Quantitative Aptitude

Cheatsheet Content

### L.C.M & H.C.F #### 1. Basic Formulas & Concepts * **Factors:** Numbers that divide a given number exactly. * **Multiples:** Numbers that are products of a given number and any integer. * **Prime Numbers:** Numbers greater than 1 with only two factors: 1 and itself. * **Composite Numbers:** Numbers greater than 1 that are not prime. * **Co-prime Numbers:** Two numbers are co-prime if their H.C.F. is 1. * **H.C.F. (Highest Common Factor) / G.C.D. (Greatest Common Divisor):** The largest number that divides two or more numbers exactly. * **Prime Factorization Method:** Express each number as a product of prime factors. H.C.F. is the product of the lowest powers of common prime factors. * **Division Method:** Divide the larger number by the smaller number. Then divide the divisor by the remainder. Repeat until the remainder is 0. The last divisor is the H.C.F. * **L.C.M. (Least Common Multiple):** The smallest number that is a multiple of two or more numbers. * **Prime Factorization Method:** Express each number as a product of prime factors. L.C.M. is the product of the highest powers of all prime factors (common and uncommon). * **Division Method:** Arrange the given numbers in a row. Divide by a prime number that divides at least two of the numbers. Bring down numbers not divisible. Repeat until no two numbers are divisible by any common prime number. L.C.M. is the product of all divisors and remaining numbers. * **Relationship between L.C.M. and H.C.F.:** * For two numbers $A$ and $B$: $A \times B = \text{L.C.M.}(A, B) \times \text{H.C.F.}(A, B)$ * For fractions: * H.C.F. of fractions = $\frac{\text{H.C.F. of Numerators}}{\text{L.C.M. of Denominators}}$ * L.C.M. of fractions = $\frac{\text{L.C.M. of Numerators}}{\text{H.C.F. of Denominators}}$ #### 2. Shortcuts & Tricks 1. **Finding H.C.F. of multiple numbers quickly:** The H.C.F. of a set of numbers must be a factor of the difference between any two numbers in the set. * *Example:* H.C.F. of 42, 63, 105. Differences: $63-42=21$, $105-63=42$. H.C.F. must be a factor of 21. Factors of 21 are 1, 3, 7, 21. Check if 21 divides all numbers. Yes. So, H.C.F. = 21. 2. **Finding L.C.M. of multiple numbers quickly:** If numbers are co-prime, their L.C.M. is their product. If one number is a multiple of another, exclude the smaller number. * *Example:* L.C.M. of 3, 6, 8. Since 6 is a multiple of 3, we can find L.C.M. of 6 and 8. L.C.M.(6, 8) = 24. 3. **H.C.F. & L.C.M. of powers:** * For $a^m, a^n, a^p$: H.C.F. = $a^{\min(m,n,p)}$, L.C.M. = $a^{\max(m,n,p)}$ * For numbers with different prime factors (e.g., $2^3 \times 3^2$, $2^2 \times 3^4 \times 5^1$): * H.C.F. = product of common prime factors with lowest powers ($2^2 \times 3^2$) * L.C.M. = product of all prime factors with highest powers ($2^3 \times 3^4 \times 5^1$) 4. **Largest number that divides A, B, C leaving same remainder R:** Find H.C.F. of $(A-R), (B-R), (C-R)$. 5. **Largest number that divides A, B, C leaving remainders $r_1, r_2, r_3$ respectively:** Find H.C.F. of $(A-r_1), (B-r_2), (C-r_3)$. 6. **Smallest number which when divided by A, B, C leaves same remainder R:** L.C.M.(A, B, C) + R. 7. **Smallest number which when divided by A, B, C leaves remainders $r_1, r_2, r_3$ where $(A-r_1) = (B-r_2) = (C-r_3) = K$ (constant difference):** L.C.M.(A, B, C) - K. 8. **Product of two numbers = Their H.C.F. $\times$ Their L.C.M.** This is crucial for problems involving two numbers. 9. **Bell/Traffic Light Problems:** To find when events will occur simultaneously again, calculate the L.C.M. of their individual timings. 10. **Finding the number of factors:** If $N = p_1^{a} \times p_2^{b} \times p_3^{c}...$, then the total number of factors is $(a+1)(b+1)(c+1)...$. #### 3. Useful Information * The H.C.F. of any two consecutive numbers is always 1. * The L.C.M. of any two consecutive numbers is always their product. * The H.C.F. of any two consecutive even numbers is 2. * The L.C.M. of any two consecutive even numbers is $2 \times (\text{product of the numbers / 2})$. * The H.C.F. is always less than or equal to the numbers, while the L.C.M. is always greater than or equal to the numbers. * If H.C.F. of two numbers is H, then the numbers can be expressed as $Hx$ and $Hy$, where $x$ and $y$ are co-prime. Their L.C.M. will be $Hxy$. ### Rational Numbers & Ordering #### 1. Basic Formulas & Concepts * **Natural Numbers (N):** $\{1, 2, 3, ...\}$ * **Whole Numbers (W):** $\{0, 1, 2, 3, ...\}$ * **Integers (Z):** $\{..., -2, -1, 0, 1, 2, ...\}$ * **Rational Numbers (Q):** Numbers that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. * Examples: $\frac{1}{2}, -3, 0.75, 5$ * Terminating decimals (e.g., 0.5, 0.25) are rational. * Non-terminating but repeating decimals (e.g., $0.333..., 0.142857142857...$) are rational. * **Irrational Numbers:** Numbers that cannot be expressed in the form $\frac{p}{q}$. Their decimal representations are non-terminating and non-repeating. * Examples: $\sqrt{2}, \pi, e$ * **Real Numbers (R):** The set of all rational and irrational numbers. * **Ordering Rational Numbers:** * **Same Denominator:** Compare numerators directly. * **Different Denominators:** 1. Find the L.C.M. of the denominators. 2. Convert each fraction to an equivalent fraction with the L.C.M. as the new denominator. 3. Compare the numerators. * **Cross-Multiplication Method:** To compare $\frac{a}{b}$ and $\frac{c}{d}$: * If $ad > bc$, then $\frac{a}{b} > \frac{c}{d}$ * If $ad \frac{5}{9}$. 2. **Comparing Fractions with Constant Difference between Numerator and Denominator:** If the difference $(q-p)$ is constant for $\frac{p}{q}$, then the fraction with the larger numerator (and denominator) is larger. This applies to proper fractions (numerator \frac{4}{5} > \frac{3}{4}$. 3. **Converting Repeating Decimals to Fractions:** * **Pure Repeating:** $0.\overline{abc} = \frac{abc}{999}$ * **Mixed Repeating:** $0.ab\overline{c} = \frac{abc - ab}{900}$ * *General Rule:* (Number formed by all digits after decimal - Number formed by non-repeating digits after decimal) / (As many 9s as repeating digits followed by as many 0s as non-repeating digits after decimal). 4. **Finding Rational Numbers between two given Rational Numbers:** * **Method 1 (Averaging):** If $x$ and $y$ are two rational numbers, then $\frac{x+y}{2}$ is a rational number between them. You can repeat this process. * **Method 2 (Equivalent Fractions):** Convert the given numbers to equivalent fractions with large common denominators. Then pick numerators between them. * *Example:* Find rational numbers between $\frac{1}{3}$ and $\frac{1}{2}$. L.C.M. of 3, 2 is 6. $\frac{1}{3} = \frac{2}{6}$, $\frac{1}{2} = \frac{3}{6}$. To find more, multiply numerator and denominator by 10: $\frac{20}{60}$ and $\frac{30}{60}$. Numbers like $\frac{21}{60}, \frac{22}{60}, ... \frac{29}{60}$ are between them. 5. **Estimating Values:** For complex fractions, sometimes it's faster to convert them to decimals for comparison if precision isn't paramount, or if the options are widely spaced. 6. **Visualizing on Number Line:** For simple cases, imagining numbers on a number line can help in ordering. #### 3. Useful Information * Every integer is a rational number (e.g., $5 = \frac{5}{1}$). * The sum, difference, product, and quotient (division by non-zero) of two rational numbers is always a rational number. * The sum or difference of a rational and an irrational number is always irrational. * The product or quotient of a non-zero rational number and an irrational number is always irrational. * The sum, difference, product, or quotient of two irrational numbers can be either rational or irrational (e.g., $\sqrt{2} + (-\sqrt{2}) = 0$ (rational), $\sqrt{2} \times \sqrt{2} = 2$ (rational), but $\sqrt{2} + \sqrt{3}$ is irrational). * Understanding the hierarchy: N $\subset$ W $\subset$ Z $\subset$ Q $\subset$ R. Irrational numbers are a separate set within R. ### Square Roots & Cube Roots #### 1. Basic Formulas & Concepts * **Square:** The product of a number with itself. $x^2 = x \times x$. * **Square Root:** A number $y$ is a square root of $x$ if $y^2 = x$. Denoted by $\sqrt{x}$. Every positive number has two square roots (one positive, one negative). $\sqrt{x}$ usually refers to the principal (positive) square root. * $\sqrt{x^2} = |x|$ * $\sqrt{xy} = \sqrt{x} \times \sqrt{y}$ (for $x,y \ge 0$) * $\sqrt{\frac{x}{y}} = \frac{\sqrt{x}}{\sqrt{y}}$ (for $x \ge 0, y > 0$) * **Perfect Square:** A number whose square root is an integer. * **Cube:** The product of a number multiplied by itself three times. $x^3 = x \times x \times x$. * **Cube Root:** A number $y$ is a cube root of $x$ if $y^3 = x$. Denoted by $\sqrt[3]{x}$. Every real number has exactly one real cube root. * $\sqrt[3]{x^3} = x$ * $\sqrt[3]{xy} = \sqrt[3]{x} \times \sqrt[3]{y}$ * $\sqrt[3]{\frac{x}{y}} = \frac{\sqrt[3]{x}}{\sqrt[3]{y}}$ * **Perfect Cube:** A number whose cube root is an integer. * **Methods to Find Square Roots:** * **Prime Factorization:** Express the number as a product of prime factors. Pair up identical factors. Take one from each pair and multiply. * **Long Division Method:** A systematic method for finding square roots of larger numbers or non-perfect squares. * **Methods to Find Cube Roots:** * **Prime Factorization:** Express the number as a product of prime factors. Group identical factors in triplets. Take one from each triplet and multiply. #### 2. Shortcuts & Tricks 1. **Estimating Square Roots (Non-perfect Squares):** * To find $\sqrt{N}$: find the nearest perfect square $P^2 N$. Then $\sqrt{N}$ is between $P$ and $Q$. * *Formula (approximation):* $\sqrt{x \pm y} \approx \sqrt{x} \pm \frac{y}{2\sqrt{x}}$ (where $x$ is a perfect square close to $N$). * *Example:* $\sqrt{26} = \sqrt{25+1} \approx \sqrt{25} + \frac{1}{2\sqrt{25}} = 5 + \frac{1}{10} = 5.1$ 2. **Finding Square Root of Perfect Squares ending in 25:** If a number ends in 25, its square root must end in 5. The digits before 25 are $N(N+1)$ for some integer $N$. The square root is $10N+5$. * *Example:* $\sqrt{625}$. Ends in 25, so root ends in 5. Remaining digits is 6. $2 \times 3 = 6$. So $N=2$. Root is $25$. * *Example:* $\sqrt{1225}$. Remaining digits is 12. $3 \times 4 = 12$. So $N=3$. Root is $35$. 3. **Unit Digit Trick for Square Roots:** * If a perfect square ends in 1, its root ends in 1 or 9. * If it ends in 4, its root ends in 2 or 8. * If it ends in 5, its root ends in 5. * If it ends in 6, its root ends in 4 or 6. * If it ends in 9, its root ends in 3 or 7. * If it ends in 00, its root ends in 0. * **Numbers ending in 2, 3, 7, 8 are NEVER perfect squares.** 4. **Finding Square Root of large perfect squares (up to 4-5 digits):** * Consider $\sqrt{ABCD}$ (4 digits). Look at the unit digit D to determine the unit digit of the root (e.g., if D=9, root ends in 3 or 7). * Look at the first two digits AB. Find the largest integer $X$ such that $X^2 \le AB$. $X$ is the tens digit of the root. * To decide between the two possibilities for the unit digit, compare $AB$ with $X(X+1)$. If $AB X(X+1)$, choose the larger. * *Example:* $\sqrt{1849}$. Ends in 9, so root ends in 3 or 7. First two digits 18. $4^2=16, 5^2=25$. So tens digit is 4. Root is 43 or 47. Compare 18 with $4 \times 5 = 20$. Since $18 \sqrt[6]{8}$, then $\sqrt[3]{3} > \sqrt{2}$. #### 3. Useful Information * A number is a perfect square if its prime factors occur in pairs (even powers). * A number is a perfect cube if its prime factors occur in triplets (powers divisible by 3). * The square of an even number is even. The square of an odd number is odd. * The cube of an even number is even. The cube of an odd number is odd. * The number of digits in the square root of an $n$-digit perfect square is $\frac{n}{2}$ if $n$ is even, and $\frac{n+1}{2}$ if $n$ is odd. * The number of digits in the cube root of an $n$-digit perfect cube is $\lceil n/3 \rceil$. * Memorizing squares up to 30 and cubes up to 15-20 is highly beneficial for speed. ### Averages #### 1. Basic Formulas & Concepts * **Average (Arithmetic Mean):** The sum of all observations divided by the number of observations. $$\text{Average} = \frac{\text{Sum of Observations}}{\text{Number of Observations}}$$ * **Sum of Observations:** Average $\times$ Number of Observations * **Weighted Average:** Used when different observations have different "weights" or frequencies. $$\text{Weighted Average} = \frac{w_1x_1 + w_2x_2 + ... + w_nx_n}{w_1 + w_2 + ... + w_n}$$ Where $x_i$ are the observations and $w_i$ are their respective weights. * **Average Speed:** Not simply the average of speeds if distances or times are different. * If a person travels distance $D_1$ at speed $S_1$ and $D_2$ at speed $S_2$: $$\text{Average Speed} = \frac{D_1 + D_2}{\frac{D_1}{S_1} + \frac{D_2}{S_2}}$$ * If a person travels at speed $S_1$ for time $T_1$ and $S_2$ for time $T_2$: $$\text{Average Speed} = \frac{S_1T_1 + S_2T_2}{T_1 + T_2}$$ * If a person travels two equal distances at speeds $S_1$ and $S_2$: $$\text{Average Speed} = \frac{2S_1S_2}{S_1+S_2}$$ (Harmonic Mean) * If a person travels three equal distances at speeds $S_1, S_2, S_3$: $$\text{Average Speed} = \frac{3S_1S_2S_3}{S_1S_2 + S_2S_3 + S_3S_1}$$ * **Properties of Averages:** * If each observation is increased/decreased by a constant $k$, the average also increases/decreases by $k$. * If each observation is multiplied/divided by a constant $k$, the average also gets multiplied/divided by $k$. * The average always lies between the smallest and largest observations. #### 2. Shortcuts & Tricks 1. **Deviation Method (Assumed Mean Method):** * Choose an "assumed mean" (preferably one of the numbers or a number close to the average). * Calculate the deviation of each number from the assumed mean. * Sum the deviations. * Average = Assumed Mean + $(\frac{\text{Sum of Deviations}}{\text{Number of Observations}})$ * *Example:* Average of 20, 25, 30, 35. Assume mean = 30. * Deviations: (20-30)=-10, (25-30)=-5, (30-30)=0, (35-30)=5. * Sum of deviations = -10 - 5 + 0 + 5 = -10. * Average = $30 + \frac{-10}{4} = 30 - 2.5 = 27.5$. 2. **Consecutive Numbers/Arithmetic Progression:** * The average of numbers in an Arithmetic Progression (AP) (e.g., consecutive integers, consecutive even/odd numbers) is the middle term if the number of terms is odd. * If the number of terms is even, the average is the average of the two middle terms. * Alternatively, for AP, Average = $\frac{\text{First Term} + \text{Last Term}}{2}$. 3. **Replacing/Adding/Removing Observations:** * **Adding a new observation:** If a new observation $x$ is added to $N$ observations with average $A$, the new average is $\frac{NA + x}{N+1}$. * **Removing an observation:** If an observation $x$ is removed from $N$ observations with average $A$, the new average is $\frac{NA - x}{N-1}$. * **Replacing an observation:** If an observation $x$ is replaced by $y$ in $N$ observations with average $A$, the new average is $A + \frac{y-x}{N}$. * *Trick:* Change in average = $\frac{\text{New Value} - \text{Old Value}}{\text{Number of items}}$. 4. **Group Averages:** If group 1 has $N_1$ items with average $A_1$, and group 2 has $N_2$ items with average $A_2$, the combined average is $\frac{N_1A_1 + N_2A_2}{N_1+N_2}$. This is a form of weighted average. 5. **Mistake in calculation:** If an observation $x$ was read incorrectly as $y$: * Correct Sum = Incorrect Sum - Incorrect Value + Correct Value * Correct Average = $\frac{\text{Correct Sum}}{\text{Number of Observations}}$ * *Trick:* Change in average = $\frac{\text{Correct Value} - \text{Incorrect Value}}{\text{Number of items}}$. Add this change to the incorrect average. 6. **Cricket Batting Average type questions:** * If a batsman scores $X$ runs in his $N^{th}$ innings and his average increases by $Y$: New Average = $X - (N-1)Y$. * If his average decreases by $Y$: New Average = $X + (N-1)Y$. * Old Average = New Average $\mp Y$. 7. **Age-related problems:** When a person leaves or joins a group, use the total sum of ages. * *Example:* If the average age of 5 members is 20. If one member leaves, the average becomes 18. Age of the person who left = (Old Total Sum) - (New Total Sum) = $(5 \times 20) - (4 \times 18) = 100 - 72 = 28$. 8. **Average of first N natural numbers:** $\frac{N+1}{2}$ 9. **Average of first N even numbers:** $N+1$ 10. **Average of first N odd numbers:** $N$ #### 3. Useful Information * The concept of average is fundamental and appears in many other topics like Profit & Loss, Time & Work, Speed, Time & Distance. * Always be careful about whether the average is for numbers, values, speed, etc. * Distinguish between arithmetic mean, geometric mean, and harmonic mean. While arithmetic mean is most common, average speed often involves the harmonic mean. * The average gives a central tendency, representing a typical value of the dataset. * When a set of numbers increases or decreases uniformly (like an AP), the average changes uniformly. ### Set Theory #### 1. Basic Formulas & Concepts * **Set:** A well-defined collection of distinct objects. * *Representation:* Roster form (listing elements: $\{1, 2, 3\}$), Set-builder form (describing elements: $\{x | x \text{ is an even integer}\}$). * **Elements:** The objects in a set. * **Cardinality ($|A|$ or $n(A)$):** The number of distinct elements in a set A. * **Types of Sets:** * **Empty Set ($\emptyset$ or \{\}):** A set with no elements. $n(\emptyset) = 0$. * **Singleton Set:** A set with exactly one element. * **Finite Set:** A set with a countable number of elements. * **Infinite Set:** A set with an uncountable number of elements. * **Universal Set (U):** The set of all elements under consideration. * **Relationships between Sets:** * **Subset ($A \subseteq B$):** Every element of A is also an element of B. * **Proper Subset ($A \subset B$):** A is a subset of B, and A is not equal to B. * **Superset ($A \supseteq B$):** B is a subset of A. * **Equality of Sets ($A = B$):** $A \subseteq B$ and $B \subseteq A$. * **Disjoint Sets:** Two sets A and B are disjoint if they have no common elements ($A \cap B = \emptyset$). * **Set Operations:** * **Union ($A \cup B$):** The set of all elements that are in A or in B (or both). * $n(A \cup B) = n(A) + n(B) - n(A \cap B)$ * If A and B are disjoint, $n(A \cup B) = n(A) + n(B)$. * **Intersection ($A \cap B$):** The set of all elements common to both A and B. * **Difference ($A - B$ or $A \setminus B$):** The set of elements that are in A but not in B. * $A - B = A \cap B^c$ * $n(A - B) = n(A) - n(A \cap B)$ * **Symmetric Difference ($A \Delta B$):** The set of elements that are in A or B but not in their intersection. * $A \Delta B = (A - B) \cup (B - A)$ * $n(A \Delta B) = n(A) + n(B) - 2n(A \cap B)$ * **Complement ($A^c$ or $A'$):** The set of all elements in the universal set U that are not in A. * $A^c = U - A$ * $n(A^c) = n(U) - n(A)$ * **De Morgan's Laws:** * $(A \cup B)^c = A^c \cap B^c$ * $(A \cap B)^c = A^c \cup B^c$ * **For Three Sets A, B, C:** * $n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$ #### 2. Shortcuts & Tricks 1. **Venn Diagrams:** Visually represent sets and their relationships. Extremely useful for solving problems involving 2 or 3 sets, especially in competitive exams. * **Region meanings for 2 sets A, B:** * Only A: $n(A) - n(A \cap B)$ * Only B: $n(B) - n(A \cap B)$ * A and B (both): $n(A \cap B)$ * A or B (at least one): $n(A \cup B)$ * Neither A nor B: $n(U) - n(A \cup B)$ * **Region meanings for 3 sets A, B, C:** * Only A: $n(A) - n(A \cap B) - n(A \cap C) + n(A \cap B \cap C)$ * Only A and B (not C): $n(A \cap B) - n(A \cap B \cap C)$ * At least two: $n(A \cap B) + n(B \cap C) + n(C \cap A) - 2n(A \cap B \cap C)$ * Exactly two: $n(A \cap B) + n(B \cap C) + n(C \cap A) - 3n(A \cap B \cap C)$ 2. **Using formulas directly:** For complex problems, write down the known values and the required value, then use the appropriate inclusion-exclusion principle formula. 3. **Complementary approach:** Sometimes it's easier to calculate the complement of what's asked and subtract it from the total. * *Example:* To find "neither A nor B", calculate $n(A \cup B)$ and subtract from $n(U)$. 4. **Power Set:** The set of all possible subsets of a set A. Denoted by $P(A)$. * If $n(A) = k$, then $n(P(A)) = 2^k$. * Number of proper subsets = $2^k - 1$. 5. **Solving survey problems:** When given percentages or numbers for various categories (e.g., students passing in different subjects), always start by filling in the most overlapping region (e.g., $A \cap B \cap C$) in a Venn diagram, then work outwards. * *Step-by-step for 3 sets:* 1. Fill $n(A \cap B \cap C)$ 2. Fill $n(A \cap B \text{ only})$, $n(B \cap C \text{ only})$, $n(C \cap A \text{ only})$ 3. Fill $n(A \text{ only})$, $n(B \text{ only})$, $n(C \text{ only})$ 4. Sum all these to get $n(A \cup B \cup C)$ 5. Subtract from total to get "none". 6. **Max/Min values of intersection/union:** * **Maximum $n(A \cap B)$:** $\min(n(A), n(B))$ * **Minimum $n(A \cap B)$:** $\max(0, n(A) + n(B) - n(U))$ * **Maximum $n(A \cup B)$:** $n(U)$ (if $A \cup B \subseteq U$) * **Minimum $n(A \cup B)$:** $\max(n(A), n(B))$ #### 3. Useful Information * The order of elements in a set does not matter. * Elements in a set are distinct (no repetitions). * Set theory is foundational to probability, logic, and computer science. * Carefully read questions to distinguish between "A only", "B only", "A and B", "A or B" etc. These have specific meanings in set theory. * Understanding the difference between $\in$ (is an element of) and $\subseteq$ (is a subset of) is crucial. $a \in A$ means $a$ is an element of set $A$. $\{a\} \subseteq A$ means the set containing $a$ is a subset of $A$. ### Decimal Fractions #### 1. Basic Formulas & Concepts * **Decimal Fraction:** A fraction whose denominator is a power of 10. Represented using a decimal point. * *Example:* $\frac{3}{10} = 0.3$, $\frac{25}{100} = 0.25$, $\frac{123}{1000} = 0.123$ * **Terminating Decimal:** A decimal that ends after a finite number of digits. These are rational numbers. * *Condition:* A fraction $\frac{p}{q}$ (in simplest form) is a terminating decimal if the prime factors of its denominator $q$ are only 2s and/or 5s. * **Non-terminating Repeating (Recurring) Decimal:** A decimal that continues infinitely but with a repeating block of digits. These are also rational numbers. * *Example:* $\frac{1}{3} = 0.333... = 0.\overline{3}$, $\frac{1}{7} = 0.142857142857... = 0.\overline{142857}$ * **Non-terminating Non-repeating Decimal:** A decimal that continues infinitely without any repeating block. These are irrational numbers. * *Example:* $\sqrt{2} = 1.41421356..., \pi = 3.14159265...$ * **Place Value:** In a decimal number, each digit holds a place value (e.g., in 12.34, 1 is tens, 2 is ones, 3 is tenths, 4 is hundredths). * **Operations on Decimals:** * **Addition/Subtraction:** Align decimal points and add/subtract as usual. * **Multiplication:** Multiply numbers as if they were integers. The number of decimal places in the product is the sum of decimal places in the multipliers. * **Division:** Convert the divisor to an integer by shifting the decimal point (and shifting the decimal point of the dividend by the same number of places). Then perform long division. #### 2. Shortcuts & Tricks 1. **Converting Fractions to Decimals:** * If the denominator has only 2s and 5s as prime factors, multiply numerator and denominator by appropriate powers of 2 or 5 to make the denominator a power of 10. * *Example:* $\frac{3}{20} = \frac{3}{2^2 \times 5} = \frac{3 \times 5}{2^2 \times 5^2} = \frac{15}{100} = 0.15$ * Otherwise, perform long division. 2. **Converting Repeating Decimals to Fractions (reiterated from Rational Numbers):** * **Pure Repeating:** $0.\overline{P} = \frac{P}{9}$, $0.\overline{PQ} = \frac{PQ}{99}$, $0.\overline{PQR} = \frac{PQR}{999}$ * **Mixed Repeating:** $0.A\overline{B} = \frac{AB - A}{90}$, $0.AB\overline{C} = \frac{ABC - AB}{900}$ * *General Rule:* $\frac{\text{All digits after decimal} - \text{Non-repeating digits after decimal}}{\text{Number of 9s (repeating) followed by number of 0s (non-repeating)}}$ 3. **Comparing Decimals:** * Compare the integer parts first. * If integer parts are same, compare the tenths digit, then hundredths, and so on. * Adding trailing zeros after the decimal point does not change the value (e.g., $0.5 = 0.50 = 0.500$). This helps in aligning and comparing. 4. **Multiplying by powers of 10:** Shift decimal point to the right. * $1.23 \times 100 = 123$ 5. **Dividing by powers of 10:** Shift decimal point to the left. * $123 \div 100 = 1.23$ 6. **Approximation:** For multiple-choice questions, sometimes quick approximation can help eliminate options. Round numbers to the nearest whole number or significant decimal place. 7. **Mental Math for Simple Operations:** Practice addition, subtraction, multiplication, and division of decimals mentally for common scenarios. * *Example:* $0.25 \times 8 = (1/4) \times 8 = 2$ * *Example:* $0.75 + 0.5 = 1.25$ 8. **Converting fractions to decimals for comparison:** For a list of fractions, converting them to decimals (even approximate) can be faster than finding a common denominator, especially if the denominators are large or prime. #### 3. Useful Information * Decimal fractions are a subset of rational numbers. * Understanding the relationship between fractions, decimals, and percentages is crucial for quantitative aptitude. * Precision matters. Pay attention to the number of decimal places required in the answer. * Be careful with division by small decimals; it results in a larger number. * *Example:* $1 \div 0.1 = 10$, $1 \div 0.01 = 100$. ### Ratio & Proportion #### 1. Basic Formulas & Concepts * **Ratio:** A comparison of two quantities of the same unit. Expressed as $a:b$ or $\frac{a}{b}$. * **Antecedent:** The first term ($a$). * **Consequent:** The second term ($b$). * A ratio is a pure number (no units). * **Compound Ratio:** If two ratios are $a:b$ and $c:d$, their compound ratio is $ac:bd$. * **Duplicate Ratio:** $a^2:b^2$ * **Sub-duplicate Ratio:** $\sqrt{a}:\sqrt{b}$ * **Triplicate Ratio:** $a^3:b^3$ * **Sub-triplicate Ratio:** $\sqrt[3]{a}:\sqrt[3]{b}$ * **Inverse Ratio:** If ratio is $a:b$, its inverse is $b:a$. * **Proportion:** An equality of two ratios. If $a:b = c:d$, then $a, b, c, d$ are in proportion. * **Extremes:** $a, d$ (first and fourth terms). * **Means:** $b, c$ (second and third terms). * **Product of Extremes = Product of Means:** $ad = bc$. * **Continued Proportion:** If $a, b, c$ are in continued proportion, then $a:b = b:c$, which means $b^2 = ac$. Here, $b$ is the **mean proportional** between $a$ and $c$. * **Third Proportional:** If $a:b = b:c$, then $c$ is the third proportional to $a$ and $b$. ($c = b^2/a$). * **Fourth Proportional:** If $a:b = c:d$, then $d$ is the fourth proportional to $a, b, c$. ($d = bc/a$). * **Componendo and Dividendo Rule:** If $\frac{a}{b} = \frac{c}{d}$, then $\frac{a+b}{a-b} = \frac{c+d}{c-d}$. * **K-Method:** If $\frac{a}{b} = \frac{c}{d} = k$, then $a=bk, c=dk$. This can simplify complex ratio problems. #### 2. Shortcuts & Tricks 1. **Combining Ratios:** * If $A:B = x:y$ and $B:C = p:q$, to find $A:B:C$: * Make the common term (B) equal. Multiply first ratio by $p$ and second by $y$. * $A:B:C = xp:yp:yq$ * Alternatively, $A:B:C = (x \times p) : (y \times p) : (y \times q)$ (N-method or Z-method). * *Example:* $A:B = 2:3$, $B:C = 4:5$. * $A:B = 2 \times 4 : 3 \times 4 = 8:12$ * $B:C = 4 \times 3 : 5 \times 3 = 12:15$ * So $A:B:C = 8:12:15$. 2. **Ratios of reciprocals:** If $a:b:c = x:y:z$, then $\frac{1}{a}:\frac{1}{b}:\frac{1}{c} = \frac{1}{x}:\frac{1}{y}:\frac{1}{z}$. To simplify, multiply by L.C.M. of denominators. * *Example:* If $2A=3B=4C$, then $A:B:C = \frac{1}{2}:\frac{1}{3}:\frac{1}{4}$. L.C.M.(2,3,4)=12. So $A:B:C = 6:4:3$. 3. **Distribution of a quantity based on ratio:** If a quantity $X$ is to be divided in ratio $a:b:c$, then the shares are $\frac{a}{a+b+c}X, \frac{b}{a+b+c}X, \frac{c}{a+b+c}X$. 4. **Age problems using ratios:** Ratios of ages at different times can be used to set up linear equations. Ensure the difference in ages remains constant. * *Trick:* If the ratio of ages of A and B is $x:y$ "now" and $p:q$ after $T$ years, the difference in parts $(q-p)$ should be proportional to $(y-x)$. Adjust ratios if needed. * Let $A_{current} = kx$, $B_{current} = ky$. * After $T$ years: $A_{new} = kx+T$, $B_{new} = ky+T$. * $\frac{kx+T}{ky+T} = \frac{p}{q}$. Solve for $k$. 5. **Income/Expenditure/Savings problems:** * Income - Expenditure = Savings * Income : Expenditure : Savings $\rightarrow$ use ratios to find individual values. 6. **Mixture Problems:** Often involve ratios. * *Example:* Mixture A has milk:water = 2:3. Mixture B has milk:water = 4:1. If they are mixed in ratio 1:1, what's the new ratio? * In A: Milk = 2/5, Water = 3/5 * In B: Milk = 4/5, Water = 1/5 * New Mixture: Total Milk = $1 \times (2/5) + 1 \times (4/5) = 6/5$. Total Water = $1 \times (3/5) + 1 \times (1/5) = 4/5$. * New Ratio Milk:Water = 6/5 : 4/5 = 6:4 = 3:2. 7. **Coin Problems:** * Value of coins = (Number of coins) $\times$ (Denomination). * Use ratios to find the number of coins of each denomination. 8. **Partnership Problems:** Profits are shared in the ratio of (Investment $\times$ Time). * If time is constant, profits are in ratio of investments. * If investments are constant, profits are in ratio of times. 9. **Inverse Proportionality:** If A is inversely proportional to B, then $A \propto \frac{1}{B}$ or $A = \frac{k}{B}$ or $AB = k$. * *Example:* More workers, less time for work. Time $\propto \frac{1}{\text{Workers}}$. 10. **Direct Proportionality:** If A is directly proportional to B, then $A \propto B$ or $A = kB$. * *Example:* More items, more cost. Cost $\propto \text{Items}$. #### 3. Useful Information * Always simplify ratios to their lowest terms. * Ensure that quantities being compared in a ratio have the same units. * Ratio and Proportion are fundamental to many quantitative aptitude topics, including percentages, time & work, speed & distance, mixtures, and partnership. * The "units" or "parts" approach is very powerful: If $A:B = 3:5$, you can assume $A=3k$ and $B=5k$ for some constant $k$. This helps in setting up equations. * Understanding the difference between "ratio" (comparison) and "fraction" (part of a whole) is important, though they are related. A ratio $a:b$ implies a fraction $\frac{a}{a+b}$ of the total. ### Simplification #### 1. Basic Formulas & Concepts * **BODMAS / PEMDAS Rule:** Order of operations * **B**rackets (Parentheses) - ( ), { }, [ ] * **O**rders (Exponents/Powers/Roots) - $x^2, \sqrt{x}$ * **D**ivision * **M**ultiplication * **A**ddition * **S**ubtraction * Note: Division and Multiplication have the same precedence, perform from left to right. Addition and Subtraction have the same precedence, perform from left to right. * **Modulus Function:** $|x|$ is the absolute value of $x$. $|x| = x$ if $x \ge 0$, and $|x| = -x$ if $x ### Number System #### 1. Basic Formulas & Concepts * **Classification of Numbers:** * **Real Numbers (R):** All numbers that can be represented on a number line. * **Rational Numbers (Q):** Can be expressed as $\frac{p}{q}$ ($q \neq 0$). Includes integers, fractions, terminating and repeating decimals. * **Irrational Numbers:** Cannot be expressed as $\frac{p}{q}$. Non-terminating, non-repeating decimals (e.g., $\sqrt{2}, \pi$). * **Integers (Z):** $\{\dots, -2, -1, 0, 1, 2, \dots\}$ * **Whole Numbers (W):** $\{0, 1, 2, 3, \dots\}$ * **Natural Numbers (N) / Counting Numbers:** $\{1, 2, 3, \dots\}$ * **Even Numbers:** Divisible by 2 ($2n$). * **Odd Numbers:** Not divisible by 2 ($2n \pm 1$). * **Prime Numbers:** Natural numbers greater than 1 with exactly two factors (1 and itself). (2, 3, 5, 7, 11, ...) * **Composite Numbers:** Natural numbers greater than 1 that are not prime (have more than two factors). (4, 6, 8, 9, 10, ...) * **Co-prime / Relatively Prime Numbers:** Two numbers whose H.C.F. is 1. (e.g., 4 and 9). * **Divisibility Rules:** (covered in Divisibility Rules section) * **Factors and Multiples:** (covered in L.C.M. & H.C.F. section) * **Place Value and Face Value:** * **Face Value:** The digit itself. * **Place Value:** The digit multiplied by its position's power of 10. (e.g., in 523, face value of 2 is 2, place value is $2 \times 10 = 20$). * **Number of Factors:** If $N = p_1^{a} \times p_2^{b} \times p_3^{c} \dots$, then the total number of factors is $(a+1)(b+1)(c+1)\dots$. * **Sum of Factors:** If $N = p_1^{a} \times p_2^{b} \times p_3^{c} \dots$, then the sum of factors is $(1+p_1+...+p_1^a)(1+p_2+...+p_2^b)(1+p_3+...+p_3^c)\dots$. * **Perfect Number:** A number that is equal to the sum of its proper positive divisors (divisors excluding the number itself). (e.g., 6: divisors 1, 2, 3; $1+2+3=6$). * **Remainder Theorem:** (covered in Remainder Theorem section) #### 2. Shortcuts & Tricks 1. **Finding Prime Numbers:** To check if a number $N$ is prime, test divisibility by prime numbers up to $\sqrt{N}$. If none divide $N$, then $N$ is prime. * *Example:* Is 101 prime? $\sqrt{101} \approx 10$. Primes up to 10 are 2, 3, 5, 7. * 101 not divisible by 2 (odd). * Sum of digits $1+0+1=2$, not div by 3. * Doesn't end in 0 or 5, not div by 5. * $101 = 7 \times 14 + 3$, not div by 7. * So, 101 is prime. 2. **Number of Trailing Zeros in $N!$:** The number of trailing zeros in $N!$ is the number of times 5 is a factor in its prime factorization. * Number of zeros = $\lfloor \frac{N}{5} \rfloor + \lfloor \frac{N}{25} \rfloor + \lfloor \frac{N}{125} \rfloor + \dots$ * *Example:* Number of zeros in $100!$: $\lfloor \frac{100}{5} \rfloor + \lfloor \frac{100}{25} \rfloor = 20 + 4 = 24$. 3. **Cyclicity of Unit Digits:** Helps find the unit digit of large powers. * **Unit digit 0, 1, 5, 6:** Always themselves for any power. * **Unit digit 4, 9:** Cyclicity of 2. * $4^1=4, 4^2=16(6), 4^3=64(4)$. Pattern: 4 (odd power), 6 (even power). * $9^1=9, 9^2=81(1), 9^3=729(9)$. Pattern: 9 (odd power), 1 (even power). * **Unit digit 2, 3, 7, 8:** Cyclicity of 4. * To find unit digit of $X^N$, find remainder of $N \div 4$. If remainder is $R$, unit digit is $X^R$. If remainder is 0, use power 4. * *Example:* Unit digit of $2^{35}$. $35 \div 4 \rightarrow$ remainder 3. Unit digit of $2^3 = 8$. * *Example:* Unit digit of $7^{42}$. $42 \div 4 \rightarrow$ remainder 2. Unit digit of $7^2 = 49 \rightarrow 9$. 4. **Divisibility Rules for Composite Numbers:** To check divisibility by a composite number (e.g., 6, 12, 15, 72), check divisibility by its co-prime factors. * Divisible by 6 if divisible by 2 and 3. * Divisible by 12 if divisible by 3 and 4. * Divisible by 72 if divisible by 8 and 9. 5. **Sum of N natural numbers:** $\frac{N(N+1)}{2}$ 6. **Sum of squares of N natural numbers:** $\frac{N(N+1)(2N+1)}{6}$ 7. **Sum of cubes of N natural numbers:** $(\frac{N(N+1)}{2})^2$ 8. **Sum of first N even numbers:** $N(N+1)$ 9. **Sum of first N odd numbers:** $N^2$ 10. **Finding the number of digits in $N^M$:** Approximately $M \times \log_{10}(N) + 1$. (Not frequently tested, but good to know). #### 3. Useful Information * 0 is neither positive nor negative. * 1 is neither prime nor composite. * 2 is the only even prime number. * Understanding the properties of different number types is crucial for solving problems efficiently. * Number theory problems often combine divisibility rules, factors, and remainders. * Familiarity with prime numbers up to 100-150 is helpful. * The concept of "base" (decimal, binary, etc.) can sometimes appear. Most problems assume base 10. ### Surds & Indices #### 1. Basic Formulas & Concepts * **Indices (Exponents):** $a^n = a \times a \times \dots \times a$ (n times), where $a$ is the base and $n$ is the index/exponent. * **Laws of Indices:** 1. $a^m \times a^n = a^{m+n}$ 2. $\frac{a^m}{a^n} = a^{m-n}$ 3. $(a^m)^n = a^{mn}$ 4. $(ab)^n = a^n b^n$ 5. $(\frac{a}{b})^n = \frac{a^n}{b^n}$ 6. $a^0 = 1$ (for $a \neq 0$) 7. $a^{-n} = \frac{1}{a^n}$ 8. $a^{\frac{m}{n}} = \sqrt[n]{a^m} = (\sqrt[n]{a})^m$ * **Surds (Radicals):** An irrational number that is a root of an integer or rational number. $\sqrt[n]{a}$ is a surd if $a$ is rational, $n$ is a positive integer, and $\sqrt[n]{a}$ is irrational. * **Order of a Surd:** The value of $n$ in $\sqrt[n]{a}$. * **Pure Surd:** A surd which has no rational factor other than 1 (e.g., $\sqrt{3}$). * **Mixed Surd:** A surd which has a rational factor other than 1 (e.g., $2\sqrt{3}$). * **Laws of Surds:** 1. $\sqrt[n]{a} = a^{1/n}$ 2. $\sqrt[n]{ab} = \sqrt[n]{a} \times \sqrt[n]{b}$ 3. $\sqrt[n]{\frac{a}{b}} = \frac{\sqrt[n]{a}}{\sqrt[n]{b}}$ 4. $(\sqrt[n]{a})^n = a$ 5. $\sqrt[m]{\sqrt[n]{a}} = \sqrt[mn]{a}$ 6. $a\sqrt[n]{x} \pm b\sqrt[n]{x} = (a \pm b)\sqrt[n]{x}$ (like surds) * **Rationalization:** The process of converting an irrational denominator to a rational number by multiplying the numerator and denominator by a suitable factor (rationalizing factor). * Rationalizing factor of $\sqrt{a}$ is $\sqrt{a}$. * Rationalizing factor of $(\sqrt{a} \pm \sqrt{b})$ is $(\sqrt{a} \mp \sqrt{b})$. * Rationalizing factor of $(a \pm \sqrt{b})$ is $(a \mp \sqrt{b})$. #### 2. Shortcuts & Tricks 1. **Comparing Surds:** * Convert surds to the same order by finding the L.C.M. of their indices. * *Example:* Compare $\sqrt{3}, \sqrt[3]{4}, \sqrt[4]{5}$. L.C.M. of 2, 3, 4 is 12. * $\sqrt{3} = 3^{1/2} = 3^{6/12} = \sqrt[12]{3^6} = \sqrt[12]{729}$ * $\sqrt[3]{4} = 4^{1/3} = 4^{4/12} = \sqrt[12]{4^4} = \sqrt[12]{256}$ * $\sqrt[4]{5} = 5^{1/4} = 5^{3/12} = \sqrt[12]{5^3} = \sqrt[12]{125}$ * So, $\sqrt{3} > \sqrt[3]{4} > \sqrt[4]{5}$. 2. **Simplifying expressions with $\sqrt{A \pm \sqrt{B}}$:** (reiterated from Square Roots) * If $\sqrt{A \pm 2\sqrt{B}}$, find two numbers $x, y$ such that $x+y=A$ and $xy=B$. Then $\sqrt{A \pm 2\sqrt{B}} = \sqrt{x} \pm \sqrt{y}$. * *Example:* $\sqrt{12 - \sqrt{140}} = \sqrt{12 - 2\sqrt{35}}$. Find $x,y$ such that $x+y=12, xy=35$. $x=7, y=5$. So $\sqrt{7} - \sqrt{5}$. 3. **Rationalization Trick for Reciprocals:** If $x = \sqrt{a} + \sqrt{b}$, then $\frac{1}{x} = \frac{1}{\sqrt{a}+\sqrt{b}} = \frac{\sqrt{a}-\sqrt{b}}{a-b}$. * If $a-b=1$, then $\frac{1}{x} = \sqrt{a}-\sqrt{b}$. * *Example:* If $x = \sqrt{5} + \sqrt{4}$, then $\frac{1}{x} = \sqrt{5} - \sqrt{4} = \sqrt{5}-2$. 4. **Infinite Series of Surds:** * $\sqrt{a\sqrt{a\sqrt{a...}}}$ (infinite) $= a$ * $\sqrt{a+\sqrt{a+\sqrt{a+...}}}$ (infinite) $= \frac{1+\sqrt{1+4a}}{2}$. If $a = n(n+1)$, then it is $n+1$. * *Example:* $\sqrt{12+\sqrt{12+\sqrt{12+...}}}$. Here $12 = 3 \times 4$, so answer is 4. * $\sqrt{a-\sqrt{a-\sqrt{a-...}}}$ (infinite) $= \frac{-1+\sqrt{1+4a}}{2}$. If $a = n(n+1)$, then it is $n$. * *Example:* $\sqrt{12-\sqrt{12-\sqrt{12-...}}}$. Here $12 = 3 \times 4$, so answer is 3. 5. **Exponents with negative numbers:** * $(-a)^{\text{even power}} = a^{\text{even power}}$ * $(-a)^{\text{odd power}} = -a^{\text{odd power}}$ 6. **Solving exponential equations:** If $a^x = a^y$, then $x=y$. If $a^x = b^x$, then $a=b$ (if $x \neq 0$). 7. **Simplifying large powers:** Use cyclicity for unit digits. For the entire number, look for patterns or factor common terms. 8. **Working with $0^0$ and $1^\infty$:** These are indeterminate forms in calculus but in basic indices, $a^0=1$ for $a \neq 0$. $0^0$ is usually undefined or 1 depending on context. In competitive exams, typically avoid such cases unless specified. 9. **Fractional Exponents:** Convert roots to fractional exponents for easier manipulation using laws of indices. * *Example:* $\sqrt[3]{x^2} \times \sqrt{x} = x^{2/3} \times x^{1/2} = x^{2/3+1/2} = x^{7/6}$. 10. **Approximation of values:** For multiple-choice questions, sometimes powers can be approximated. $2^{10} \approx 10^3 = 1000$. #### 3. Useful Information * Indices and surds are fundamental to algebra and often appear in conjunction with number system, simplification, and algebraic problems. * Memorizing the laws of indices and surds is essential for speed and accuracy. * Always simplify expressions as much as possible before performing final calculations. * Be careful with negative signs and fractional exponents. * Understanding the relationship between roots and fractional exponents is key. ### Divisibility Rules #### 1. Basic Formulas & Concepts * **Divisibility:** A number $a$ is divisible by a number $b$ if $a \div b$ yields an integer remainder of 0. * **Fundamental Principle:** If a number is divisible by two co-prime numbers, it is divisible by their product. (e.g., if a number is divisible by 2 and 3, it is divisible by 6). * **Divisibility Rules for Common Numbers:** * **2:** A number is divisible by 2 if its last digit is even (0, 2, 4, 6, 8). * **3:** A number is divisible by 3 if the sum of its digits is divisible by 3. * **4:** A number is divisible by 4 if the number formed by its last two digits is divisible by 4. * **5:** A number is divisible by 5 if its last digit is 0 or 5. * **6:** A number is divisible by 6 if it is divisible by both 2 and 3. * **7:** (More complex, often easier to divide for smaller numbers) * Double the last digit and subtract it from the remaining number. If the result is divisible by 7, the original number is divisible by 7. Repeat if necessary. * *Example:* 343. $34 - (2 \times 3) = 34 - 6 = 28$. 28 is div by 7, so 343 is div by 7. * **8:** A number is divisible by 8 if the number formed by its last three digits is divisible by 8. * **9:** A number is divisible by 9 if the sum of its digits is divisible by 9. * **10:** A number is divisible by 10 if its last digit is 0. * **11:** A number is divisible by 11 if the difference between the sum of the digits at odd places and the sum of the digits at even places (from the right) is either 0 or a multiple of 11. * *Example:* 12345. Sum of odd place digits (from right): $5+3+1=9$. Sum of even place digits: $4+2=6$. Difference $9-6=3$. Not div by 11. * *Example:* 1331. Sum of odd place digits: $1+3=4$. Sum of even place digits: $3+1=4$. Difference $4-4=0$. Div by 11. * **12:** A number is divisible by 12 if it is divisible by both 3 and 4. * **13:** (More complex) * Multiply the last digit by 4 and add it to the remaining number. If the result is divisible by 13, the original number is divisible by 13. Repeat if necessary. * *Example:* 273. $27 + (4 \times 3) = 27 + 12 = 39$. 39 is div by 13, so 273 is div by 13. * **25:** A number is divisible by 25 if the number formed by its last two digits is 00, 25, 50, or 75. * **125:** A number is divisible by 125 if the number formed by its last three digits is 000 or divisible by 125. #### 2. Shortcuts & Tricks 1. **Combining Co-prime Rules:** To check divisibility by $N$, find two co-prime factors $a, b$ such that $a \times b = N$. If the number is divisible by both $a$ and $b$, it's divisible by $N$. * *Example:* Divisibility by 72: Check divisibility by 8 and 9. * *Example:* Divisibility by 15: Check divisibility by 3 and 5. 2. **Missing Digit Problems:** Use divisibility rules to find missing digits in a number. * *Example:* If $32X5$ is divisible by 3, what is $X$? Sum of digits $3+2+X+5 = 10+X$. For $10+X$ to be divisible by 3, $X$ can be 2, 5, 8. 3. **Large Numbers and Repeated Digits:** * A number formed by repeating a digit 6 times (e.g., 111111, 222222) is always divisible by 7, 11, 13, 37. * A number formed by repeating a 2-digit number three times (e.g., 121212) is divisible by 7, 11, 13, 101. * A number formed by repeating a 3-digit number twice (e.g., 123123) is divisible by 7, 11, 13, 1001. (Since $123123 = 123 \times 1001$, and $1001 = 7 \times 11 \times 13$). 4. **Divisibility by $10^N$:** A number is divisible by $10^N$ if its last $N$ digits are 0. 5. **Divisibility by $2^N$:** A number is divisible by $2^N$ if its last $N$ digits form a number divisible by $2^N$. 6. **Divisibility by $5^N$:** A number is divisible by $5^N$ if its last $N$ digits form a number divisible by $5^N$. 7. **General Rule for Divisibility by Prime $P$ (other than 2, 3, 5):** * To check divisibility by $P$, find $k$ such that $10k+1$ or $10k-1$ is a multiple of $P$. * Or, use the oscillating method: $N = 10a+b$. * For $P=7$: $a-2b$ * For $P=11$: $a-b$ (sum of alternate digits) * For $P=13$: $a+4b$ * For $P=17$: $a-5b$ * For $P=19$: $a+2b$ * For $P=23$: $a+7b$ * *Example (19):* Is 361 divisible by 19? $36 + 2 \times 1 = 38$. Yes, 38 is divisible by 19. 8. **Prime Factorization:** Divisibility rules are essentially quick ways of checking for specific prime factors (or products of prime factors like 4, 6, 8, 9, 10, 12). #### 3. Useful Information * Divisibility rules are essential for quick calculations, finding factors, and solving problems related to number properties. * They are often used in conjunction with L.C.M. & H.C.F., Number System, and Remainder Theorem questions. * Memorizing the rules up to 12 (and 25, 125) is highly recommended. * Practice applying these rules to various numbers to improve speed and accuracy. * When a number has multiple conditions (e.g., divisible by both 3 and 4), check each condition individually. ### Percentages #### 1. Basic Formulas & Concepts * **Percentage:** A fraction with 100 as the denominator. Represented by the symbol '%'. * $X\% = \frac{X}{100}$ * To convert a fraction/decimal to percentage, multiply by 100. * To convert a percentage to a fraction/decimal, divide by 100. * **Calculating Percentage of a Quantity:** * $X\%$ of $Y = \frac{X}{100} \times Y$ * **Percentage Increase/Decrease:** * Percentage Increase = $\frac{\text{Increase Amount}}{\text{Original Amount}} \times 100$ * Percentage Decrease = $\frac{\text{Decrease Amount}}{\text{Original Amount}} \times 100$ * **Result after % Change:** * If a quantity $Q$ is increased by $X\%$, New Quantity = $Q \times (1 + \frac{X}{100})$ * If a quantity $Q$ is decreased by $X\%$, New Quantity = $Q \times (1 - \frac{X}{100})$ * **Percentage Point Change vs. Percentage Change:** * If a rate goes from 4% to 5%, it's a 1 percentage point increase, but a $\frac{5-4}{4} \times 100 = 25\%$ increase. * **Successive Percentage Changes:** If a quantity is changed by $X\%$ and then by $Y\%$: * Final Change = $(X + Y + \frac{XY}{100})\%$ * This is known as the **effective percentage change** or **net percentage change**. * Use positive $X, Y$ for increase, negative for decrease. * *Example:* Price increased by 10%, then decreased by 10%. Net change = $(10 - 10 + \frac{10 \times (-10)}{100})\% = -\frac{100}{100}\% = -1\%$. (1% decrease). #### 2. Shortcuts & Tricks 1. **Fraction-Percentage Equivalents:** Memorize common fraction-percentage conversions for speed. * $1/2 = 50\%$ * $1/3 = 33.33\%$ * $1/4 = 25\%$ * $1/5 = 20\%$ * $1/6 = 16.66\%$ * $1/7 = 14.28\%$ * $1/8 = 12.5\%$ * $1/9 = 11.11\%$ * $1/10 = 10\%$ * $1/11 = 9.09\%$ * $1/12 = 8.33\%$ * And their multiples (e.g., $3/4 = 75\%$). 2. **"Percentage of" is Commutative:** $X\%$ of $Y = Y\%$ of $X$. * *Example:* $16\%$ of $25 = 25\%$ of $16 = \frac{1}{4} \times 16 = 4$. This is often easier. 3. **Base Change Problems:** If A is $X\%$ more than B, then B is how much less than A? * If A is $X\%$ more than B, then $A = B(1 + \frac{X}{100})$. * B is less than A by $\frac{X}{100+X} \times 100 \%$. * If A is $X\%$ less than B, then $B$ is more than A by $\frac{X}{100-X} \times 100 \%$. * *Example:* Price of petrol increases by 25%. By what % should consumption be reduced to keep expenditure same? Here $X=25$. Reduction = $\frac{25}{100+25} \times 100 = \frac{25}{125} \times 100 = \frac{1}{5} \times 100 = 20\%$. 4. **Percentage Point vs. Percentage:** Always be clear about the base for percentage calculations. 5. **Multiplication Factor Method:** * Increase by $X\% \rightarrow$ multiply by $(1 + \frac{X}{100})$ * Decrease by $X\% \rightarrow$ multiply by $(1 - \frac{X}{100})$ * This is very useful for successive changes: Final Value = Original Value $\times (1 \pm \frac{X}{100}) \times (1 \pm \frac{Y}{100}) \times \dots$ 6. **Population Growth/Depreciation:** These are applications of successive percentage changes. * After $N$ years, if growth rate is $R\%$: Population $= \text{Initial} \times (1 + \frac{R}{100})^N$. * Depreciation: Population $= \text{Initial} \times (1 - \frac{R}{100})^N$. 7. **Venn Diagrams for Percentage Problems:** Similar to set theory, Venn diagrams can be used for problems involving percentages of people passing/failing in subjects, or liking different things. * $n(A \cup B) = n(A) + n(B) - n(A \cap B)$. If numbers are percentages, divide by 100. * *Example:* 70% pass in English, 60% pass in Math, 20% fail in both. * Pass in at least one = Total - Fail in both = $100\% - 20\% = 80\%$. * $80\% = 70\% + 60\% - \text{Pass in both}$. * Pass in both = $130\% - 80\% = 50\%$. 8. **Profit & Loss and Simple/Compound Interest:** These topics are direct applications of percentages. 9. **Approximation:** For competitive exams, often options are spaced out enough that rounding off percentages or numbers to the nearest convenient value can save time. 10. **Unitary Method:** If $X\%$ corresponds to a value $V$, then $1\%$ corresponds to $V/X$, and $100\%$ (total) corresponds to $(V/X) \times 100$. #### 3. Useful Information * Percentages are a core concept in quantitative aptitude, forming the basis for many other topics. * Always identify the "base" or "original value" for percentage calculations. A common mistake is to calculate percentage increase/decrease based on the wrong base. * Practice problems involving different scenarios: income-expenditure, population, elections, marks, etc. * Be clear about what "of" and "by" mean in percentage context. "Increased by 10%" means original + 10% of original. "10% of total" means $10/100 \times \text{total}$. * Understanding the relationship between fractions, decimals, and percentages is crucial for flexibility in problem-solving. ### Remainder Theorem #### 1. Basic Formulas & Concepts * **Division Algorithm:** When a number (dividend) is divided by another number (divisor), we get a quotient and a remainder. * Dividend = Divisor $\times$ Quotient + Remainder * $N = D \times Q + R$ * Where $0 \le R ### Probability #### 1. Basic Formulas & Concepts * **Experiment:** An operation that can produce some well-defined outcomes. * **Random Experiment:** An experiment whose outcome cannot be predicted with certainty. * **Sample Space (S):** The set of all possible outcomes of a random experiment. * **Event (E):** A subset of the sample space. * **Probability of an Event:** $$P(E) = \frac{\text{Number of Favorable Outcomes}}{\text{Total Number of Possible Outcomes}} = \frac{n(E)}{n(S)}$$ * **Range of Probability:** $0 \le P(E) \le 1$. * $P(E) = 0$ for an impossible event. * $P(E) = 1$ for a sure event. * **Complementary Event ($E'$ or $E^c$):** The event that E does not occur. * $P(E') = 1 - P(E)$ * **Mutually Exclusive Events:** Events that cannot occur simultaneously ($E_1 \cap E_2 = \emptyset$). * $P(E_1 \cup E_2) = P(E_1) + P(E_2)$ * **Exhaustive Events:** A set of events whose union is the entire sample space ($E_1 \cup E_2 \cup \dots \cup E_n = S$). * **Independent Events:** The occurrence of one event does not affect the probability of the other event. * $P(E_1 \cap E_2) = P(E_1) \times P(E_2)$ * **Conditional Probability:** The probability of event A occurring given that event B has already occurred. * $P(A|B) = \frac{P(A \cap B)}{P(B)}$ (provided $P(B) > 0$) * **Addition Theorem (for any two events):** * $P(E_1 \cup E_2) = P(E_1) + P(E_2) - P(E_1 \cap E_2)$ * **Permutations and Combinations:** Often used to count favorable and total outcomes. * **Permutation ($^n P_r$):** Number of ways to arrange $r$ items from $n$ distinct items, where order matters. $^n P_r = \frac{n!}{(n-r)!}$. * **Combination ($^n C_r$):** Number of ways to select $r$ items from $n$ distinct items, where order does not matter. $^n C_r = \frac{n!}{r!(n-r)!}$. #### 2. Shortcuts & Tricks 1. **"At least one" problems:** Often easier to calculate $1 - P(\text{none})$. * *Example:* Probability of getting at least one head in 3 coin tosses. * Total outcomes = $2^3 = 8$ (HHH, HHT, HTH, THH, HTT, THT, TTH, TTT). * None (all tails) = TTT (1 outcome). * $P(\text{at least one head}) = 1 - P(\text{all tails}) = 1 - \frac{1}{8} = \frac{7}{8}$. 2. **Dice Problems:** * Sum of two dice: $n(S) = 36$. Max sum = 12, Min sum = 2. * Sums 2, 12 (1 way each) * Sums 3, 11 (2 ways each) * Sums 4, 10 (3 ways each) * Sums 5, 9 (4 ways each) * Sums 6, 8 (5 ways each) * Sum 7 (6 ways) * For $N$ dice, $n(S) = 6^N$. 3. **Card Problems:** Standard deck of 52 cards. * 4 suits (Hearts, Diamonds, Clubs, Spades) * 13 cards per suit (A, K, Q, J, 10, 9, 8, 7, 6, 5, 4, 3, 2) * 26 Red (H, D), 26 Black (C, S) * 12 Face Cards (K, Q, J of each suit) * 4 Aces. * Remember combinations for drawing cards: $^n C_r$. * *Example:* Probability of drawing 2 Kings from a deck of 52 cards. * $n(S) = ^{52}C_2 = \frac{52 \times 51}{2} = 1326$. * $n(E) = ^4 C_2 = \frac{4 \times 3}{2} = 6$. * $P(E) = \frac{6}{1326} = \frac{1}{221}$. 4. **Bag/Ball Problems:** * Always use combinations for selecting balls if order doesn't matter. * *Example:* A bag has 3 Red, 4 Blue balls. Probability of drawing 2 Red balls. * $n(S) = ^7 C_2 = \frac{7 \times 6}{2} = 21$. * $n(E) = ^3 C_2 = \frac{3 \times 2}{2} = 3$. * $P(E) = \frac{3}{21} = \frac{1}{7}$. 5. **Conditional Probability (without direct formula):** If $P(A|B)$ is asked, sometimes it's easier to restrict the sample space to event B and then count favorable outcomes for A within that reduced space. 6. **Bayes' Theorem (for advanced conditional probability):** $$P(A|B) = \frac{P(B|A)P(A)}{P(B)}$$ Where $P(B) = P(B|A)P(A) + P(B|A')P(A')$. (Often used in medical testing, reliability problems). 7. **Odd Against / Odd in Favor:** * Odds in favor of E = $P(E) : P(E')$ * Odds against E = $P(E') : P(E)$ 8. **Geometric Probability:** Involves ratio of lengths, areas, or volumes. * *Example:* A point is chosen randomly on a line segment of length L. What is the probability it is within distance $x$ from one end? $2x/L$. 9. **Complementary Events for complex situations:** When a problem involves "at least", "not more than", "not less than", consider using the complementary event. 10. **Tree Diagrams:** Useful for visualizing sequential events and their probabilities, especially when events are dependent. #### 3. Useful Information * Probability is fundamentally about counting. Mastering permutations and combinations is critical. * Clearly define your sample space and the event in question. * Distinguish between "with replacement" and "without replacement" when drawing items. * Understand the difference between mutually exclusive and independent events. * "AND" generally implies multiplication of probabilities (for independent events), while "OR" implies addition (for mutually exclusive events). * For problems involving multiple trials, binomial probability might be relevant: $P(X=k) = ^n C_k p^k (1-p)^{n-k}$. (Probability of exactly $k$ successes in $n$ independent Bernoulli trials). ### Trains #### 1. Basic Formulas & Concepts * **Speed, Time, Distance Relationship:** * Distance = Speed $\times$ Time * Speed = Distance / Time * Time = Distance / Speed * **Unit Conversion:** * $1 \text{ km/hr} = \frac{5}{18} \text{ m/s}$ * $1 \text{ m/s} = \frac{18}{5} \text{ km/hr}$ * **Relative Speed:** * **Same Direction:** $S_{rel} = |S_1 - S_2|$ (subtract speeds) * **Opposite Direction:** $S_{rel} = S_1 + S_2$ (add speeds) * **Distance Covered by a Train:** * **Passing a Pole/Man/Tree (point object):** Distance = Length of the train ($L_T$) * **Passing a Platform/Bridge/Tunnel (object with length):** Distance = Length of the train ($L_T$) + Length of the object ($L_O$) * **Passing another Train:** * If trains are moving in **same direction**: Distance = $L_{T1} + L_{T2}$. Time = $\frac{L_{T1} + L_{T2}}{|S_1 - S_2|}$ * If trains are moving in **opposite direction**: Distance = $L_{T1} + L_{T2}$. Time = $\frac{L_{T1} + L_{T2}}{S_1 + S_2}$ * **Man in a Train:** When a man sitting in a moving train observes another train: * The length of the first train is irrelevant. The distance covered is the length of the second train. * Relative speed is used. #### 2. Shortcuts & Tricks 1. **Direct Unit Conversion:** Immediately convert speeds to m/s if distances are in meters and time in seconds, or vice versa. Avoid mixing units. 2. **Relative Speed - Distance Sum:** In almost all train problems involving two trains or a train passing a stationary object with length, the *total distance covered* is the sum of the lengths of the entities involved. 3. **For a train passing a pole/man/tree:** Time taken = $\frac{\text{Length of Train}}{\text{Speed of Train}}$ 4. **For a train passing a platform/bridge/tunnel:** Time taken = $\frac{\text{Length of Train} + \text{Length of Platform}}{\text{Speed of Train}}$ 5. **Train passing a man *walking* in the same direction:** * Distance = Length of Train. * Relative Speed = Speed of Train - Speed of Man. * Time = $\frac{\text{Length of Train}}{\text{Speed of Train} - \text{Speed of Man}}$ 6. **Train passing a man *walking* in the opposite direction:** * Distance = Length of Train. * Relative Speed = Speed of Train + Speed of Man. * Time = $\frac{\text{Length of Train}}{\text{Speed of Train} + \text{Speed of Man}}$ 7. **Two trains crossing each other (either direction):** * Time = $\frac{\text{Sum of Lengths}}{\text{Relative Speed}}$ 8. **Finding Length of Train when passing two different stationary objects:** * If a train of length $L$ passes a pole in $T_1$ seconds and a platform of length $P$ in $T_2$ seconds, and its speed is $S$: * $S = L/T_1$ * $S = (L+P)/T_2$ * Equate S: $L/T_1 = (L+P)/T_2 \implies LT_2 = LT_1 + PT_1 \implies L(T_2-T_1) = PT_1 \implies L = \frac{PT_1}{T_2-T_1}$ 9. **Meeting Point Problems (Trains moving towards each other):** * If two trains start simultaneously from A and B towards each other and meet after time T, then $T = \frac{\text{Distance between A and B}}{S_A + S_B}$. 10. **After Meeting Point Problems:** * If two trains start from A and B simultaneously, travel towards each other, and after meeting, take $t_1$ and $t_2$ hours respectively to reach B and A, then $\frac{S_A}{S_B} = \sqrt{\frac{t_2}{t_1}}$. #### 3. Useful Information * Train problems are a specific application of Speed, Time, Distance. A strong understanding of the basics of S-T-D is essential. * Always draw a mental picture or a simple diagram of the scenario to correctly identify the distance covered and whether to add or subtract speeds. * Pay close attention to units and ensure consistency throughout the problem. Convert everything to m/s and meters, or km/hr and km. * The length of the train itself is a crucial part of the "distance" in most scenarios. * When a train passes a point object, it covers its own length. When it passes a linear object, it covers its own length plus the length of the object. ### Boats & Streams #### 1. Basic Formulas & Concepts * **Speed of Boat/Man in Still Water ($S_B$ or $u$):** The speed of the boat or swimmer when there is no current. * **Speed of Stream/Current ($S_C$ or $v$):** The speed of the water flow. * **Downstream Speed ($S_D$):** Speed of boat/man with the current. * $S_D = S_B + S_C$ * **Upstream Speed ($S_U$):** Speed of boat/man against the current. * $S_U = S_B - S_C$ (Note: $S_B$ must be greater than $S_C$ for the boat to move upstream) * **Deriving $S_B$ and $S_C$ from $S_D$ and $S_U$:** * $S_B = \frac{S_D + S_U}{2}$ * $S_C = \frac{S_D - S_U}{2}$ * **Distance, Speed, Time Relationship:** * Distance = Speed $\times$ Time * Time = Distance / Speed * Distance = $S_D \times T_D = S_U \times T_U$ #### 2. Shortcuts & Tricks 1. **Immediate Calculation of $S_D$ and $S_U$:** If $S_B$ and $S_C$ are given, immediately calculate $S_D$ and $S_U$. 2. **Immediate Calculation of $S_B$ and $S_C$:** If $S_D$ and $S_U$ are given, immediately calculate $S_B$ and $S_C$. These four values are the core of most problems. 3. **Equal Distance Problems:** * If a boat travels a certain distance upstream and the same distance downstream: * Time taken for upstream ($T_U$) and downstream ($T_D$) journey: $\frac{D}{S_U}$ and $\frac{D}{S_D}$. * Total Time = $\frac{D}{S_U} + \frac{D}{S_D} = D \left( \frac{1}{S_B-S_C} + \frac{1}{S_B+S_C} \right) = D \frac{2S_B}{S_B^2 - S_C^2}$. * Average Speed for round trip = $\frac{2S_D S_U}{S_D + S_U} = \frac{2(S_B+S_C)(S_B-S_C)}{2S_B} = \frac{S_B^2 - S_C^2}{S_B}$. 4. **Ratio Method:** If $S_B:S_C = x:y$, then $S_D:S_U = (x+y):(x-y)$. This can simplify problems involving ratios. * *Example:* Speed of boat in still water is twice the speed of the current. Find ratio of downstream to upstream speed. * $S_B:S_C = 2:1$. * $S_D:S_U = (2+1):(2-1) = 3:1$. 5. **Finding Time to Cover a Distance (Upstream/Downstream):** Directly use $T = D/S$. 6. **Distance Covered by a Floating Object:** A floating object (like a leaf or a message in a bottle) only moves with the speed of the stream. So, its speed is $S_C$. 7. **Meeting Point Problems (Boats moving towards each other in a stream):** * If two boats start from A and B (distance D apart) and move towards each other, one upstream and one downstream. Their relative speed is $S_U + S_D$. * Time to meet = $\frac{D}{S_U + S_D}$. Be careful to identify which boat is going upstream and which downstream. 8. **Man swimming against current, then with current:** Similar to boat and stream problems. The man's speed in still water is $S_B$. 9. **Circular Track in Water:** If a boat travels on a circular track in a river, the concept remains the same: use $S_D$ for one direction and $S_U$ for the other. 10. **Problems where time is constant:** If Time is constant, then Distance is directly proportional to Speed. * $\frac{D_D}{D_U} = \frac{S_D}{S_U}$. * If a person rows to a certain place and back, and the total time is given, and $S_B, S_C$ are known, calculate $D$. * $T_{total} = \frac{D}{S_D} + \frac{D}{S_U}$. Solve for $D$. #### 3. Useful Information * Boats & Streams is a direct application of relative speed concepts from Speed, Time, Distance. * Always clearly identify the direction of travel (upstream or downstream) for each part of the journey. * The key is to correctly set up the equations using $S_D = S_B + S_C$ and $S_U = S_B - S_C$. * Be very careful with units (km/hr vs m/s) and convert them consistently. * Often, the problem will give you two of the four variables ($S_B, S_C, S_D, S_U$) or enough information to find them, and then ask for distance or time. ### Time & Work #### 1. Basic Formulas & Concepts * **Work = Rate $\times$ Time** * **Rate (Efficiency):** Amount of work done per unit of time. Rate = Work / Time. * **Total Work:** Usually assumed as 1 unit, or by finding L.C.M. of individual times. * **Individual Rate:** If a person can do a work in $N$ days, their rate is $1/N$ work per day. * **Combined Rate:** If A's rate is $R_A$ and B's rate is $R_B$, their combined rate is $R_A + R_B$. * **Combined Time:** If A takes $T_A$ days and B takes $T_B$ days to complete a work, their combined time is $\frac{1}{\frac{1}{T_A} + \frac{1}{T_B}} = \frac{T_A T_B}{T_A + T_B}$. * **Work done by M1 men in D1 days working H1 hours/day with E1 efficiency doing W1 work:** $$\frac{M_1 D_1 H_1 E_1}{W_1} = \frac{M_2 D_2 H_2 E_2}{W_2}$$ (This is the Man-Days-Hours-Work formula) * **Negative Work:** Some entities (like leaks in pipes or destructive forces) can do negative work. Their rate is subtracted. #### 2. Shortcuts & Tricks 1. **L.C.M. Method (Unitary Method):** * Assume total work as the L.C.M. of the individual times taken by workers. * Calculate individual efficiencies (units of work per day/hour). * Add efficiencies for combined work, subtract for negative work. * Total Time = Total Work / Combined Efficiency. * *Example:* A takes 10 days, B takes 15 days. L.C.M.(10, 15) = 30 units. * A's efficiency = 30/10 = 3 units/day. * B's efficiency = 30/15 = 2 units/day. * Combined efficiency = 3+2 = 5 units/day. * Time to complete together = 30/5 = 6 days. 2. **Ratio of Time and Efficiency:** Time and Efficiency are inversely proportional. * If A is twice as efficient as B, then A takes half the time B takes. * If ratio of efficiencies of A:B is $x:y$, then ratio of times taken by A:B is $y:x$. 3. **Work done by 'n' men in 'D' days:** Total work = $n \times D$ (man-days). 4. **Pipes and Cisterns:** This is essentially a Time & Work problem. * Inlet pipes do positive work (filling), outlet pipes/leaks do negative work (emptying). * Rate of filling = $1/T_{fill}$. Rate of emptying = $1/T_{empty}$. * Net rate = Sum of filling rates - Sum of emptying rates. * Time to fill = Total Capacity / Net Rate. 5. **Alternating Work:** If A and B work on alternate days: * Calculate work done in 2 days (A's work + B's work). * Find how many such 2-day cycles are needed to complete most of the work. * Calculate remaining work and who finishes it. 6. **Leaving/Joining Work:** * Calculate work done together for a certain period. * Subtract this from total work to find remaining work. * Calculate time for remaining work by remaining workers. 7. **Wages Distribution:** Wages are distributed in proportion to the work done (or efficiency if time is constant). * Total Wage = Sum of individual shares. 8. **Chain Rule (Man-Days-Hours formula application):** * $\frac{M_1 D_1 H_1}{W_1} = \frac{M_2 D_2 H_2}{W_2}$ (If efficiency is same, $E_1=E_2=1$) * This is very useful for comparing two scenarios. 9. **Fraction Method:** * A completes $1/T_A$ of work in 1 day. * If they work for $x$ days, they complete $x/T_A$ of work. * Total work is 1. 10. **Efficiency Percentage:** If A is $X\%$ more efficient than B, A's efficiency is $(1+X/100)$ times B's efficiency. #### 3. Useful Information * Time & Work problems are very common in competitive exams. * The L.C.M. method is generally the most robust and easiest to visualize for multiple workers. * Always ensure consistency of units (days, hours, minutes). * Be careful with negative work (e.g., pipes emptying a tank). * Understand the relationship between work, rate (efficiency), and time. * Practice various types of problems, including those with varying work schedules, wages, and groups of workers. ### Partnership #### 1. Basic Formulas & Concepts * **Partnership:** When two or more people invest money in a business with the intention of sharing profits or losses. * **Types of Partners:** * **Working Partner:** Contributes capital and also manages the business. * **Sleeping Partner:** Contributes capital but does not manage the business. * **Profit Sharing:** * **Simple Partnership (Capital for same duration):** Profits are divided in the ratio of their investments. * If $I_A, I_B, I_C$ are investments of A, B, C, then Profit Ratio = $I_A : I_B : I_C$. * **Compound Partnership (Capital for different durations):** Profits are divided in the ratio of the product of their investments and the time period for which the investment was made. * If $I_A, I_B, I_C$ are investments for time $T_A, T_B, T_C$ respectively, then Profit Ratio = $(I_A \times T_A) : (I_B \times T_B) : (I_C \times T_C)$. * **Loss Sharing:** Losses are also shared in the same ratio as profits. #### 2. Shortcuts & Tricks 1. **Investment $\times$ Time = Equivalent Capital (or Profit Share Unit):** This is the core principle. Calculate this product for each partner. The ratio of these products is the profit-sharing ratio. * *Example:* A invests Rs. 5000 for 8 months. B invests Rs. 4000 for 10 months. * A's equivalent capital = $5000 \times 8 = 40000$ * B's equivalent capital = $4000 \times 10 = 40000$ * Profit Ratio = $40000 : 40000 = 1:1$. 2. **Changing Investments during the year:** * If a partner changes their investment (adds or withdraws money) during the year, calculate the equivalent capital by summing up (Investment $\times$ Time) for each period. * *Example:* A invests $I_1$ for $T_1$ months, then adds $I_2$ (total $I_1+I_2$) for $T_2$ months. * A's equivalent capital = $(I_1 \times T_1) + ((I_1+I_2) \times T_2)$. 3. **Time Period Consistency:** Ensure all time periods are in the same unit (months or years). If the business runs for a year, total time is 12 months. 4. **Working vs. Sleeping Partner:** Sometimes a working partner takes a fixed salary or percentage of profit first, and then the remaining profit is distributed as per investment ratio. * *Steps:* 1. Calculate total profit. 2. Subtract working partner's salary/fixed share. 3. Distribute remaining profit based on investment ratios. 5. **Ratio Simplification:** Always simplify the ratio of equivalent capitals to its lowest terms before calculating individual shares. 6. **Finding Total Capital / Total Profit:** If one partner's share of profit and their ratio is known, you can find the total profit/capital. * If A's share is $P_A$ and profit ratio is $R_A:R_B:R_C$, then Total Profit = $P_A \times \frac{R_A+R_B+R_C}{R_A}$. 7. **Multiple Partners:** The principles extend to any number of partners. 8. **When one partner leaves or joins:** Adjust the time periods for their investments accordingly. 9. **Calculation of time for investment:** If a partner invests for 'x' months and the total duration is 'T' months, then their investment is considered for 'x' months. If they join 'm' months after start, their investment is for 'T-m' months. 10. **Profit = Revenue - Expenses:** Sometimes, problems might include expenses, in which case the net profit is calculated first. #### 3. Useful Information * Partnership problems are a direct application of ratios. * The key is to correctly calculate the "effective investment" for each partner, which is the product of their capital and the time it was invested. * Pay close attention to the time duration of each investment. Months are typically used, so convert years to months if necessary. * Clearly distinguish between total profit, individual profit shares, and how expenses/salaries are handled before distribution. * Practice problems with varying complexities: simple, compound, and those involving working partners or changes in investment. ### SI and CI (Simple Interest and Compound Interest) #### 1. Basic Formulas & Concepts * **Principal (P):** The initial sum of money borrowed or invested. * **Rate (R):** The interest rate per annum (per year). Expressed as a percentage. * **Time (T):** The duration for which the money is borrowed/invested. Usually in years. * **Amount (A):** The total money at the end of the time period (Principal + Interest). * **Simple Interest (SI):** Interest calculated only on the principal amount. * $\text{SI} = \frac{P \times R \times T}{100}$ * $\text{Amount (A)} = P + \text{SI} = P \left(1 + \frac{R \times T}{100}\right)$ * Simple interest is constant for each year. * **Compound Interest (CI):** Interest calculated on the principal amount AND on the accumulated interest of previous periods. * **Compounded Annually:** * $\text{Amount (A)} = P \left(1 + \frac{R}{100}\right)^T$ * $\text{CI} = A - P = P \left[\left(1 + \frac{R}{100}\right)^T - 1\right]$ * **Compounded Half-Yearly:** Rate becomes $R/2$, Time becomes $2T$. * $\text{A} = P \left(1 + \frac{R/2}{100}\right)^{2T}$ * **Compounded Quarterly:** Rate becomes $R/4$, Time becomes $4T$. * $\text{A} = P \left(1 + \frac{R/4}{100}\right)^{4T}$ * **Compounded Monthly:** Rate becomes $R/12$, Time becomes $12T$. * $\text{A} = P \left(1 + \frac{R/12}{100}\right)^{12T}$ * **Compounded Annually, but rate different for each year:** * $\text{A} = P \left(1 + \frac{R_1}{100}\right) \left(1 + \frac{R_2}{100}\right) \dots$ * **Difference between CI and SI for 2 years:** * $\text{CI} - \text{SI} = P \left(\frac{R}{100}\right)^2$ * **Difference between CI and SI for 3 years:** * $\text{CI} - \text{SI} = P \left(\frac{R}{100}\right)^2 \left(3 + \frac{R}{100}\right)$ #### 2. Shortcuts & Tricks 1. **Ratio Method for CI:** * If rate is $R\%$, then the amount becomes $(1 + R/100)$ times the principal each year. * *Example:* $R=10\% = 1/10$. Amount becomes $11/10$ times. * For 2 years: $P \xrightarrow{\times 11/10} A_1 \xrightarrow{\times 11/10} A_2$. * This is useful for finding the ratio of P, A, CI. 2. **Effective Rate for CI:** For compounding periods other than annually, calculate the effective annual rate if needed, or simply adjust R and T. 3. **Percentage Change Method (for CI):** CI can be viewed as successive percentage increases. * For 2 years: Effective percentage increase = $(R + R + \frac{R \times R}{100})\% = (2R + \frac{R^2}{100})\%$. This is the CI as a percentage of Principal. * *Example:* $R=10\%$. Effective rate for 2 years CI = $(2 \times 10 + \frac{10^2}{100})\% = (20 + 1)\% = 21\%$. * CI = $21\%$ of P. 4. **Tree Method / Waterfall Method (for CI):** Visualizes interest earned each year. * *Year 1:* P $\times R\%$ (SI for 1st year = CI for 1st year) * *Year 2:* P $\times R\%$ (on original P) + (P $\times R\%$) $\times R\%$ (interest on 1st year's interest). * This clearly shows the difference between CI and SI. * *Example:* P=1000, R=10%, T=2 years. * Year 1: Interest = $1000 \times 10\% = 100$. * Year 2: Interest on P = $1000 \times 10\% = 100$. Interest on Year 1 Interest = $100 \times 10\% = 10$. * Total CI = $100 + 100 + 10 = 210$. * Total SI = $100 + 100 = 200$. * Difference CI - SI = 10. 5. **Doubling/Tripling Time:** * **Simple Interest:** If money doubles in $N$ years at SI, then $R = \frac{100}{N}$. If it becomes $X$ times, $R = \frac{(X-1)100}{N}$. * **Compound Interest:** If money doubles in $N$ years at CI, it will become 4 times in $2N$ years, 8 times in $3N$ years, etc. (Geometric progression). * **Rule of 72 (Approximation for CI doubling time):** Time to double $\approx \frac{72}{R}$ years (for R in percentage). 6. **Installment Problems (Loan Repayment):** * **Simple Interest:** If a loan of P is repaid in $n$ equal annual installments of $x$, with interest R. * $P = \frac{x}{1+RT_1} + \frac{x}{1+RT_2} + \dots$ (where $T_i$ are times from repayment to end of total loan term). This is complex. * Simpler approach: Total principal + Total interest = $n \times x$. * Total Interest = Interest on P for 1 year + Interest on (P-x) for 1 year + ... * **Compound Interest:** For installments $x$ and rate $R$: * $P = \frac{x}{(1 + R/100)^1} + \frac{x}{(1 + R/100)^2} + \dots + \frac{x}{(1 + R/100)^n}$ 7. **Finding P, R, T when others are given:** Rearrange the formulas. 8. **When interest is paid back after specific intervals:** The principal reduces. Calculate interest for each interval separately on the *new* principal. 9. **Approximation:** For multiple-choice questions, if options are far apart, use approximation. #### 3. Useful Information * SI and CI are fundamental for financial calculations. * Understand the core difference: SI is constant on principal, CI is interest on interest. * Pay close attention to the compounding frequency (annually, semi-annually, quarterly, etc.) and adjust R and T accordingly. * Memorize the formulas for CI-SI difference for 2 and 3 years. * Practice problems that combine these concepts, such as finding the difference between CI and SI, or finding the principal given the amount and interest. * Be careful with time units (years, months, days). Convert everything to years for the formula. ### Mensuration & Area #### 1. Basic Formulas & Concepts * **Mensuration:** Deals with the measurement of geometric figures (lengths, areas, volumes). * **Perimeter:** The total distance around the boundary of a 2D shape. * **Area:** The amount of surface enclosed by a 2D shape. * **Volume:** The amount of space occupied by a 3D object. * **Surface Area:** The total area of all surfaces of a 3D object. #### 1.1. 2D Shapes (Area & Perimeter) * **Square:** * Side: $a$ * Perimeter = $4a$ * Area = $a^2$ * Diagonal = $a\sqrt{2}$ * **Rectangle:** * Length: $l$, Breadth: $b$ * Perimeter = $2(l+b)$ * Area = $l \times b$ * Diagonal = $\sqrt{l^2+b^2}$ * **Triangle:** * Base: $b$, Height: $h$ * Area = $\frac{1}{2} \times b \times h$ * **Heron's Formula:** For sides $a, b, c$. Semi-perimeter $s = \frac{a+b+c}{2}$. Area = $\sqrt{s(s-a)(s-b)(s-c)}$ * **Equilateral Triangle:** Side $a$. Area = $\frac{\sqrt{3}}{4} a^2$. Height = $\frac{\sqrt{3}}{2} a$. * **Right-angled Triangle:** Base $b$, Height $h$. Area = $\frac{1}{2} b h$. Hypotenuse = $\sqrt{b^2+h^2}$. * **Circle:** * Radius: $r$, Diameter: $d = 2r$ * Circumference (Perimeter) = $2\pi r = \pi d$ * Area = $\pi r^2$ * Area of Sector = $\frac{\theta}{360^\circ} \pi r^2$ (where $\theta$ is angle in degrees) * Length of Arc = $\frac{\theta}{360^\circ} 2\pi r$ * **Semi-circle:** * Radius: $r$ * Perimeter = $\pi r + 2r$ * Area = $\frac{1}{2} \pi r^2$ * **Parallelogram:** * Base: $b$, Height: $h$ * Area = $b \times h$ * **Rhombus:** * Diagonals: $d_1, d_2$ * Area = $\frac{1}{2} d_1 d_2$ * **Trapezium (Trapezoid):** * Parallel sides: $a, b$, Height: $h$ * Area = $\frac{1}{2} (a+b) h$ #### 1.2. 3D Shapes (Volume & Surface Area) * **Cube:** * Side: $a$ * Volume = $a^3$ * Lateral Surface Area (LSA) = $4a^2$ * Total Surface Area (TSA) = $6a^2$ * Diagonal = $a\sqrt{3}$ * **Cuboid:** * Length: $l$, Breadth: $b$, Height: $h$ * Volume = $l \times b \times h$ * LSA = $2h(l+b)$ * TSA = $2(lb+bh+hl)$ * Diagonal = $\sqrt{l^2+b^2+h^2}$ * **Cylinder:** * Radius: $r$, Height: $h$ * Volume = $\pi r^2 h$ * Curved Surface Area (CSA) = $2\pi r h$ * TSA = $2\pi r h + 2\pi r^2 = 2\pi r (h+r)$ * **Cone:** * Radius: $r$, Height: $h$, Slant Height: $l = \sqrt{r^2+h^2}$ * Volume = $\frac{1}{3} \pi r^2 h$ * CSA = $\pi r l$ * TSA = $\pi r l + \pi r^2 = \pi r (l+r)$ * **Sphere:** * Radius: $r$ * Volume = $\frac{4}{3} \pi r^3$ * Surface Area = $4\pi r^2$ * **Hemisphere:** * Radius: $r$ * Volume = $\frac{2}{3} \pi r^3$ * CSA = $2\pi r^2$ * TSA = $3\pi r^2$ * **Prism:** Volume = Base Area $\times$ Height * **Pyramid:** Volume = $\frac{1}{3} \times$ Base Area $\times$ Height #### 2. Shortcuts & Tricks 1. **Pythagorean Triplets:** Memorize common triplets (3,4,5; 5,12,13; 7,24,25; 8,15,17) to quickly find missing sides in right-angled triangles for area/perimeter/diagonal calculations. 2. **Ratio of Areas/Volumes:** * If sides of two similar figures are in ratio $a:b$, then their areas are in ratio $a^2:b^2$. * If sides of two similar 3D figures are in ratio $a:b$, then their volumes are in ratio $a^3:b^3$. * *Example:* If radius of circle is doubled, area becomes 4 times ($2^2$). 3. **Percentage Change in Area/Volume:** * For 2D shapes: If length changes by $X\%$ and breadth by $Y\%$, then area changes by $(X+Y+\frac{XY}{100})\%$. (Successive percentage change, as in Percentages section). * For 3D shapes: Can be extended, e.g., if side of cube increases by $X\%$, volume increases by $(3X + \frac{3X^2}{100} + \frac{X^3}{10000})\%$. Or use multiplication factors $(1+X/100)^3$. 4. **Area of path around/inside a rectangle:** * Path outside: Area $= 2w(l+b+2w)$ * Path inside: Area $= 2w(l+b-2w)$ (where $w$ is width of path) 5. **Area of cross-roads in a rectangle:** Area $= w(l+b-w)$ (where $w$ is width of road). 6. **Relationship between Sphere and Cylinder:** If a sphere is inscribed in a cylinder, then $H_{cyl} = 2R_{sphere}$ and $R_{cyl} = R_{sphere}$. * Volume of cylinder : Volume of sphere : Volume of cone (with same base and height) = $3:2:1$. 7. **Melt and Recast Problems:** When a solid is melted and recast into another shape, the volume remains constant. * Volume of original solid = Volume of new solid. 8. **Costing Problems:** * Cost of fencing = Perimeter $\times$ Rate per unit length. * Cost of painting/tiling = Area $\times$ Rate per unit area. * Cost of filling = Volume $\times$ Rate per unit volume. 9. **Units:** Be extremely careful with units. Convert everything to a consistent unit (e.g., cm, $cm^2, cm^3$ or m, $m^2, m^3$). $1 \text{ m} = 100 \text{ cm}$. $1 \text{ m}^2 = 10000 \text{ cm}^2$. $1 \text{ m}^3 = 10^6 \text{ cm}^3$. 10. **Approximation of $\pi$:** Use $22/7$ or $3.14$ as needed. If options are far apart, rounding $\pi$ to 3 can sometimes quickly eliminate options. #### 3. Useful Information * Mensuration requires memorizing a lot of formulas. Create a formula sheet and practice regularly. * Drawing diagrams for each problem helps in visualization and correctly applying formulas. * Understand the difference between LSA/CSA and TSA for 3D shapes. * Many competitive exams feature questions that combine two or more geometric shapes (e.g., a cone on top of a cylinder). Break these down into simpler parts. * Focus on understanding how changes in dimensions affect area and volume (scale factors). * For problems involving cutting, folding, or transformation of shapes, the key is to identify what remains constant (e.g., volume in melting, surface area in some cases of unfolding). ### Time and Distance #### 1. Basic Formulas & Concepts * **Speed, Time, Distance Relationship:** * Distance (D) = Speed (S) $\times$ Time (T) * Speed (S) = Distance (D) / Time (T) * Time (T) = Distance (D) / Speed (S) * **Unit Conversion:** * $1 \text{ km/hr} = \frac{5}{18} \text{ m/s}$ * $1 \text{ m/s} = \frac{18}{5} \text{ km/hr}$ * **Relative Speed:** * **Same Direction:** $S_{rel} = |S_1 - S_2|$ (subtract speeds) * **Opposite Direction:** $S_{rel} = S_1 + S_2$ (add speeds) * **Average Speed:** * **General:** Average Speed = $\frac{\text{Total Distance}}{\text{Total Time}}$ * **Equal Distances:** If $D_1=D_2=D$ at speeds $S_1, S_2$, Average Speed = $\frac{2S_1S_2}{S_1+S_2}$ (Harmonic Mean). * **Equal Times:** If $T_1=T_2=T$ at speeds $S_1, S_2$, Average Speed = $\frac{S_1+S_2}{2}$ (Arithmetic Mean). #### 2. Shortcuts & Tricks 1. **Constant Distance:** If Distance is constant, Speed is inversely proportional to Time (S $\propto 1/T$). * If $S_1/S_2 = a/b$, then $T_1/T_2 = b/a$. * *Example:* If a car covers a distance at $S_1$ speed in $T_1$ time and $S_2$ speed in $T_2$ time, then $S_1 T_1 = S_2 T_2$. 2. **Constant Time:** If Time is constant, Distance is directly proportional to Speed (D $\propto S$). * If $S_1/S_2 = a/b$, then $D_1/D_2 = a/b$. 3. **Constant Speed:** If Speed is constant, Distance is directly proportional to Time (D $\propto T$). * If $T_1/T_2 = a/b$, then $D_1/D_2 = a/b$. 4. **Meeting Point Problems (Moving Towards Each Other):** * If two objects start simultaneously from A and B (distance D apart) and move towards each other with speeds $S_1, S_2$. * Time to Meet = $\frac{D}{S_1+S_2}$. 5. **Meeting Point Problems (Moving in Same Direction):** * If two objects start simultaneously from A and B (distance D apart), with $S_1 > S_2$, and move in the same direction (e.g., A chasing B). * Time to Meet/Overtake = $\frac{D}{S_1-S_2}$. 6. **Problems with Lateness/Earliness:** * If a person travels at speed $S_1$ and is $t_1$ late, and at speed $S_2$ and is $t_2$ early/late. * Distance = $\frac{S_1 S_2}{|S_1 - S_2|} \times \text{Time Difference}$ (in hours). * Time Difference needs to be converted to hours if given in minutes. * *Example:* A man travels at 30 km/hr and reaches 10 min late. He travels at 40 km/hr and reaches 5 min early. * Time difference = 10 min (late) + 5 min (early) = 15 min = $15/60 = 1/4$ hr. * Distance = $\frac{30 \times 40}{|30-40|} \times \frac{1}{4} = \frac{1200}{10} \times \frac{1}{4} = 120 \times \frac{1}{4} = 30$ km. 7. **Stopping Time:** If a vehicle travels at $S_1$ km/hr without stopping and $S_2$ km/hr with stopping, then: * Time per hour it stops = $\frac{S_1 - S_2}{S_1}$ hours. * *Example:* A bus travels at 60 km/hr without stops and 40 km/hr with stops. * Stopping time per hour = $\frac{60-40}{60} = \frac{20}{60} = \frac{1}{3}$ hour = 20 minutes. 8. **Ratio of Time and Speed:** Use the inverse relationship directly. If speeds are in ratio $a:b:c$, times are in ratio $1/a:1/b:1/c$. Convert to integers by multiplying by L.C.M. 9. **Circular Track Problems:** * When two people run on a circular track: * **Same Direction:** They meet for the first time after Time = $\frac{\text{Circumference}}{|S_1-S_2|}$. * **Opposite Direction:** They meet for the first time after Time = $\frac{\text{Circumference}}{S_1+S_2}$. * Number of distinct meeting points. 10. **Escalator Problems:** * Speed of man (on still escalator) = $S_M$. Speed of escalator = $S_E$. * Upstream (against escalator): Effective speed = $S_M - S_E$. * Downstream (with escalator): Effective speed = $S_M + S_E$. * Number of steps visible = Total steps. * Total steps = (Effective speed of person) $\times$ (Time taken). #### 3. Useful Information * Time and Distance is a very broad topic, often combined with Trains and Boats & Streams. * Always ensure units are consistent. This is the most common source of error. * Drawing diagrams for complex scenarios (e.g., two people meeting) can be very helpful. * Understanding relative speed is crucial for solving problems involving multiple moving objects. * Memorize the formulas for average speed, especially for equal distances and equal times. * Practice a wide variety of problems to become adept at identifying which formula or trick to apply. ### Profit & Loss #### 1. Basic Formulas & Concepts * **Cost Price (CP):** The price at which an article is purchased. * **Selling Price (SP):** The price at which an article is sold. * **Marked Price (MP) / List Price:** The price at which an article is listed for sale, usually higher than CP. * **Profit (Gain):** When SP > CP. * Profit = SP - CP * Profit % = $\frac{\text{Profit}}{\text{CP}} \times 100$ * **Loss:** When SP Y$, it's a profit. If $X ### Work & Wages #### 1. Basic Formulas & Concepts * **Wages are Proportional to Work Done:** The fundamental principle is that payment received is directly proportional to the amount of work completed. * **Wages are Proportional to Efficiency if Time is Constant:** If multiple individuals work for the same amount of time, their wages are distributed according to their individual efficiencies (rate of work). * **Wages are Proportional to (Efficiency $\times$ Time) if Work is Constant:** If total work is fixed, and individuals work for different durations or have different efficiencies, their wages are proportional to the product of their efficiency and the time they spent working. * **Combined Work and Wages:** If a group of people complete a task and receive a total wage, individual shares are determined by their contribution to the total work. #### 2. Shortcuts & Tricks 1. **L.C.M. Method for Work Done:** (Reiterated from Time & Work) * Find the L.C.M. of the individual times taken to complete the work. This L.C.M. represents the "Total Units of Work". * Calculate the efficiency (units of work per day/hour) for each person. * If they work for the same number of days, their wages are in the ratio of their efficiencies. * If they work for different number of days, their wages are in the ratio of (Efficiency $\times$ Days Worked). * *Example:* A takes 10 days, B takes 15 days. They work together for 5 days and get paid Rs. 500. * Total Work = L.C.M.(10, 15) = 30 units. * A's efficiency = 3 units/day. B's efficiency = 2 units/day. * In 5 days: A does $3 \times 5 = 15$ units. B does $2 \times 5 = 10$ units. * Work done ratio A:B = 15:10 = 3:2. * A's share = $\frac{3}{3+2} \times 500 = \frac{3}{5} \times 500 = 300$. * B's share = $\frac{2}{5} \times 500 = 200$. 2. **Involvement of a Third Person (e.g., C):** * If A and B can do a work in $T_{AB}$ days, and A, B, C together in $T_{ABC}$ days, then C's work rate can be found by $1/T_{ABC} - (1/T_A + 1/T_B)$. * Then distribute wages based on the ratio of work done by A, B, and C. * *Example:* A and B work together for 10 days to complete a work. They are joined by C and finish the remaining work in 2 days. Total wage Rs. 1000. If A takes 20 days alone, B takes 30 days alone. * A's rate = 1/20, B's rate = 1/30. * L.C.M.(20, 30) = 60 units. A's eff = 3, B's eff = 2. * Work by A and B in 10 days = $(3+2) \times 10 = 50$ units. * Remaining work = $60 - 50 = 10$ units. * This 10 units is done by A, B, C in 2 days. Combined eff of A+B+C = $10/2 = 5$ units/day. * C's efficiency = $5 - (3+2) = 0$ units/day. This implies C did no work. (This is a hypothetical example, usually C would have an efficiency). * Let's rephrase: A, B, C together finish work in 6 days. A takes 10, B takes 15. * Total Work = 30 units. A eff = 3, B eff = 2. A+B+C eff = $30/6 = 5$. * C's eff = $5 - (3+2) = 0$. Still 0. * Let's make C contribute: A takes 10, B takes 15, C takes 30. * Total Work = 30 units. A eff = 3, B eff = 2, C eff = 1. * Ratio of efficiencies = 3:2:1. * A's share = $\frac{3}{3+2+1} \times \text{Total Wage} = \frac{3}{6} \times \text{Total Wage}$. 3. **Wages for partial work:** If a person leaves after doing partial work, they get paid for the work they actually completed. 4. **Comparing work done to total work:** * If A completes $1/3$ of the work, B completes $1/2$ of the work, and C completes the rest. * Work by C = $1 - (1/3 + 1/2) = 1 - (2/6 + 3/6) = 1 - 5/6 = 1/6$. * Wages are distributed in ratio $1/3 : 1/2 : 1/6 \implies 2:3:1$. 5. **Man-Days-Hours Formula for Wages:** While the formula $\frac{M_1 D_1 H_1 E_1}{W_1} = \frac{M_2 D_2 H_2 E_2}{W_2}$ is for work comparison, it implies that $W_1$ and $W_2$ are proportional to work done, which in turn is proportional to wages. So, if total wages are given, the wages can be split based on the work done by each group. 6. **Combined efficiency and time:** If $M$ men and $W$ women work together, and their individual efficiencies are known, calculate the total efficiency and then the work done. 7. **Ratio of work done:** Always calculate the ratio of work done by each individual or group to determine the distribution of wages. 8. **Understanding the "unit" of work:** Whether it's units per day, per hour, etc., be consistent. 9. **Total amount of work:** If not specified, assume it to be 1 or use the LCM method. 10. **Consider the scenario:** Are they working together for the entire duration? Do some leave early? Does a new person join? These factors influence the actual work done by each. #### 3. Useful Information * Work & Wages is essentially an extension of Time & Work problems, with an added layer of financial distribution. * The core principle is direct proportionality of wages to work done. * The L.C.M. method is highly effective for these problems as it provides concrete "units of work" to distribute. * Ensure that you are calculating the actual work done by each person/group, not just their potential to do work. * Practice problems that involve different scenarios of work completion and varying efficiencies. ### Pipes & Cisterns #### 1. Basic Formulas & Concepts * **Pipes & Cisterns:** A specialized application of Time & Work problems, where filling pipes are "workers" doing positive work, and emptying pipes/leaks are "workers" doing negative work. * **Total Work (Capacity):** Represented by the total volume/capacity of the tank/cistern. Often assumed as 1 unit, or by taking the L.C.M. of individual pipe times. * **Rate of Filling/Emptying:** * If a pipe can fill a tank in $T_{fill}$ hours, its filling rate is $1/T_{fill}$ of the tank per hour. * If a pipe can empty a tank in $T_{empty}$ hours, its emptying rate is $1/T_{empty}$ of the tank per hour. * **Net Rate:** * If multiple pipes are working simultaneously, the net rate is the sum of filling rates minus the sum of emptying rates. * Net Rate = $\sum (\text{Filling Rates}) - \sum (\text{Emptying Rates})$ * **Time to Fill/Empty:** * Time = Total Capacity / Net Rate * If Net Rate is positive, the tank fills. If negative, it empties. * **Partial Filling/Emptying:** If a pipe is open for a specific duration, calculate the fraction of the tank filled/emptied. #### 2. Shortcuts & Tricks 1. **L.C.M. Method (Unitary Method):** (Most efficient for these problems) * Assume the total capacity of the tank as the L.C.M. of the individual times taken by all pipes (filling and emptying). * Calculate the efficiency (units of tank filled/emptied per hour/minute) for each pipe. Assign positive for filling, negative for emptying. * Sum the efficiencies to get the net efficiency. * Time to fill/empty = Total Capacity / Net Efficiency. * *Example:* Pipe A fills in 10 hours, Pipe B fills in 15 hours, Pipe C empties in 12 hours. * Total Capacity = L.C.M.(10, 15, 12) = 60 units. * A's efficiency = $+60/10 = +6$ units/hour. * B's efficiency = $+60/15 = +4$ units/hour. * C's efficiency = $-60/12 = -5$ units/hour. * Net efficiency (A+B+C) = $6+4-5 = +5$ units/hour. * Time to fill the tank = $60/5 = 12$ hours. 2. **Fraction Method:** * Tank filled by A in 1 hour = $1/T_A$. * Tank emptied by B in 1 hour = $1/T_B$. * Net fraction filled in 1 hour = $1/T_A - 1/T_B$. * Total time = $1 / (\text{Net fraction filled in 1 hour})$. 3. **Alternating Pipes:** If pipes work in turns: * Calculate the net work done in one cycle (e.g., 2 hours if two pipes alternate). * Determine how many such cycles are needed to fill most of the tank. * Calculate remaining work and the time taken by the pipe that starts next. 4. **Leak Problems:** A leak is essentially an emptying pipe. Treat it as such. * If a pipe fills in $T_P$ and a leak empties in $T_L$, then net time to fill is $\frac{1}{1/T_P - 1/T_L}$. * If a pipe fills in $T_P$ and due to a leak, it fills in $T_{actual}$, then the leak alone can empty the tank in time $T_L = \frac{T_P T_{actual}}{T_{actual} - T_P}$. 5. **Opening/Closing Pipes at Different Times:** * Calculate the work done (filled/emptied) during each phase where different combinations of pipes are open. * Keep track of the current level of water in the tank. * Sum up the times taken for each phase. 6. **Partial Filling/Emptying:** If only a fraction of the tank needs to be filled (e.g., $2/3$ full), multiply the total time to fill by that fraction. 7. **Relation to Time & Work:** Remember that all Time & Work principles apply. If a pipe's efficiency is twice another, it takes half the time. 8. **Units of Time:** Be careful to use consistent units of time (minutes, hours). Convert if necessary. 9. **Capacity of tank:** If the capacity is given, use it. Otherwise, assume LCM. 10. **Water level in tank:** Sometimes problems involve the water level in the tank. This is usually directly proportional to the volume. #### 3. Useful Information * Pipes & Cisterns problems are a direct variation of Time & Work. If you are comfortable with Time & Work, this topic will be straightforward. * The L.C.M. method is almost always the most intuitive and error-proof approach. * Always clearly differentiate between filling pipes (positive work) and emptying pipes/leaks (negative work). * Carefully read the problem to understand the sequence of events (which pipes open when, for how long). * Practice problems involving various combinations of pipes, leaks, and timing. ### P & C (Permutations & Combinations) #### 1. Basic Formulas & Concepts * **Factorial (!):** The product of all positive integers less than or equal to a given positive integer. $n! = n \times (n-1) \times \dots \times 2 \times 1$. $0! = 1$. * **Fundamental Principle of Counting:** * **Multiplication Principle:** If an event can occur in $m$ ways and another independent event can occur in $n$ ways, then both events can occur in $m \times n$ ways. * **Addition Principle:** If an event can occur in $m$ ways OR another event can occur in $n$ ways (and they are mutually exclusive), then the total number of ways is $m+n$. * **Permutation:** An arrangement of objects in a specific order. Order matters. * **Number of permutations of $n$ distinct objects taken $r$ at a time:** $$^n P_r = \frac{n!}{(n-r)!}$$ * **Number of permutations of $n$ distinct objects:** $n!$ * **Number of permutations of $n$ objects where $p_1$ are of one type, $p_2$ of another, etc.:** $$\frac{n!}{p_1! p_2! \dots p_k!}$$ * **Circular Permutations:** * Number of circular permutations of $n$ distinct objects = $(n-1)!$ * If clockwise and anticlockwise arrangements are considered same (e.g., necklace, beads), then $\frac{(n-1)!}{2}$. * **Combination:** A selection of objects where the order does not matter. * **Number of combinations of $n$ distinct objects taken $r$ at a time:** $$^n C_r = \frac{n!}{r!(n-r)!}$$ * **Properties of Combinations:** * $^n C_r = ^n C_{n-r}$ * $^n C_0 = 1, ^n C_n = 1, ^n C_1 = n$ * $^n C_r + ^n C_{r-1} = ^{n+1} C_r$ #### 2. Shortcuts & Tricks 1. **"AND" vs. "OR":** * "AND" problems typically involve multiplication (e.g., select a shirt AND a pant). * "OR" problems typically involve addition (e.g., select a shirt OR a t-shirt). 2. **Repetition Allowed vs. Not Allowed:** * If repetition is allowed, the number of choices for each position remains constant. * *Example:* 3-digit number from {1,2,3,4} with repetition: $4 \times 4 \times 4 = 64$. * If repetition is not allowed, the number of choices decreases for subsequent positions. * *Example:* 3-digit number from {1,2,3,4} without repetition: $4 \times 3 \times 2 = 24$. (which is $^4 P_3$) 3. **Specific Positions/Restrictions:** Handle restrictions first. * *Example:* How many 5-digit numbers can be formed using 0,1,2,3,4 if the first digit cannot be 0? * First digit: 4 choices (1,2,3,4). * Remaining 4 digits: $4 \times 3 \times 2 \times 1 = 24$ ways. * Total = $4 \times 24 = 96$. 4. **Objects Always Together / Never Together:** * **Always Together (Block Method):** Treat the objects that must stay together as a single unit. Arrange this unit with other objects, then arrange the objects within the unit. * *Example:* Arrange A,B,C,D,E such that A and B are always together. * Treat (AB) as one unit. Arrangements of (AB),C,D,E = $4! = 24$. * Arrangements of A,B within (AB) = $2! = 2$. * Total = $24 \times 2 = 48$. * **Never Together:** Total arrangements - Arrangements where they are always together. * *Example:* A,B,C,D,E such that A and B are never together. * Total = $5! = 120$. * Always together = 48. * Never together = $120 - 48 = 72$. 5. **Gap Method (for "never together"):** Arrange the other items first, then place the restricted items in the gaps created. * *Example:* Arrange B,C,D,E (4 items) in $4!$ ways. This creates 5 gaps ( _ B _ C _ D _ E _ ). Place A in any of these 5 gaps. $4! \times 5 = 24 \times 5 = 120$. Wait, this is wrong. This is for A not being next to any specific letter, not A and B not together. * Gap method is more complex than (Total - Together). * **Correct Gap Method for "never together":** Arrange the "non-restricted" items. This creates gaps. Place the "restricted" items in these gaps. * *Example:* Arrange A,B,C,D,E such that A and B are never together. * Arrange C,D,E: $3! = 6$ ways. (e.g., C D E) * Gaps created: _ C _ D _ E _ (4 gaps). * Place A and B in 2 of these 4 gaps: $^4 P_2 = 4 \times 3 = 12$ ways. * Total = $6 \times 12 = 72$. (Matches previous result). 6. **Committee/Selection Problems:** Use combinations ($^n C_r$) as order doesn't matter. * *Example:* Select a committee of 3 men and 2 women from 5 men and 4 women. * Men: $^5 C_3 = 10$. Women: $^4 C_2 = 6$. * Total ways = $10 \times 6 = 60$. 7. **At Least/At Most Problems:** Use complementary approach ($1 - P(\text{none})$) or sum individual cases. * *Example:* Select at least 1 girl from 3 girls and 4 boys. * Total selections = $^7 C_x$. (Assuming a total selection size is specified, e.g., 3 people). * Select 3 people: Total = $^7 C_3 = 35$. * No girls (all boys) = $^4 C_3 = 4$. * At least 1 girl = $35 - 4 = 31$. 8. **Digit Problems with 0:** When forming numbers, the first digit cannot be 0. Account for this restriction first. 9. **Geometrical Problems:** * Number of lines formed by $N$ points: $^N C_2$. If $K$ points are collinear, subtract $^K C_2$ and add 1 (for the single line). * Number of triangles formed by $N$ points: $^N C_3$. If $K$ points are collinear, subtract $^K C_3$. * Number of diagonals in a polygon with $N$ sides: $^N C_2 - N$. 10. **Divisibility:** When forming numbers divisible by 2, 5, 10, etc., place the restricted digit (unit digit) first. #### 3. Useful Information * The biggest challenge in P&C is discerning whether a problem requires permutation (order matters) or combination (order doesn't matter). * Always break down complex problems into smaller, manageable choices using the multiplication and addition principles. * Clearly identify any restrictions or conditions (e.g., repetition, specific positions, items together/apart). * Practice a wide range of problems, especially those involving words, numbers, committees, and geometric figures. * Memorize factorial values for small numbers ($0! - 7!$) to save time. * P&C is foundational for Probability. ### Alligation & Mixtures #### 1. Basic Formulas & Concepts * **Alligation Rule:** A rule that helps find the ratio in which two or more ingredients at different prices (or concentrations) must be mixed to produce a mixture of a desired mean price (or concentration). * Let $C_1$ be the cost of the first ingredient, and $C_2$ be the cost of the second ingredient. * Let $C_m$ be the mean cost of the mixture. * The ratio of the quantities of the two ingredients is: $$\frac{\text{Quantity of 1st Ingredient}}{\text{Quantity of 2nd Ingredient}} = \frac{C_2 - C_m}{C_m - C_1}$$ * This can be visualized as a cross-diagram: ``` C1 C2 \ / \ / \ / ------ Cm ------ / \ / \ / \ (C2 - Cm) (Cm - C1) ``` * The ratio of quantities is $(C_2 - C_m) : (C_m - C_1)$. * **Important:** All three values ($C_1, C_2, C_m$) must be in the same units (e.g., cost/kg, percentage concentration, speed, etc.). $C_m$ must lie between $C_1$ and $C_2$. * **Mixture:** A blend of two or more substances. * **Concentration:** The amount of a substance in a given volume or mass of a mixture (often expressed as a percentage or ratio). #### 2. Shortcuts & Tricks 1. **Direct Application of Alligation Rule:** For problems involving mixing two types of items (e.g., rice at Rs. X/kg and rice at Rs. Y/kg to get a mixture at Rs. Z/kg), directly apply the alligation diagram. * *Example:* Rice A at Rs. 10/kg, Rice B at Rs. 15/kg. Mixture at Rs. 12/kg. ``` 10 15 \ / \ / \ / ------ 12 ------ / \ / \ / \ (15 - 12) (12 - 10) 3 2 ``` Ratio of quantities A:B = 3:2. 2. **Alligation with Percentages:** Can be used for mixing solutions of different concentrations. * *Example:* 20% alcohol solution mixed with 50% alcohol solution to get 30% alcohol solution. ``` 20% 50% \ / \ / \ / ------ 30% ------ / \ / \ / \ (50 - 30) (30 - 20) 20 10 ``` Ratio of quantities = 20:10 = 2:1. 3. **Alligation with Averages/Speeds:** * If a person covers a distance at speed $S_1$ for time $T_1$ and at speed $S_2$ for time $T_2$, the average speed is $S_{avg}$. * Here, speeds $S_1, S_2, S_{avg}$ are the "costs", and the ratio obtained is the ratio of "times" $T_1:T_2$. ``` S1 S2 \ / \ / \ / ----- S_avg ----- / \ / \ / \ (S2 - S_avg) (S_avg - S1) ``` Ratio of times = $(S_2 - S_{avg}) : (S_{avg} - S_1)$. 4. **Replacing a part of the Mixture:** * If a container contains $X$ units of mixture, and $Y$ units are withdrawn and replaced by pure liquid. This process is repeated $N$ times. * Final quantity of original liquid = $X \left(1 - \frac{Y}{X}\right)^N$. * *Example:* A 50L milk container. 10L is removed and replaced by water. This is done twice. * Original milk = 50L. $Y=10, X=50, N=2$. * Milk remaining = $50 \left(1 - \frac{10}{50}\right)^2 = 50 \left(1 - \frac{1}{5}\right)^2 = 50 \left(\frac{4}{5}\right)^2 = 50 \times \frac{16}{25} = 2 \times 16 = 32$ L. 5. **Mixing more than two ingredients:** Alligation rule can be extended or applied in stages. * Mix two ingredients, find the mean. Then mix this mean with the third ingredient. 6. **Ratio Method for Mixtures:** * If a mixture has components A and B in ratio $x:y$. * When a quantity is removed, the ratio of A and B remains the same in the removed quantity and in the remaining mixture. * When a pure component is added, the quantity of the other component remains constant. * *Example:* Milk:Water = 3:1. Remove 4L, add 4L water. New ratio 2:1. * Let total be $4k$. Milk = $3k$, Water = $k$. * After 4L removed: Milk = $3k-3$, Water = $k-1$. (Ratio is still 3:1) * Add 4L water: Milk = $3k-3$, Water = $k-1+4 = k+3$. * New ratio: $\frac{3k-3}{k+3} = \frac{2}{1} \implies 3k-3 = 2k+6 \implies k=9$. * Original mixture = $4k = 36$ L. 7. **Solving using percentages:** Convert everything to percentages of one component. 8. **Weighted Average approach:** Alligation is a special case of weighted average. * If $Q_1$ of $C_1$ and $Q_2$ of $C_2$ are mixed, $C_m = \frac{Q_1 C_1 + Q_2 C_2}{Q_1 + Q_2}$. * This formula can be rearranged to derive the alligation rule. #### 3. Useful Information * Alligation is a powerful tool for quickly solving mixture problems that would otherwise require solving system of equations. * Ensure that the "values" ($C_1, C_2, C_m$) are comparable and in the same units. * The alligation method yields the ratio of *quantities* (volumes, weights, etc.) of the ingredients. * Practice problems involving different types of "costs" or "concentrations" to master the application of alligation. * Be careful when repeatedly removing and replacing parts of a mixture; the formula for remaining quantity of original liquid is very useful. * Mixture problems often combine with ratio & proportion concepts. ### Geometry #### 1. Basic Formulas & Concepts * **Lines and Angles:** * **Point:** A location in space. * **Line:** A straight path extending infinitely in both directions. * **Line Segment:** A part of a line with two endpoints. * **Ray:** A part of a line with one endpoint, extending infinitely in one direction. * **Angle:** Formed by two rays sharing a common endpoint (vertex). * Acute ($ 90^\circ, 180^\circ,