac{iN}{10}\right)^{\text{th}}$ observation.

• Percentiles ($P_1, P_2, \dots, P_{99}$)

  • Percentiles divide the data into 100 equal parts, each containing 1% of the total observations.
  • Ungrouped Data Formula:     $P_i = \text{value of } i\left(\frac{n+1}{100}\right)^{\text{th}} \text{ observation}, \quad i \in \{1, 2, \dots, 99\}$
  • Grouped Data Formula:     $P_i = L + \frac{h}{f} \left(\frac{iN}{100} - \text{c.f.}\right), \quad i \in \{1, 2, \dots, 99\}$
  • $P_i$ class is the class containing the $Libraries\left(\frac{iN}{100}\right)^{\text{th}} observation.\n\n* **Key Equivalence Relations**\n * \text{Median} = Q_2 = D_5 = P_{50}\n * Q_1 = P_{25}\n * Q_3 = P_{75}\n * D_1 = P_{10}, \quad D_3 = P_{30}, \quad D_7 = P_{70}\n\n* **Graphical Location of Partition Values (Ogives)**\n * **Less than Ogive**: Plot upper class boundaries on X-axis against less-than cumulative frequencies (l.c.f.) on Y-axis. Join the points with a freehand smooth curve.\n * **More than Ogive**: Plot lower class boundaries on X-axis against more-than cumulative frequencies (m.c.f.) on Y-axis. Join the points with a freehand smooth curve.\n * **Locating Values**:\n * Locate \frac{N}{2}$on Y-axis for Median ($Q_2$),$\frac{iN}{4}$for$Q_i$,$\frac{iN}{10}$for$D_i$, and$\frac{iN}{100}$for$P_i.\n * Draw a line parallel to X-axis from this Y-value to intersect the Ogive at point P.\n * Drop a perpendicular from P to X-axis; the coordinate on X-axis gives the partition value.\n * Alternatively, the X-coordinate of the intersection point of Less-than Ogive and More-than Ogive gives the Median.\n\n# Measures of Dispersion\n\n* **Concept and Definition**\n * An average or measure of central tendency does not reveal the degree of scatter or spread of observations around the central value.\n * According to Spiegel, "The degree to which numerical data tend to spread about an average value is called the variation or dispersion of the data."\n\n* **Range**\n * Simplest measure of dispersion defined as the difference between the largest value (L$) and the smallest value ($S) in a dataset.\n * \text{Range} = L - S\n * For grouped data: L$is the upper limit of the highest class and$S is the lower limit of the lowest class.\n\n* **Quartile Deviation (Semi-Interquartile Range)**\n * Measures the spread of the middle 50% of the data and is unaffected by extreme values or open-ended classes.\n * \text{Interquartile Range} = Q_3 - Q_1\n * \text{Quartile Deviation (Q.D.)} = \frac{Q_3 - Q_1}{2}\n\n* **Variance and Standard Deviation**\n * Based on all observations and measures deviations from the arithmetic mean.\n * **Variance (\sigma^2)**: The arithmetic mean of the squares of deviations of all observations taken from their arithmetic mean.\n * Raw Data:\n      \text{Var}(X) = \sigma^2 = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})^2 = \frac{1}{n} \sum_{i=1}^n x_i^2 - (\bar{x})^2\n * Grouped/Ungrouped Frequency Distribution:\n      \text{Var}(X) = \sigma^2 = \frac{1}{N} \sum_{i=1}^n f_i (x_i - \bar{x})^2 = \frac{1}{N} \sum_{i=1}^n f_i x_i^2 - (\bar{x})^2\n      where N = \sum_{i=1}^n f_i$and$\bar{x} = \frac{1}{N}\sum_{i=1}^n f_i x_i$.
  • Standard Deviation ($\sigma$): The positive square root of the variance.     $\sigma = \sqrt{\text{Var}(X)}$
  • Change of Origin and Scale:
  • If $u_i = \frac{x_i - A}{h}$, where $A$ is assumed mean and $h$ is class width/scale:       $\bar{x} = A + h \bar{u}$$\text{Var}(u) = \sigma_u^2 = \frac{1}{N} \sum f_i u_i^2 - (\bar{u})^2$$\text{Var}(X) = h^2 \cdot \text{Var}(u) \implies \sigma_x = h \cdot \sigma_u$
  • Variance and S.D. are independent of change of origin ($A$), but dependent on change of scale ($h$).

• Combined Standard Deviation

  • For two datasets of sizes $n_1$ and $n_2$, with means $\bar{x}_1, \bar{x}_2$ and standard deviations $\sigma_1, \sigma_2$:
  • Combined Mean:       $\bar{x}_c = \frac{n_1 \bar{x}_1 + n_2 \bar{x}_2}{n_1 + n_2}$
  • Deviations from Combined Mean:       $d_1 = \bar{x}_1 - \bar{x}_c, \quad d_2 = \bar{x}_2 - \bar{x}_c$
  • Combined Standard Deviation:       $\sigma_c = \sqrt{\frac{n_1(\sigma_1^2 + d_1^2) + n_2(\sigma_2^2 + d_2^2)}{n_1 + n_2}}$

• Coefficient of Variation (C.V.)

  • A relative measure of dispersion expressed as a percentage, independent of units of measurement.
  • $\text{C.V.} = \frac{\sigma}{\bar{x}} \times 100$
  • Interpretation: A distribution with a smaller C.V. is more consistent, homogeneous, stable, or compact. A larger C.V. indicates greater variability or heterogeneity.

Skewness

• Concept of Skewness

  • Skewness indicates the lack of symmetry in a frequency distribution and specifies the direction and extent of asymmetry.

• Symmetric Distribution (Zero Skewness)

  • Frequencies are symmetrically distributed around the mean such that values equidistant from the mean have equal frequencies.
  • $\text{Mean} = \text{Median} = \text{Mode}$
  • $(Q_3 - Q_2) = (Q_2 - Q_1) \implies 2Q_2 = Q_1 + Q_3$
  • Skewness = 0.

• Positively Skewed Distribution

  • Frequencies are concentrated at lower values of the variable; the right tail of the curve is longer.
  • $\text{Mean} > \text{Median} > \text{Mode}$
  • $(Q_3 - Q_2) > (Q_2 - Q_1)$

• Negatively Skewed Distribution

  • Frequencies are concentrated at higher values of the variable; the left tail of the curve is longer.
  • $\text{Mode} > \text{Median} > \text{Mean}$
  • $(Q_3 - Q_2) < (Q_2 - Q_1)$

• Karl Pearson's Coefficient of Skewness ($Sk_p$)

  • Based on the difference between mean and mode relative to standard deviation:     $Sk_p = \frac{\text{Mean} - \text{Mode}}{\sigma}$
  • When mode is ill-defined or indeterminate, using the empirical relationship $\text{Mean} - \text{Mode} = 3(\text{Mean} - \text{Median})$:     $Sk_p = \frac{3(\text{Mean} - \text{Median})}{\sigma}$
  • Properties:
  • Unitless relative measure.
  • $Sk_p = 0$ for symmetric, $Sk_p > 0$ for positive skew, $Sk_p < 0$ for negative skew.
  • Most values lie between $-1$ and $1$; theoretical limits are $-3$ and $3$

• Bowley's Coefficient of Skewness ($Sk_b$)

  • Based on quartiles, useful for distributions with open-ended classes:     $Sk_b = \frac{(Q_3 - Q_2) - (Q_2 - Q_1)}{(Q_3 - Q_2) + (Q_2 - Q_1)} = \frac{Q_3 + Q_1 - 2Q_2}{Q_3 - Q_1}$
  • Properties:
  • Unitless relative measure.
  • $Sk_b = 0$ for symmetric, $Sk_b > 0$ for positive skew, $Sk_b < 0$ for negative skew.
  • Limits: $-1 \le Sk_b \le 1$.

Bivariate Frequency Distribution and Chi-Square Statistic

• Bivariate Frequency Distribution

  • Statistical data involving simultaneous measurement of two variables $X$ and $Y$ on each unit of observation is called bivariate data.
  • A two-way table summarizing bivariate data into $m$ classes of $X$ and $n$ classes of $Y$ is called a bivariate frequency table (containing $m \times n$ cells).
  • Cell frequency $f_{ij}$ represents the frequency of observations falling simultaneously in the $i^{\text{th}}$ class of $X$ and $j^{\text{th}}$ class of $Y$
  • Total frequency N = \sum_{i=1}^m \sum_{j=1}^n f_{ij} = \sum f_x = \sum f_y$.\n\n* **Marginal Distributions**\n * Marginal frequency distribution of X$: Individual frequency distribution of$X$obtained by summing frequencies across all rows for each column ($f_i = \sum_j f_{ij}).\n * Marginal frequency distribution of Y$: Individual frequency distribution of$Y$obtained by summing frequencies across all columns for each row ($f_{.j} = \sum_i f_{ij}).\n\n* **Conditional Distributions**\n * Frequency distribution of one variable for a fixed value or class of the other variable.\n * Conditional distribution of X$given$Y = y_j$: Uses row$y_j$frequencies with row total$f_{.j}.\n * Conditional distribution of Y$given$X = x_i$: Uses column$x_i$frequencies with column total$f_{i.}.\n\n* **Categorical Variables and Contingency Tables**\n * A categorical variable takes non-numerical qualitative attributes (e.g., Gender, Blood Group, Preferences).\n * A two-way frequency table representing two categorical variables is called a contingency table.\n\n* **Chi-Square Statistic (\chi^2)**\n * Used to test association or independence between two categorical variables.\n * Formula:\n    \chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}\n * O_{ij}$: Observed frequency in cell$(i, j)\n * E_{ij}$: Expected frequency in cell$(i, j)\n * Expected Frequency Formula:\n    E_{ij} = \frac{R_i \times C_j}{N}\n    where R_i$is the$i^{\text{th}}$row total,$C_j$is the$j^{\text{th}}$column total, and$N is the grand total.\n * \chi^2 \ge 0 always.\n\n# Correlation\n\n* **Concept of Correlation**\n * Correlation measures the degree of linear association or relationship between two variables X$and$Y in a bivariate distribution.\n\n* **Covariance**\n * Joint variation between two variables X$and$Y:\n    \text{Cov}(X, Y) = \frac{1}{n} \sum_{i=1}^n (x_i - \bar{x})(y_i - \bar{y}) = \frac{1}{n} \sum_{i=1}^n x_i y_i - \bar{x}\bar{y}\n * Properties of Covariance:\n 1. \text{Cov}(X, Y) = \text{Cov}(Y, X)\n 2. \text{Cov}(X, c) = 0$for any constant$c\n 3. \text{Cov}(X, X) = \text{Var}(X)\n 4. Change of origin and scale: If U = \frac{X - a}{h}$and$V = \frac{Y - b}{k}, then:\n       \text{Cov}(X, Y) = h k \cdot \text{Cov}(U, V)\n\n* **Karl Pearson's Coefficient of Correlation (r_{xy})**\n * Definition:\n    r_{xy} = \frac{\text{Cov}(X, Y)}{\sigma_x \sigma_y}\n * Computational Direct Formula:\n    r_{xy} = \frac{n \sum x_i y_i - (\sum x_i)(\sum y_i)}{\sqrt{\left[n \sum x_i^2 - (\sum x_i)^2\right] \left[n \sum y_i^2 - (\sum y_i)^2\right]}}\n * Step-deviation / Change of variables formula:\n    r_{uv} = \frac{n \sum u_i v_i - (\sum u_i)(\sum v_i)}{\sqrt{\left[n \sum u_i^2 - (\sum u_i)^2\right] \left[n \sum v_i^2 - (\sum v_i)^2\right]}}\n    where u_i = \frac{x_i - a}{h}$and$v_i = \frac{y_i - b}{k}.\n * r_{uv} = r_{xy}$if$h$and$k have the same algebraic sign.\n * r_{uv} = -r_{xy}$if$h$and$k have opposite algebraic signs.\n * Properties of Correlation Coefficient:\n 1. -1 \le r_{xy} \le 1\n 2. r_{xy} = r_{yx}\n 3. Independent of change of origin and scale (magnitude preserved).\n\n* **Scatter Diagram and Types of Correlation**\n * **Perfect Positive Correlation (r = +1)**: Points lie exactly on a single straight line rising from left to right.\n * **High Degree Positive Correlation (0.8 < r < 1)**: Points form a narrow band rising from left to right.\n * **Low Degree Positive Correlation (0 < r < 0.8)**: Points form a wide band rising from left to right.\n * **Perfect Negative Correlation (r = -1)**: Points lie exactly on a single straight line falling from left to right.\n * **High Degree Negative Correlation (-1 < r < -0.8)**: Points form a narrow band falling from left to right.\n * **Low Degree Negative Correlation (-0.8 < r < 0)**: Points form a wide band falling from left to right.\n * **Zero / No Correlation (r = 0)**: Points are scattered randomly without any discernible trend or form a symmetric non-linear pattern.\n\n# Permutations and Combinations\n\n* **Fundamental Principles of Counting**\n * **Addition Principle**: If one event can occur in m$ways and a second event with no common outcomes can occur in$n$ways, then either of the two events can occur in$m + n ways.\n * **Multiplication Principle**: If one operation can be carried out in m$ways, and following it a second operation can be carried out in$n$ways, then both operations together can be carried out in$m \times n ways.\n * **Invariance Principle**: The total number of choices or arrangements remains unchanged regardless of the order in which counting steps are performed.\n\n* **Factorial Function (n!)**\n * For a natural number n$,$n!$is the product of all positive natural numbers from$1$to$n:\n    n! = n \times (n-1) \times (n-2) \times \dots \times 2 \times 1\n * Special Values: 0! = 1$,$1! = 1\n * Recurrence relation: n! = n \times (n-1)!\n\n* **Permutations (^nP_r)**\n * An ordered arrangement of r$objects selected from$n distinct objects.\n * **Distinct Objects, No Repetition**:\n    ^nP_r = \frac{n!}{(n-r)!} = n(n-1)(n-2)\dots(n-r+1), \quad r \le n\n * ^nP_n = n!\n * ^nP_0 = 1\n * ^nP_1 = n\n * \frac{^nP_r}{^nP_{r-1}} = n - r + 1\n * **Repetitions Allowed**:\n * Number of arrangements of n$distinct objects taken$r$at a time with repetition =$n^r\n * **Conditional Permutations**:\n * m$specified objects always together:$m! \times (n - m + 1)!\n * m$specified objects never together: Total permutations minus together =$(n - m)(n - m + 1)!\n * Particular object always included: r \cdot ^{n-1}P_{r-1}\n * Particular object never included: ^{n-1}P_r\n * **Objects Not All Distinct**:\n * If among n$objects,$n_1$are of type 1,$n_2$of type 2, ...,$n_k$of type$k:\n      \text{Number of Permutations} = \frac{n!}{n_1! \, n_2! \dots n_k!}\n * **Circular Permutations**:\n * Arrangement of n$distinct objects around a circle =$(n-1)!\n * If clockwise and anti-clockwise arrangements are indistinguishable (e.g., beads in a necklace): \frac{(n-1)!}{2}\n * If m$objects are alike out of$n$:$\frac{(n-1)!}{m!}\n * Arrangement of r$objects from$n around a circle:\n * Distinguishable clockwise/anti-clockwise: \frac{^nP_r}{r}\n * Indistinguishable clockwise/anti-clockwise: \frac{^nP_r}{2r}\n\n* **Combinations (^nC_r)**\n * An unordered selection of r$objects from$n distinct objects.\n * Formula:\n    ^nC_r = \frac{n!}{r!(n-r)!} = \frac{^nP_r}{r!}\n * **Properties of Combinations**:\n 1. ^nC_r = ^nC_{n-r}\n 2. ^nC_0 = ^nC_n = 1\n 3. ^nC_1 = n\n 4. If ^nC_x = ^nC_y$, then either$x = y$or$x + y = n\n 5. Pascal's Identity: ^nC_r + ^nC_{r-1} = ^{n+1}C_r\n 6. Total number of subsets of a set of size n$:$\sum_{r=0}^n ^nC_r = ^nC_0 + ^nC_1 + \dots + ^nC_n = 2^n\n 7. Sum of even/odd combination terms: ^nC_0 + ^nC_2 + ^nC_4 + \dots = ^nC_1 + ^nC_3 + ^nC_5 + \dots = 2^{n-1}\n 8. ^nC_r = \frac{n}{r} \cdot ^{n-1}C_{r-1}\n\n# Probability\n\n* **Basic Concepts & Definitions**\n * **Random Experiment**: An experiment whose all possible outcomes are known in advance, but the exact outcome cannot be predicted beforehand.\n * **Sample Space (S$)**: The set of all possible outcomes of a random experiment. Number of elements is$n(S).\n * **Event (A$)**: Any subset of the sample space ($A \subseteq S).\n * **Types of Events**:\n * Elementary/Simple Event: Contains a single sample point (n(A) = 1).\n * Sure Event: Event equal to the whole sample space (A = S \implies P(A) = 1).\n * Impossible Event: Event containing no sample points (A = \emptyset \implies P(A) = 0).\n * Complementary Event (A'$or$A^c$): Set of all sample points in$S$not in$A$.$P(A') = 1 - P(A).\n * Union (A \cup B$): Occurrence of at least one of the events$A$or$B.\n * Intersection (A \cap B$): Simultaneous occurrence of both events$A$and$B.\n * Exhaustive Events: Events A$and$B$such that$A \cup B = S\n * Mutually Exclusive Events: Events A$and$B$such that$A \cap B = \emptyset (cannot occur together).\n\n* **Classical Probability**\n * For equally likely outcomes:\n    P(A) = \frac{n(A)}{n(S)}\n * Axiomatic Rules:\n * 0 \le P(A) \le 1\n * P(\emptyset) = 0, \quad P(S) = 1\n * If A \subseteq B$, then$P(A) \le P(B)\n * P(A \cap B') = P(A) - P(A \cap B)\n * De Morgan's Law in Probability: P(A' \cap B') = P((A \cup B)') = 1 - P(A \cup B)\n\n* **Addition Theorem of Probability**\n * For any two events A$and$B:\n    P(A \cup B) = P(A) + P(B) - P(A \cap B)\n * If A$and$B$are mutually exclusive ($A \cap B = \emptyset):\n    P(A \cup B) = P(A) + P(B)\n\n* **Conditional Probability**\n * Probability of event A$given that event$B$has already occurred ($P(B) > 0):\n    P(A|B) = \frac{P(A \cap B)}{P(B)}\n * Probability of event B$given that event$A$has already occurred ($P(A) > 0):\n    P(B|A) = \frac{P(A \cap B)}{P(A)}\n\n* **Multiplication Theorem**\n * P(A \cap B) = P(A) \cdot P(B|A) = P(B) \cdot P(A|B)\n\n* **Independent Events**\n * Two events A$and$B are independent if occurrence of one does not affect the probability of occurrence of the other.\n * Condition: P(A|B) = P(A)$or$P(B|A) = P(B)\n * Equivalent Condition: P(A \cap B) = P(A) \cdot P(B)\n * If A$and$B are independent, then:\n * A$and$B' are independent.\n * A'$and$B are independent.\n * A'$and$B' are independent.\n\n# Linear Inequations\n\n* **Linear Inequalities**\n * Statements involving algebraic expressions and relation symbols
  • Income Comparison:
  • If A's income is $r\%$ more than B's, B's income is less than A's by $\frac{r}{100+r} \times 100\%$
  • If A's income is $r\%$ less than B's, B's income is more than A's by $\frac{r}{100-r} \times 100\%$
  • Price-Consumption Tradeoff:
  • If price increases by $x\%$ with constant expenditure, consumption must be reduced by $\frac{x}{100+x} \times 100\%$
  • If price decreases by $x\%$ with constant expenditure, consumption can be increased by $\frac{x}{100-x} \times 100\%$

• Profit and Loss

  • Cost Price (C.P.): Purchase price of the article.
  • Selling Price (S.P.): Price at which the article is sold.
  • $\text{Gain / Profit} = \text{S.P.} - \text{C.P.} \quad (\text{when S.P.} > \text{C.P.})$
  • $\text{Loss} = \text{C.P.} - \text{S.P.} \quad (\text{when C.P.} > \text{S.P.})$
  • $\text{Gain \%} = \frac{\text{Gain}}{\text{C.P.}} \times 100$
  • $\text{Loss \%} = \frac{\text{Loss}}{\text{C.P.}} \times 100$
  • $\text{S.P.} = \frac{100 + \text{Gain \%}}{100} \times \text{C.P.} = \frac{100 - \text{Loss \%}}{100} \times \text{C.P.}$
  • $\text{C.P.} = \frac{100}{100 + \text{Gain \%}} \times \text{S.P.} = \frac{100}{100 - \text{Loss \%}} \times \text{S.P.}$
  • Special Cases:
  • If two items are sold at the same price, one at a gain of $x\%$ and the other at a loss of $x\%$, the transaction always results in an overall loss:       $\text{Overall Loss \%} = \left(\frac{x}{10}\right)^2$
  • False Weight Profit:       $\text{Profit \%} = \frac{\text{Error}}{\text{True Value} - \text{Error}} \times 100$

• Interest (Simple and Compound)

  • Simple Interest (S.I.):     $I = \frac{P \cdot n \cdot r}{100}, \quad A = P + I = P\left(1 + \frac{n \cdot r}{100}\right)$     where $P$ = Principal, $n$ = time in years, $r$ = rate per annum.
  • Compound Interest (C.I.):     $A = P\left(1 + \frac{r}{100}\right)^n, \quad \text{C.I.} = A - P$
  • Compounded Half-Yearly: $A = P\left(1 + \frac{r/2}{100}\right)^{2n}$
  • Compounded Quarterly: $A = P\left(1 + \frac{r/4}{100}\right)^{4n}$
  • Fractional Time ($n = k + \frac{a}{b}$ years):       $A = P\left(1 + \frac{r}{100}\right)^k \left(1 + \frac{\frac{a}{b}r}{100}\right)$
  • Different Annual Rates ($R_1, R_2, R_3$):       $A = P\left(1 + \frac{R_1}{100}\right)\left(1 + \frac{R_2}{100}\right)\left(1 + \frac{R_3}{100}\right)$
  • Difference between C.I. and S.I. for 2 years:       $\text{Difference} = P\left(\frac{r}{100}\right)^2$

• Depreciation

  • Monetary value reduction of an asset over time due to wear, tear, and usage.
  • Present Value after $n$ years:     $V_n = V\left(1 - \frac{r}{100}\right)^n$
  • Value $n$ years ago:     $V_{\text{ago}} = \frac{V}{\left(1 - \frac{r}{100}\right)^n}$     where $V$ = initial value, $r$ = rate of depreciation per annum, $n$ = duration in years.

• Partnership

  • Working Partner: Invests capital and manages business operations.
  • Sleeping Partner: Invests capital only.
  • Division of Profit/Loss:
  • Same investment duration: Distributed in ratio of capitals ($X_1 : X_2 : \dots$).
  • Same capital: Distributed in ratio of time periods ($t_1 : t_2 : \dots$).
  • Different capital and time: Distributed in ratio of products of capital and time ($X_1 t_1 : X_2 t_2 : \dots$).

• Goods and Services Tax (GST)

  • Indirect comprehensive tax replacing multiple indirect taxes.
  • Components:
  • CGST (Central GST) - Levied by Central Govt.
  • SGST (State GST) - Levied by State Govt.
  • UTGST (Union Territory GST) - Levied by UT Govt.
  • IGST (Integrated GST) - Levied on inter-state trade by Central Govt.
  • Relationships:     $\text{CGST} = \text{SGST} = \frac{1}{2} \text{GST}$
  • Identification Codes: HSN (Harmonized System of Nomenclature) for goods; SAC (Service Accounting Code) for services.
  • Input Tax Credit (ITC):     $\text{Net GST Payable} = \text{Output Tax (collected on sales)} - \text{ITC (paid on purchases)}$

• Shares and Dividends

  • Face Value (FV) / Nominal Value / Par Value: Printed value on share certificate.
  • Market Value (MV): Open market price at stock exchange.
  • Share Status:
  • At Premium / Above Par: $\text{MV} > \text{FV}$
  • At Par: $\text{MV} = \text{FV}$
  • At Discount / Below Par: $\text{MV} < \text{FV}$
  • Dividend: Annual profit distributed to shareholders, calculated as a percentage of Face Value.     $\text{Annual Dividend per Share} = \text{Dividend \%} \times \text{FV}$$\text{Total Annual Dividend} = \text{Number of Shares} \times \text{Dividend per Share}$
  • Rate of Return / Yield \%:     $\text{Rate of Return \%} = \frac{\text{Annual Income}}{\text{Total Investment}} \times 100 = \frac{\text{Dividend per Share}}{\text{Market Value per Share}} \times 100$
  • Brokerage:
  • When Buying Shares: $\text{Cost per Share} = \text{MV} + \text{Brokerage}$
  • When Selling Shares: $\text{Selling Price per Share} = \text{MV} - \text{Brokerage}$