TNPSC GROUP 2 APTITUDE AND MENTAL ABILITY TEST IMPORTANT FORMULAS 1 Page
Range: The difference between largest and smallest value of the observation is called The Range and is denoted by R ie R = Largest value Smallest value R=L-S Lorenz Curve: Lorenz curve is a graphical method of studying dispersion It was introduced by MaxOLorenz, a great Economist and a statistician, to study the distribution of wealth and income It is also used to study the variability in the distribution of profits, wages, revenue, etc Arithmetic mean or mean : WEIGHTED ARITHMETIC MEAN: - 2 Page
Harmonic mean (HM) : Median : The median is that value of the variate which divides the Vgroup into two equal parts, one part comprising all values greater, and the other, all values less than median Ungrouped or Raw data : Arrange the given values in the increasing or decreasing order If the number of values are odd, median is the middle value If the number of values are even, median is the mean of middle two values By formula Mode : The mode refers to that value in a distribution, which occur most frequently It is an actual value, which has the highest concentration of items in and around it Co-efficient of Range : L- LARGE NUMBER S SMALL NUMBER MENTAL ABILITY: 3 Page
IMPORTANT FORMULAS TOPIC WISE SIMPLIFICATION FORMULAS What is BODMAS rule? BODMAS rule defines the correct sequence in which operations are to be performed in a given mathematical expression to find its value In BODMAS, B = Bracket O = Order (Powers, Square Roots, etc) DM = Division and Multiplication (left-to-right) AS = Addition and Subtraction (left-to-right) Some Basic Formulae: (a + b)(a - b) = (a2 - b2) (a + b)2 = (a2 + b2 + 2ab) (a - b)2 = (a2 + b2-2ab) (a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca) (a3 + b3) = (a + b)(a2 - ab + b2) (a3 - b3) = (a - b)(a2 + ab + b2) (a3 + b3 + c3-3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ac) When a + b + c = 0, then a3 + b3 + c3 = 3abc Percentage IMPORTANT FORMULAS Values to memorize for quick calculations 4 Page
Highest Common Factor (HCF) Lowest Common Multiple (LCM) FORMULAS Highest Common Factor (HCF) or Greatest Common Measure (GCM) or Greatest Common Divisor (GCD) The HCF of two or more than two numbers is the greatest number that divided each of them exactly There are two methods of finding the HCF of a given set of numbers: 1 Factorization Method 2 Division method Factorization Method: Express the each one of the given numbers as the product of prime factors The product of least powers of common prime factors gives HCF 2 Division Method: Suppose we have to find the HCF of two given numbers, divide the larger by the smaller one Now, divide the divisor by the remainder Repeat the process of dividing the preceding number by the remainder last obtained till zero is obtained as remainder The last divisor is required HCF Least Common Multiple (LCM) The least number which is exactly divisible by each one of the given numbers is called their LCM There are two methods of finding the LCM of a given set of numbers: 1 Factorization Method, 2 Division Method (Division Method is short cut method of LCM) 1 Factorization Method: Resolve each one of the given numbers into a product of prime factors Then, LCM is the product of highest powers of all the factors 2 Division Method : Arrange the given numbers in a row in any order Divide by a number which divided exactly at least two of the given numbers and carry forward the numbers which are not divisible Repeat the above process till no two of the numbers are divisible by the same number except 1 The product of the divisors and the undivided numbers is the required LCM of the given numbers Product of two numbers Product of two numbers = Product of their HCF and LCM Co prime numbers Two numbers are said to be co primes if their HCF is 1 HCF and LCM of Fractions 5 Page
Ratio and Proportion FORMULAS Ratio: The ratio of two quantities a and b in the same units, is the fraction and we write it as a : b In the ratio a : b, we call a as the first term or antecedent and b, the second term or consequenteg The ratio 5 : 9 represents 5 with antecedent = 5, consequent = 9 Rule: The multiplication or division of each term of a ratio by the same non-zero number does not affect the ratio Eg 4 : 5 = 8 : 10 = 12 : 15 Also, 4 : 6 = 2 : 3 Proportion: The equality of two ratios is called proportion If a : b = c : d, we write a : b :: c : d and we say that a, b, c, d are in proportion Here a and d are called extremes, while b and c are called mean terms Product of means = Product of extremes Thus, a : b :: c : d (b x c) = (a x d) Fourth Proportional: If a : b = c : d, then d is called the fourth proportional to a, b, c Third Proportional: a : b = b : c, then c is called the third proportion to a and b Mean Proportional: Mean proportional between a and b is Comparison of Ratios: Compounded Ratio: The compounded ratio of the ratios: (a : b), (c : d), (e : f) is (ace : bdf) Duplicate Ratios: Duplicate ratio of (a : b) is (a2 : b2) Sub-duplicate ratio of (a : b) is (a : b) Triplicate ratio of (a : b) is (a3 : b3) Sub-triplicate ratio of (a : b) is (a1/3 : b1/3) a c a+b c+d If =, then = [componendo and dividendo] b d a-b c-d Variations: We say that x is directly proportional to y, if x = ky for some constant k and we write, x y We say that x is inversely proportional to y, if xy = k for some constant k and 1 we write, x y 6 Page
Simple interest Compound interest IMPORTANT FORMULAS Simple Interest - Important Formulas Principal: The money borrowed or lent out for a certain period is called the principal or the sum Interest: Extra money paid for using other's money is called interest Simple Interest (SI): If the interest on a sum borrowed for certain period is reckoned uniformly, then it is called simple interest Let Principal = P, Rate = R% per annum (pa) and Time = T years Then PxRxT (i) Simple Intereest = x SI x SI x SI (ii) P = ;R= and T = RxT PxT PxR Compound Interest - Important Formulas Let Principal = P, Rate = R% per annum, Time = n years When interest is compound Annually: R n Amount = P 1 + When interest is compounded Half-yearly: (R/2) 2n Amount = P 1 + When interest is compounded Quarterly: (R/4) 4n Amount = P 1 + When interest is compounded Annually but time is in fraction, say 3 years 3 Amount = P 1 + R x 1+ R When Rates are different for different years, say R1%, R2%, R3% for 1st, 2nd and 3rd year respectively R1 R2 R3 Then, Amount = P 1 + 1+ 1+ Present worth of Rs x due n years hence is given by: x Present Worth = 1+ R 7 Page
Area Volume IMPORTANT FORMULAS FUNDAMENTAL CONCEPTS Results on Triangles: Sum of the angles of a triangle is 180 The sum of any two sides of a triangle is greater than the third side Pythagoras Theorem: 1 In a right-angled triangle, (Hypotenuse)2 = (Base)2 + (Height)2 2 The line joining the mid-point of a side of a triangle to the positive vertex is called the median 3 The point where the three medians of a triangle meet, is called centroid The centroid divided each of the medians in the ratio 2 : 1 4 In an isosceles triangle, the altitude from the vertex bisects the base 5 The median of a triangle divides it into two triangles of the same area The area of the triangle formed by joining the mid-points of the sides of a given triangle is one-fourth of the area of the given triangle Results on Quadrilaterals: 1 The diagonals of a parallelogram bisect each other 2 Each diagonal of a parallelogram divides it into triangles of the same area 3 The diagonals of a rectangle are equal and bisect each other 4 The diagonals of a square are equal and bisect each other at right angles 5 The diagonals of a rhombus are unequal and bisect each other at right angles 6 A parallelogram and a rectangle on the same base and between the same parallels are equal in area 7 Of all the parallelogram of given sides, the parallelogram which is a rectangle has the greatest area IMPORTANT FORMULAE 1 Area of a rectangle = (Length x Breadth) Area Area Length = and Breadth = Breadth Length 2 Perimeter of a rectangle = 2(Length + Breadth) Area of a square = (side)2 = (diagonal)2 Area of 4 walls of a room = 2 (Length + Breadth) x Height 1 2 3 4 5 6 Area of a triangle = x Base x Height Area of a triangle = s(s-a)(s-b)(s-c) where a, b, c are the sides of the triangle and s = (a + b + c) 3 Area of an equilateral triangle = x (side)2 4 a Radius of incircle of an equilateral triangle of side a = 23 a Radius of circumcircle of an equilateral triangle of side a = 3 Radius of incircle of a triangle of area and semi-perimeter r = s 8 Page
1 Area of parallelogram = (Base x Height) 2 Area of a rhombus = 3 Area of a trapezium = 1 2 Area of a circle = R2, where R is the radius Circumference of a circle = 2 R 2 R Length of an arc =, where is the central angle 360 1 R2 Area of a sector = (arc x R) = 2 360 3 4 1 2 x (Product of diagonals) x (sum of parallel sides) x distance between them Circumference of a semi-circle = R R2 Area of semi-circle = 2 9 Page
VOLUME AND SURFACE AREA CUBOID Let length = l, breadth = b and height = h units Then Volume = (l x b x h) cubic units Surface area = 2(lb + bh + lh) sq units Diagonal = l2 + b2 + h2 units CUBE Let each edge of a cube be of length a Then, Volume = a3 cubic units Surface area = 6a2 sq units Diagonal = 3a units CYLINDER Let radius of base = r and Height (or length) = h Then, Volume = ( r2h) cubic units Curved surface area = (2 rh) sq units Total surface area = 2 r(h + r) sq units CONE Let radius of base = r and Height = h Then, Slant height, l = h2 + r2 units Volume = r2h cubic units Curved surface area = ( rl) sq units Total surface area = ( rl + r2) sq units SPHERE Let the radius of the sphere be r Then, Volume = r3 cubic units Surface area = (4 r2) sq units HEMISPHERE Let the radius of a hemisphere be r Then, Volume = r3 cubic units Curved surface area = (2 r2) sq units Total surface area = (3 r2) sq units Note: 1 litre = 0 cm3 10 P a g e
TIME AND WORK IMPORTANT FORMULAS If A can do a piece of work in n days, work done by A in 1 day = 1/n If A does 1/n work in a day, A can finish the work in n days If M1 men can do W1 work in D1 days working H1 hours per day and M2 men can do W2 work in D2 days working H2 hours per day (where all men work at the same rate), then M1 D1 H1 / W1 = M2 D2 H2 / W2 If A can do a piece of work in p days and B can do the same in q days, A and B together can finish it in pq / (p+q) days If A is thrice as good as B in work, then Ratio of work done by A and B = 3 : 1 Ratio of time taken to finish a work by A and B = 1 : 3 11 P a g e
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