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Operations on Integers The set of natural numbers, zero and the negatives of natural numbers form the set of Integers. The set of integers includes all the whole numbers. There is no smallest integer. Addition of Integers The sum of two positive integers results in a positive integer. Ex: 8 + 2 = 10 The sum of two negative integers results in a negative integer. Ex: (– 6) + (– 3) = – 9 The sum of a positive and a negative integer is the difference of the numbers with the sign of the larger integer of the two. Ex: 45 + (– 25) = 20 and (– 45) + 20 = – 25 The additive inverse of any integer 'a' is '– a', and the additive inverse of '– a' is 'a'. Ex: Additive inverse of (– 12) = – (– 12) = 12 Subtraction of Integers Subtraction is the opposite of addition, Therefore, to subtract two integers, we add the additive inverse of the integer that is being subtracted to the other integer. Ex: 23 – 43 = 23 + (Additive inverse of 43) = 23 + (– 43) = – 20 Multiplication of Integers The product of two positive integers is a positive integer. The product of a positive and a negative integer is a negative integer. The product of two negative integers is a positive integer. If the number of negative integers in a product is even, then the product is a positive integer. Similarly, if the number of negative integers in a product is odd, then the product is a negative integer. Division of Integers Division is the inverse operation of multiplication. The division of a positive integer by a positive integer results in a positive integer. The division of a negative integer by a positive integer results in a negative integer. The division of a positive integer by a negative integer results in a negative integer. The division of a negative integer by a negative integer results in a positive integer. For any integer 'a', a × 0 = 0 × a = 0. a ÷ 0 is not defined. 0 ÷ a = 0, where a is not equal to zero. Properties of Integers Closure property Closure property under addition: Integers are closed under addition, i.e. for any two integers a and b, a + b is an integer. Ex: 3 + 4 = 7; (– 9) + 7 = – 2. Closure property under subtraction: Integers are closed under subtraction, i.e. for any two integers a and b, a – b is an integer. Ex: (– 21) – (– 9) = (– 12); 8 – 3 = 5. Closure property under multiplication: Integers are closed under multiplication, i.e. for any two integers a and b, ab is an integer. Ex: 5 × 6 = 30; (– 9) × (– 3) = 27. Closure property under division: Integers are not closed under division, i.e. for any two integers a and b, ababmay not be an integer. Ex:(– 2) ÷ (– 4) = 1212 Commutative property Commutative property under addition: Addition is commutative for integers. For any two integers a and b, a + b = b + a. Ex: 5 + (– 6) = 5 – 6 = – 1; (– 6) + 5 = – 6 + 5 = –1 ∴ 5 + (– 6) = (– 6) + 5. Commutative property under subtraction: Subtraction is not commutative for integers. For any two integers a and b, a – b ≠ b – a. Ex: 8 – (– 6) = 8 + 6 = 14; (– 6) – 8 = – 6 – 8 = – 14 ∴ 8 – (– 6) ≠ – 6 – 8. Commutative property under multiplication: Multiplication is commutative for integers. For any two integers a and b, ab = ba. Ex: 9 × (– 6) = – (9 × 6) = – 54; (– 6) × 9 = – (6 × 9) = – 54 ∴ 9 × (– 6) = (– 6) × 9. Commutative property under division: Division is not commutative for integers. For any two integers a and b, a ÷ b ≠ b ÷ a. Ex: (– 14) ÷ 2 = – 7 2 ÷ (–14) = – 1717 (– 14) ÷ 2 ≠ 2 ÷ (–14). Associative property Associative property under addition: Addition is associative for integers. For any three integers a, b and c, a + (b + c) = (a + b) + c Ex: 5 + (– 6 + 4) = 5 + (– 2) = 3; (5 – 6) + 4 = (– 1) + 4 = 3 ∴ 5 + (– 6 + 4) = (5 – 6) + 4. Associative property under subtraction: Subtraction is associative for integers. For any three integers a, b and c, a – (b – c) ≠ (a – b) – c Ex: 5 – (6 – 4) = 5 – 2 = 3; (5 – 6) – 4 = – 1 – 4 = – 5 ∴ 5 – (6 – 4) ≠ (5 – 6) – 4. Associative property under multiplication: Multiplication is associative for integers. For any three integers a, b and c, (a × b) × c = a × (b × c) Ex: [(– 3) × (– 2)] × 4 = (6 × 4) = 24 (– 3) × [(– 2) × 4] = (– 3) × (– 8) = 24 ∴ [(– 3) × (– 2)] × 4 = [(– 3) × (– 2) × 4]. Associative property under division: Division is not associative for integers. Distributive property Distributive property of multiplication over addition: For any three integers a, b and c, a × (b + c) = (a × b) + (a × c). Ex: – 2 (4 + 3) = –2 (7) = –14 = (– 2 × 4) + (– 2 × 3) = (– 8) + (– 6) = – 14. Distributive property of multiplication over subtraction: For any three integers, a, b and c, a × (b - c) = (a × b) – (a × c). Ex: – 2 (4 – 3) = – 2 (1) = – 2 = (–2 × 4) – (– 2 × 3) = (– 8) – (– 6) = – 2. The distributive property of multiplication over the operations of addition and subtraction is true in the case of integers. Identity under addition: Integer 0 is the identity under addition. That is, for any integer a, a + 0 = 0 + a = a. Ex: 4 + 0 = 0 + 4 = 4. Identity under multiplication: The integer 1 is the identity under multiplication. That is, for any integer a, 1 × a = a × 1 = a. Ex: (– 4) × 1 = 1 × (– 4) = – 4. When an integer is multiplied by –1, the result is the integer with sign changed i.e. the additive identity of the integer. For any integer a, a × –1 = –1 × a = –a. Multiplication and Division of Fraction Types of fractions Proper fraction A fraction whose numerator is less than the denominator is called the proper fraction. A proper fraction is a fraction that represents a part of a whole. Improper fraction A fraction whose numerator is greater than the denominator is called an improper fraction. An improper fraction is a combination of a whole and a proper fraction. Mixed fraction An improper fraction can be expressed as a mixed fraction. The numerator of the improper fraction is divided by the denominator to obtain the quotient and the remainder. The mixed fraction is written as Quotient RemainderDivisorQuotient RemainderDivisor. Multiplication of fractions To multiply a fraction by a whole number, multiply the whole number by the numerator of the fraction. Ex: 2 × 2323 = 4343. While multiplying a whole number by a mixed fraction, change the mixed fraction into an improper fraction and multiply the whole number by the numerator of the fraction. Ex: 3 × 2 3434 = 3 × 114114 = 334334. To multiply two fractions, multiply their numerators and denominators. Ex: 3434 × 2525 = 3 × 24 × 53 × 24 × 5 = 620620. When two proper fractions are multiplied, the product is less than each of the individual fractions. When two improper fractions are multiplied, the product is greater than each of the individual fractions. Fraction as an operator ‘of ’ Fraction acts as an operator of. 'of' represents multiplication. Ex: 1313 of 90 = 1313 × 90 = 30. Reciprocal of a fraction To obtain the reciprocal of a fraction, interchange the numerator with the denominator. Ex: The reciprocal of 3737 is 7373. Division of fractions To divide a whole number by a fraction, find the reciprocal of the fraction and then multiply it by the whole number. Ex: 2 ÷ 3434 = 2 × 4343 = 8383. To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number. Ex: 2727 ÷ 6 = 2727 × 1616 = 121121. To divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second fraction. Ex: 1212 ÷ 3434 = 1212 × 4343 = 4646. To divide a whole number by a mixed fraction, convert the mixed fraction into an improper fraction and multiply the whole number by the reciprocal of the improper fraction. Ex: 5 ÷ 61414 = 5 ÷ 254254 = 5 × 425425 = 4545. Multiplication and Division on Decimals Multiplication of decimals Steps to multiply a whole number by a decimal number: Step 1: Ignore the decimals and multiply the two numbers. Step 2: Count the number of digits to the right of decimal point in the original decimal number. Step 3: Insert the decimal in the answer by the same count from right to left. e.g. (i) 3 × 0.2 = 0.6 (ii) 3 × 0.4 = 1.2 Steps to multiply a decimal number by a decimal number: Step 1: Ignore the decimals and multiply the two numbers. Step 2: Count the number of digits to the right of decimal point in both the decimal numbers Step 3: Add the number of digits counted and insert the decimal in the answer by the same count from right to left. e.g. (i) 0.2 × 0.7 = 0.14 (ii) 0.9 × 0.02 = 0.018 Multiplication of a decimal number by 10, 100 or 1000: For multiplying a decimal number by 10, retain the original number and shift the decimal to the right by one place For multiplying a decimal number by 100, retain the original number and shift the decimal to the right by two places For multiplying a decimal number by 1000, retain the original number and shift the decimal to the right by three places Division of decimals Steps to divide a decimal number by a whole number: Step 1: Convert the decimal number into a fraction. Step 2: Find the reciprocal of the divisor. Step 3: Multiply the fraction by the reciprocal. e.g. 0.14 ÷ 3 = 1410014100 × 1313 = 1430014300 Steps to divide a decimal number by another decimal number: Step 1: Convert both the decimal numbers into fractions. Step 2: Find the reciprocal of the divisor. Step 3: Multiply the fraction by the reciprocal. e.g. 0.45 ÷ 0.15 = 4510045100 ÷ 1510015100= 4510045100× 1001510015= 3 Data Representation Any information collected can be first arranged in a frequency distribution table, and this information can be put as a visual representation in the form of pictographs or bar graphs. Data collected is organised in a table for easy understanding and interpreting. Graphs are a visual representation of organised data. Bar graphs A bar graph is the representation of data using rectangular bars of uniform width, and with their lengths depending on the frequency and the scale chosen. The bars can be plotted vertically or horizontally. You can look at a bar graph and make deductions about the data. Bar graphs are used for plotting discrete or discontinuous data, i.e. data that has discrete values and is not continuous. Some examples of discontinuous data are 'shoe size' and 'eye colour', for which you can use a bar chart. On the other hand, examples of continuous data include 'height' and 'weight'. A bar graph is very useful if you are trying to record certain information, whether the data is continuous or not. Bar graphs can also be used for comparative analysis. Double bar graphs are used for comparing data between two different things. The difference between a bar graph and a double bar graph is that a bar graph displays one set of data, and a double bar graph compares two different sets of information or data. Scale of a bar graph If numbers in units are to be shown in a bar graph, the graph represents one unit of ength for one observation and if it has to show numbers in tens or hundreds, one unit length can represent 10 or 100 observations. This is called the scale of the graph. Length of a bar in a bar graph depends on the frequency and the scale chosen. Data Value Different forms of representative or central value are used to represent and describe different forms of data. One of the representative values is the “Arithmetic Mean”. Arithmetic mean Arithmetic mean is a number that lies between the highest and the lowest values of the data. Arithmetic mean or Average = Sum of observationsNumber of observationsSum of observationsNumber of observations Note that we need not arrange the data in ascending or descending order to calculate arithmetic mean. Range Range is calculated by subtracting the lowest observation from the highest observation of a data. It gives an idea of the spread of the observations. Range = Highest observation – Lowest observation Mode Mode of a set of observations refers to the observation that occurs most often. Steps to calculate the mode of a data: Step 1: Arrange the data in ascending order. Step 2: Tabulate the data in a frequency distribution table. Step 3: The observation with a greatest frequency is the mode. Median Median refers to the value that lies in the middle of the data with half of the observations above it and the other half of the observations below it. Steps to calculate the median of a data: Step 1: Arrange the data in the ascending order. Step 2: The value that lies in the middle such that half of the observations lie above it and the other half below it will be the median. The mean, mode and median are the representative values of a group of observations or data, and lie between the minimum and maximum values of the data. They are also called the measures of the central tendency. Introduction to Simple Equations Variable A variable is something that varies. A variable can take different numerical values; its value is not fixed. Variables are usually denoted by letters of the alphabet, such as x, y, z, l, m, n, p, q etc. Expressions are formed using variables. Expression Expressions are formed by performing the operations like addition, subtraction, multiplication and division on the variables.Value of an expression depends on the value of the variable using which the expression is formed. Equation An equation is a condition of equality between two mathematical expressions. It is a condition on a variable. e.g. 2x - 3 = 5, 3x + 9 = 11, 4y + 2 = 12 Two expressions on both sides of the equation should have equal value and atleast one of the expressions must have a variable. An equation remains the same if the LHS and the RHS of the equation are interchanged. The equality sign in an equation shows that the value of the expression to the left hand side is equal to the value of the expression to the right hand side. If there is some other sign other than the equality sign between the LHS and the RHS, it is not an equation. e.g. 3x + 5 > 6 is not an equation. Solution of an equation The value of the variable for which the left hand side of an equation is equal to its right hand side is called the solution of that equation. e.g. For the equation, 5x + 5 = 15, x = 2 is a solution. When the same number is added to or subtracted from both the sides of a balanced equation, the value of the left hand side remains equal to its value on the right hand side. If the same mathematical operation is not done on both sides of a balanced equation, the balance is disturbed. e.g. (1) 5x + 3 = 13 On adding 2 to both sides of the equation, we get 5x + 3 + 2 = 13 + 2 5x + 5 = 15 (2) On subtracting 2 from both sides of the equation, we get 5x + 3 - 2 = 13 - 2 5x + 1 = 11 When an equation is divided or multiplied on both the sides by a non-zero number, the value of the left hand side remains equal to its value on the right hand side. e.g. (1) 5x + 3 = 13 On dividing both sides of the equation by 4, we get (5x + 3) ÷ 4 = 13 ÷ 4 2) 5x + 1 = 13 On multiplying both sides of the equation by 4, we get 4(5x + 1) = 4(13) 20x + 4 = 52 Application of Simple Equations Solving an equation To find the solution of an equation, a series of identical mathematical operations are performed on both the sides of the equation so that only the variable remains on one side. On simplifying all the numbers, the result obtained is the solution of the equation. If the operation is performed on only one side of the equation, the balance of the equation is disturbed. Ex: 3x + 8 = 83 3x + 8 - 8 = 83 - 8 3x = 75 x = 753753 x = 25. Transposition Moving a term of an equation from one side to the other side is called transposing. Transposing a number is same as adding to or subtracting the same number from both sides of the equation. Ex: Solve 2x + 8 = 24 Given, 2x + 8 = 24 Transposing 8 to the right hand side, we get ⇒ 2x = 24 - 8 ⇒ 2x = 16 ⇒ x = 162162 ⇒ x = 8. Hence, the value of x is 8. The sign of a number changes when it is transposed from one side of the equation to the other. To solve puzzles/problems from practical situations equations are formed corresponding to such situations and then those equations are solved to give the solution to the puzzles/problems. Solution to an equation An equation can be built from the solution of the equation using the property of doing the same mathematical operation on both sides of an equation. Angles Angle An angle is formed when lines, rays or line segments meet. The lines that form an angle are called the sides or the arms of the angle. The common end point is called the vertex of the angle. Types of angles based on their measure The angle whose measure is less than 90° is called an acute angle. The angle whose measure is 90° is called a right angle. The angle whose measure is greater than 90° and less than 180° is called an obtuse angle. The angle whose measure is equal to 180° is called a straight angle. The angle whose measure is greater than 180° and less than 360° is called a reflex angle. Related angles Two angles sum of whose measures is 90° are called complementary angles. When two angles are complementary, each angle is said to be the complement of the other. Two angles sum of whose measures is 180° are called supplementary angles. When two angles are supplementary, each angle is said to be the supplement of the other. Angles with a common vertex and a common arm are called adjacent angles. Non–common arms of adjacent angles are on either side of the common arm. Adjacent angles have no common interior points. Two adjacent angles can be either supplementary or complementary. A pair of adjacent angles whose non–common arms are opposite rays is called a linear pair. The angles of a linear pair are supplementary. Two angles that are not adjacent but have a common vertex are called vertically opposite angles. Vertically opposite angles are opposite to each other and are equal. Two pairs of vertically opposite angles are formed when two lines intersect. If one pair of vertically opposite angles are acute angles, then the other pair of angles are obtuse. Pairs of Lines Intersecting lines Lines that meet at a point are called intersecting lines. The point where they meet is known as the point of intersection. Parallel lines Lines that always remain the same distance apart and never meet are called parallel lines. Transversal A line that intersects two or more lines at distinct points is called a transversal. Angles formed by lines intersected by a transversal • Eight angles are formed when a transversal intersects two lines. The angles that lie between the lines are called interior angles. The angles that lie on the outer sides of the lines are called exterior angles. • Angles formed on the same side of a transversal, on the same side of the two lines and at corresponding vertices are called corresponding angles. • Angles formed on the opposite sides of the transversal at the two distinct points of intersection and between the two lines are called alternate interior angles. • Angles formed on the opposite sides of the transversal at the two distinct points of intersection but outside the two lines are called alternate exterior angles. • Angles that have different vertices, lie on the same side of the transversal and are interior angles are called consecutive interior angles or allied or co-interior angles. If two parallel lines are cut by a transversal then each pair of: • Corresponding angles is equal in measure • Alternate interior angles is equal in measure • Interior angles on the same side of the transversal are supplementary Two lines are said to be parallel if each pair of: • Corresponding angles are equal • Alternate interior angles are equal • Interior angles on the same side of the transversal are supplementary Triangles A triangle is a closed figure made of three line segments. Every triangle has three sides, three angles, and three vertices. These are known as the parts of a triangle. Classification of triangles The types of triangles classified by their sides are: Equilateral triangle Isosceles triangle Scalene triangle The types of triangles classified by their angles are: Right angled triangle Obtuse angled triangle Acute angled triangle Equilateral triangle: A triangle in which all the sides are equal is called an equilateral triangle. All the three angles of an equilateral triangle are also equal, and each measures 60°. Isosceles triangle: A triangle in which any two sides are equal is called an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are called the base angles, and they are equal. Scalene triangle: A triangle in which no two sides are equal is called a Scalene triangle. Acute-angled triangle: A triangle with all its angles less than 90° is known as an acute-angled triangle. Obtuse-angled triangle: A triangle with one of its angles more than 90° and less than 180° is known as an obtuse-angled triangle. Right-angled triangle: A triangle with one of its angles equal to 90° is known as a right-angled triangle. The side opposite the vertex with 90° angle is called the hypotenuse, and is the longest side of the triangle. Median The line segment drawn from a vertex of a triangle to the midpoint of the opposite side is called a median of the triangle. Three medians can be drawn from each vertex of a triangle. Medians of a triangle are concurrent. The point of concurrence is called the centroid of the traingle, and is denoted by G. The centroid and medians of a triangle always lie inside the triangle. The centroid of a triangle divides the median in the ratio 2:1. Altitude The line segment drawn from the vertex of a triangle which is perpendicular to the opposite side of that triangle is called an altitude of the triangle. A triangle has three altitudes. The altitudes of a triangle are concurrent. The point of concurrence is called the orthocentre, and is denoted by O. The altitude and orthocentre of a triangle need not lie inside the triangle. Properties of Triangles Angle sum property The sum of the three angles of a triangle is 180°. e.g. If A, B and C are the angles of a triangle ABC, then ∠A + ∠B + ∠C = 180°. Proof: Consider a triangle ABC. Let line XY be parallel to side BC at A. AB is a transversal that cuts the line XY and AB, at A and B, respectively. As the alternate interior angles are equal, ∠1 = ∠4 and ∠2 = ∠5. ∠4, ∠3 and ∠5 form linear angles, and their sum is equal to 180°. ⇒ ∠4 + ∠3 + ∠5 = 180° ⇒ ∠1 + ∠2 + ∠3 = 180° Hence, the sum of the three angles of a triangle is 180°. Exterior angle property An exterior angle of a triangle is equal to the sum of its opposite interior angles. e.g. If ∠4 is an exterior angle of ΔABC, ∠1 and ∠2 are the interior opposite angles, then ∠4 = ∠1 + ∠2. The sum of the lengths of any two sides of a triangle is greater than the third side. In a right-angled triangle, the side opposite the right angle is called the hypotenuse, and the other two sides are called its legs. Pythagorean theorem In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. If b and c are legs and a is the hypotenuse of a right angled triangle then, a2 = b2 + c2. Converse of Pythagorean theorem If the sum of the squares on two sides of a triangle is equal to the square of the third side, then the triangle must be a right-angled triangle. Congruence of Plane Figures If two objects are of exactly the same shape and size, they are said to be congruent. The relation between two congruent objects being congruent is called congruence. A plane figure is any shape that can be drawn in two dimensions e.g. rectangle, square, triangle, rhombus, etc. To check if two figures drawn on a paper are congruent, make a traced copy of one of the figures on a tracing paper and place it over the other. The other method is to cut out one of these figures and place it over the The method of superposition examines the congruence of plane figures, line segments and angles. Two plane figures are congruent if each, when superimposed on the other, covers it exactly. Congruence is denoted by ≅. e.g. Two plane figures, say, P1 and P2, are congruent if the trace copy of P1 fits exactly on that of P2. We write P1 ≅ P2 If two line segments have the same or equal length, they are congruent. Also, if two line segments are congruent, then they have the same length. e.g. Two line segments, say, AB−−−AB_ and EF−−−EF_ are congruent if they have equal lengths. We write this as AB−−−AB_ ≅ EF−−−EF_ or AB = EF. If two angles have the same measure, they are congruent. Also, if two angles are congruent, their measures are the same. If two angles are congruent, then the lengths of their arms do not matter. e.g. Two angles, say, ∠PQR and ∠XYZ, are congruent if their measures are equal. We write this as ∠PQR ≅ ∠XYZ or as ∠PQR = ∠XYZ. However, commonly, we write ∠PQR ≅ ∠XYZ. Two circles of equal radii are congruent. Two squares of equal sides are congruent. Criteria for Congruence of Triangles Congruence of triangles Consider triangles ABC and XYZ. Cut triangle ABC and place it over XYZ. The two triangles cover each other exactly, and they are of the same shape and size. Also notice that A falls on X, B on Y, and C on Z. Also, side AB falls along XY, side BC along YZ, and side AC along XZ. So, we can say that triangle ABC is congruent to triangle XYZ. Symbolically, it is represented as ΔABC ≅ ΔXYZ. So, in general, we can say that two triangles are congruent if all the sides and all the angles of one triangle are equal to the corresponding sides and angles of the other triangle. Order of the letters in the names of the congruent triangles shows the corresponding relationship. In two congruent triangles ABC and XYZ, the corresponding vertices are A and X, B and Y, and C and Z, that is, A corresponds to X, B to Y, and C to Z. Similarly, the corresponding sides are AB and XY, BC and YZ, and AC and XZ. Also, angle A corresponds to X, B to Y, and C to Z. So, we write ABC corresponds to XYZ. Criteria for congruence of triangles Two triangles can be congruent if three of their corresponding parts are equal. SSS congruence criterion Two triangles are congruent if three sides of one triangle are equal to the three corresponding sides of the other triangle. SAS congruence criterion Two triangles are congruent if two sides and the included angle of one triangle are equal to the corresponding two sides and the included angle of the other triangle. RHS congruence criterion Two right-angled triangles are congruent if the hypotenuse and a side of one triangle are equal to the hypotenuse and the corresponding side of the other triangle. ASA congruence criterion Two triangles are congruent if two angles and the included side of one triangle are equal to the corresponding two angles and the included side of the other triangle. Two triangles with equal corresponding angles may not be congruent. So, there is no such thing as AAA congruence of triangles. Two congruent triangles have equal areas and equal perimeters. Ratios and Proportions Ratio Ratios are used to compare quantities. To compare two quantities, the units of the quantities must be the same. Ratios help us to compare quantities and determine the relation between them. We write ratios in the form of fractions and then compare them by converting them to like fractions. If these like fractions are equal, then the ratios are said to be equivalent. e.g. Cost of 6 pens is Rs 90. What would be the cost of 10 such pens? Solution: Cost of 6 pens = Rs 90 Cost of 1 pen = 90 ÷ 6 = Rs 15 Hence, cost of 10 pens = 10 × 15 = Rs 150. Proportion When two ratios are equivalent, the four quantities are said to be in proportion. Ratio and proportion problems can be solved by using two methods, the unitary method and equating the ratios to make proportions, and then solving the equation. Unitary method Unitary method is the method of finding the value of one unit (unit rate) at first and then the value of required number of units. Percentages Percentage is another method used to compare quantities. Percent is derived from the Latin word ‘per centum’, which means per hundred. Percentage is the numerator, of a fraction, whose denominator is hundred. Percent is represented by the symbol - %. e.g. 2110021100 or 21% Percentages Percentage is the numerator of a fraction, whose denominator is hundred. If the denominator in the fraction is not hundred we multiply the fraction with hundred percent to get percentage. Percentages are also used for comparing quantities. Converting fractional numbers into percentage To convert a fraction into a percentage, multiply it by hundred and then place the % symbol. e.g. 4545 = 4545 × 100 = 80% Percentages related to proper fractions are less than 100, whereas percentages related to improper fractions are more than 100. Converting decimals into percentage To convert a decimal into percentage, multiply the decimal by hundred and then place the % symbol. e.g. 0.25 = 2510025100 × 100 = 25% Conversions A given percentage can be converted into fractions and decimals. Also, a decimal can be converted as percentage. To convert a percentage into a fraction Remove the percent sign, and then divide the number by hundred. e.g. 20% = 2010020100 = 1515. To convert a percentage into a decimal Step 1: Remove the percent sign. Step 2: Divide the number by 100, or move the decimal point two places to the left in the numerator. e.g. 4% = 41004100 = 0.04 To convert a decimal into a percentage Step1: Convert the decimal into a fraction. Step 2: Multiply the fraction by hundred. Step 3: Put a percent sign next to the number. Else, shift the decimal point two places to the right. e.g. 0.25 = 2510025100 = 2510025100 × 100 = 25 % Application of Percentages Percentages are helpful in comparison. The increase or decrease in a certain quantity can be expressed as percentage increase or decrease. Percentage increase/decrease = Amount of change (Increase or Decrease)Original amount or baseAmount of change (Increase or Decrease)Original amount or base ×100 Profit and loss Profit is the money gained on selling an item. Loss is the money lost on selling an item. If the selling price of an item is more than the cost price, then a profit is incurred. If the cost price of an item is more than the selling price, then a loss is incurred. When the cost price is equal to the selling price, it is called a no-profit no-loss situation. Profit = Selling price – Cost price Loss = Cost price – Selling price CP = SP ⇒ No Profit no Loss Profit percentage or loss percentage is always calculated on the cost price. Profit percentage = ProfitCost priceProfitCost price × 100 Loss percentage = LossCost priceLossCost price × 100 Simple interest Interest is the fee paid as the cost of borrowing on the borrowed money. The way of calculating interest, where the principal is not changed, is known as "simple interest". As the number of years increases, interest will also increase. If P denotes the principal, R denotes the rate of interest and T denotes the time period, then the simple interest I paid for T years is P × T × R100P × T × R100. Sum that is returned at the end of the time period is called the amount (A). Amount can be obtained by adding the sum borrowed (principal) and the interest. Amount = Principal + Interest. Introduction to Rational Numbers All numbers, including whole numbers, integers, fractions and decimal numbers, can be written in the NumeratorDenominator NumeratorDenominator form. Rational number A rational number is a number that can be written in the form p/q, where p and q are integers and q ≠ 0. The denominator of a rational number can never be zero. e.g. 911911, 5858, 712712 A rational number is positive if its numerator and denominator are both either positive integers or negative integers. e.g. 2525, 3434, -7-10-7-10, -5-11-5-11 If either the numerator or the denominator of a rational number is a negative integer, then it is a negative rational number. e.g. -25-25, 3-43-4, 7-107-10, 5-115-11 Representation of rational numbers on the number line The rational number zero is neither negative nor positive. Positive rational numbers are represented to the right of zero on the number line. Negative rational numbers are represented to the left of zero on the number line. A rational number obtained by multiplying or dividing both the numerator and the denominator of a rational number by the same non-zero integer, is said to be the equivalent form of the given rational number. Rational numbers in Standard form A rational number is said to be in its standard form if its numerator and denominator have no common factor other than 1, and its denominator is a positive integer. To reduce a rational number to its standard form, divide its numerator and denominator by their highest common factor (HCF). To find the standard form of a rational number with a negative integer as the denominator, divide its numerator and denominator by their HCF with a minus sign. Comparison of Rational Numbers Among the positive rational numbers with the same denominator, the number with the greatest numerator is the largest. It is easy to compare the rational numbers with same denominators. e.g. 28302830 > 26302630 > 21302130. A negative rational number is to the left of zero whereas a positive rational number is to the right of zero on a number line. So, a positive rational number is always greater than a negative rational number. To compare two negative rational numbers with the same denominator, their numerators are compared ignoring the minus sign. The number with the greatest numerator is the smallest. e.g. –710710 < – 310310; – 6767 < – 4747 To compare rational numbers with different denominators, they are converted into equivalent rational numbers with the same denominator, which is equal to the LCM of their denominators. There are unlimited number of rational numbers between two rational numbers. To find a rational number between the given rational numbers, they are converted to rational numbers with same denominators. Operations on Rational Numbers Addition of Rational numbers The sum of two rational numbers with the same denominator is a rational number whose numerator is the sum of the numerators of the rational numbers with the same denominator. To add rational numbers with different denominators, they are converted into equivalent rational numbers with the same denominator. Additive inverse of a Rational number Two rational numbers whose sum is zero are called the additive inverses of each other. e.g. -720-720 is the additive inverse of 720720 and 720720 is the additive inverse of -720-720. Subtraction of Rational numbers The difference between two rational numbers with the same denominator is a rational number whose numerator is the difference of the numerators of the rational numbers with the same denominator. To subtract rational numbers with different denominators, they are converted into equivalent rational numbers with the same denominator. Multiplication of Rational numbers The numerator and denominator of the product of two rational numbers are equal to the product of their individual numerators and denominators. The numerator of the product of a rational number and an integer is equal to the product of the numerator and the integer with the same denominator. Reciprocal of a Rational number Two rational numbers whose product is 1 are called reciprocals of each other. A rational number and its reciprocal will always have the same sign. e.g. -136-136 × 6-136-13 = 1 Division of Rational numbers To divide one rational number by another, first number is multiplied with the reciprocal of the second number. e.g. 518518 ÷ 2929 = 518518 × 9292 = 54 Construction of Triangles Any one of the following sets of measurements are required to construct a triangle• Length of the three sides • Two sides and the included angle • Two angles and the included side • Length of the hypotenuse and one side in case of a right-angled triangle. Construction of a triangle when measurements of its three sides are given Construct ΔABC, when AB = 6 cm, BC = 7 cm and CA = 9 cm. Steps of construction: Step 1: Draw line segment BC = 7 cm. Step 2: Draw an arc with B as the centre and the radius equal to 6 cm. Step 3: Draw an arc with C as the centre and the radius equal to 9 cm. Step 4: Name the point of intersection of these two arcs as A. Step 5: Join points A and B, and points A and C. Triangle ABC is the required triangle. Construction of a triangle when measurements of two sides and the included angle are given Construct ΔPQR, when PQ = 4 cm, QR = 6 cm and ∠PQR = 60°. Steps of construction: Step 1: Draw line segment QR = 6 cm. Step 2: Construct an angle of 60° at point Q. Step 3: Draw an arc on the ray QX with Q as the centre and the radius equal to 4 cm. Step 4: Name the point where the arc cuts ray QX, as P. Step 5: Join points P and R. Triangle PQR is the required triangle. Construction of a triangle, when two angles and the included side are given Construct ΔXYZ, when ∠ZXY = 40°, ∠XYZ = 95° and the included side XY = 8 cm. Steps of construction: Step 1: Draw line segment XY = 8 cm. Step 2: Construct an angle of 40° at X with XY. Step 3: Construct another angle of 95° at Y with YX. Step 4: Name the point of intersection of the two rays as Z. Triangle XYZ is the required triangle. Construction of a right-angled triangle, when the length of one side and the hypotenuse are given Construct a right-angled triangle LMN, with hypotenuse LN = 8 cm and side MN = 5 cm. Steps of construction: Step 1: Draw line 'l'. Step 2: Mark a point on 'l' and name it M. Step 3: Draw a line segment MN = 5 cm on 'l' . Step 4: Construct a right angle XMN at M. Step 5: Draw an arc with N as the centre and radius equal to 8 cm, such that it intersects MX. Step 6: Mark the point of intersection as L. Step 7: Join points L and N. Triangle LMN is the required triangle. Construction of Parallel Lines Two lines in a plane that never meet each other at any point are said to be parallel to each other. Any line intersecting a pair of parallel lines is called a transversal. Properties of angles formed by parallel lines and transversal: • All pairs of alternate interior angles are equal. • All pairs of corresponding angles are equal. • All pairs of alternate exterior angles are equal. • The interior angles formed on the same side of the transversal are supplementary (the sum of their measures is 180°). Construction of a parallel line using the alternate interior angle property Step 1: Draw line 'l' and point A outside it. Step 2: Mark point B on line 'l'. Step 3: Draw line 'n' joining point A and point B. Step 4: Draw an arc with B as the centre, such that it intersects line 'l' at D and line 'n' at E. Step 5: Draw another arc with the same radius and A as the centre, such that it intersects line 'n' at F. Ensure that arc drawn from A cuts the line 'n' between A and B. Step 6: Draw another arc with F as the centre and distance DE as the radius. Step 7: Mark the point of intersection of this arc and the previous arc as G. Step 8: Draw line 'm' passing through points A and G. Line 'm' is the required parallel line. Verification of the construction If the pair of alternate interior angles are equal in measure, then line 'm' is parallel to line 'l'. Construction of a parallel line using the corresponding angle property Step 1: Draw line 'l' and point P outside it. Step 2: Mark point Q on line 'l'. Step 3: Draw line 'n' joining point P and point Q. Step 4: Draw an arc with Q as the centre, such that it intersects line 'l' at R and line 'n' at S. Step 5: Draw another arc with the same radius and P as the centre, such that it intersects line 'n' at X. Ensure that arc drawn from P cuts the line 'n' outside QP. Step 6: Draw another arc with X as the centre and distance RS as the radius, such that it intersects the previous at Y. Step 7: Draw line 'm' passing through points P and Y. Line 'm' is the required parallel line. Verification of the construction If the pair of corresponding angles are equal in measure, then line 'm' is parallel to line 'l'. Plane Figures Perimeter of a closed figure is the distance around it, whereas area is the region enclosed by closed figure. Perimeter of a regular polygon = number of sides x length of one side. Area and perimeter of a rectangle The perimeter of a rectangle is twice the sum of the lengths of its adjacent sides. Perimeter of a rectangle of length 'l' units and breadth 'b' units = 2(l + b). The area of a rectangle is the product of its length and breadth. Area of a rectangle of length 'l' units and breadth 'b' units = l × b. The perimeter of rectangle ABCD = 2(AB + BC). Area of rectangle ABCD = AB x BC. Each diagonal of a rectangle divides it into two triangles that are equal in area. Area and perimeter of a square The perimeter of a square with side s units is the four times the length of its side. Perimeter of a square with side s units = 4 × s The area of a square with side s is is equal to side multiplied by side. Area of a square with side s units = s × s The perimeter of square ABCD = 4AB or 4BC or 4CD or 4DA. Area of square ABCD = AB2 or BC2 or CD2 or DA2. The diagonals of a square divide it into four triangles that are equal in area. A rectangle and a square having the same perimeter need not have the same area. If the perimeter of a figure increases it is not necessary that its area also increases. Area and perimeter of a triangle The perimeter of a triangle is the sum of the lengths of its sides. Perimeter of a triangle with sides a, b and c = (a + b + c). The area of a triangle is the space enclosed by its three sides. Area of a triangle is half of the product of its base and the corresponding altitude. Area of a triangle with b as the base and h as the altitude = 1212 × bh. Triangles equal in area need not be congruent, but all congruent triangles are equal in area. Area and perimeter of a parallelogram The perimeter of a parallelogram is twice the sum of the lengths of the adjacent sides. The area of a parallelogram is the product of its base and the corresponding altitude. Area of a parallelogram with b as the base and h as the altitude = b × h. Any side of a parallelogram can be considered as the base. The perpendicular drawn on that side from the opposite vertex is known as the height (altitude). The perimeter of parallelogram ABCD = 2(AB + BC) Area of parallelogram ABCD = (AB x DE) or (AD x BF). A parallelogram in which the adjacent sides are equal is called a rhombus. The perimeter and area of a rhombus can be calculated using the same formulae as that for a parallelogram. Conversion of units 1cm = 10 mm 1 cm2 = 100 mm2 1 m2 = 10000 cm2 1 hectare = 10,000 m2 Circles A circle is defined as a collection of points on a plane that are at an equal distance from a fixed point on the plane. The fixed point is called the centre of the circle. Any line segment that passes through the centre of a circle and whose end points are on the circle is called its diameter. Any line segment from the centre of the circle to its circumference is called the radius of the circle. The diameter of a circle is two times the radius. Circumference of a circle The distance around a circular region is known as its circumference. Ratio of circumference and diameter of a circle is denoted by the Greek symbol π. π is an irrational number, whose value is approximately equal to 227227 or 3.14 Circumference of a circle = 2πr, where r is the radius of the circle or Circumference of a circle = πd, where d is the diameter of the circle. Circumference = Diameter x 3.14 Area of a circle The area of a circle is the region enclosed in the circle. The area of a circle can be calculated by using the formula is πr2, if radius r is given; πd24πd24, if diameter d is given; C24πC24π, if circumference C is given. Concentric circles Circles with the same centre but different radii are called concentric circles. Area between two concentric circles = Area of outer circle – Area of inner circle. Conversion of units To convert from a unit of area to its smaller unit of area, we multiply. To convert from a unit of area to a larger unit of area, we divide. 1 cm = 10 mm 1 m = 100 cm 1 km = 1000 m 1 cm2 = 100 mm2 1 m2 = 10000 cm2 1 km2 = 1000000 m2 1 hectare = 10000 m2 Understanding Algebraic Expressions Expressions that contain only constants are called numeric or arithmetic expressions. It is a set of numerical values that are separated by the four mathematical operations, addition, subtraction, multiplication and division. e.g. 9 + 8, 5 – 3 Expressions that contain constants and variables, or just variables, are called algebraic expressions. Variables and constants are combined using mathematical operations to form an algebraic expression. e.g. x – 5, 3b – 6 While writing algebraic expressions, we do not write the sign of multiplication. An algebraic expression containing only variables also has the constant 1 associated with it. The parts of an algebraic expression added to form the expression are called its terms. e.g. The number of terms in the expression x2 + 3x + 5 is 3. A term that contains variables is called a variable term. A term that contains only a number is called a constant term. The constants and the variables whose product makes a term of an algebraic expression, are called the factors of the term. The factors of a constant term in an algebraic expression are not considered. The numerical factor of a variable term is called its coefficient. If the coefficient of a term is 1, it is usually omitted. If the coefficient of a term is –1, it is indicated by only the minus sign. The variable factors of a term are called its algebraic factors. Terms that have different algebraic factors are called unlike terms. Terms that have the same algebraic factors are called like terms. We can compare only the like terms.The unlike terms can not be compared. Algebraic expressions that contain only one term are called monomials. Algebraic expressions that contain two unlike terms are called binomials. Algebraic expressions that contain three unlike terms are called trinomials. All algebraic expressions that have one or more terms are called polynomials. Therefore, monomials, binomials and trinomials are also polynomials. Operations on Algebraic Expressions Addition of algebraic expressions To add algebraic expressions, rearrange the terms in the sum of the given algebraic expressions, so that their like terms and constants are grouped together. While rearranging terms, move them with the correct plus (+) or minus (–) sign before them. To add like terms in an algebraic expression, multiply the sum of their coefficients with their common algebraic factors. e.g. Add 5x2y + 6 and 2x2y – 11. Sol: (5x2y + 6) + (2x2y – 11) = 5x2y + 6 + 2x2y – 11 = 5x2y + 2x2y + 6 – 11 = (5 + 2)x2y + 6 – 11 = 7x2y – 5. Subtraction of algebraic expressions To subtract algebraic expressions Change the signs of the terms of the expression being subtracted. Rearrange the terms in the difference of the given algebraic expressions, so that their like terms and constants are grouped together. While rearranging terms, move them with the correct signs before them. Multiply the difference of their coefficients with their common algebraic factors. Unlike terms remain unchanged in the sum or difference of algebraic expressions. e.g. Subtract 2xy – 3x 2y – 4 from 2x2y – 3xy + 4y + 5. = (2x2y – 3xy + 4y + 5) – (2xy – 3x2y – 4) = 2x2y – 3xy + 4y + 5 – 2xy + 3x2y + 4 = 2x2 y + 3x2y – 3xy – 2xy + 4y + 5 + 4 = (2 + 3)x2y – 3xy – 2xy + 4y + 5 + 4 = 5x2 y – 5xy + 4y + 9. Application of Algebraic Expressions Algebraic expressions can be used to represent number patterns. Ex: Table showing the relation between the number of cones and the number of ice-cream scoops. Number of cones(n) Number of ice-cream scoops (2n) 1 2 2 4 3 6 8 16 15 30 Thus, we can find the value of an algebraic expression if the values of all the variables in the expression are known. e.g. Find the value of the expression 3x2y – 2xy2 + 2xy for x = 2 and y = –2. Sol: 3x2y – 2xy2 + 2xy . Putting x = 2 and y = –2 in the given expression, 3x2y – 2xy2 + 2xy = 3×(2)2×(–2) – 2×(2)×(–2)2 + 2×(2)×(–2) = 3×4×(–2) – 4×4 + 4×(–2) = – 24 –16 – 8 = – 48. Formulas and rules such as the perimeter and area for different geometrical figures are written in a concise and general form using simple, and easy-to-remember algebraic expressions. If 's' represents the side of a square, then its perimeter is '4s' and area is 's2'. If 'l' represents the length and 'b' represents the breadth of a rectangle, then its perimeter is '2(l + b)' and area is 'l × b'. Area of a triangle with base 'b' and the corresponding altitude 'h' is '1212 × base × height'. Perimeter of an equilateral triangle with the length of the side as 'a' units is '3a'. Exponents and Powers An exponent or power is a mathematical representation that indicates the number of times that a number is multiplied by itself. If a number is multiplied by itself m times, then it can be written as: a x a x a x a x a...m times = am. Here, a is called the base, and m is called the exponent or power or index. Numbers raised to the power of two are called square numbers. Square numbers are also read as two-square, three-square, four-square, five-square, and so on. Numbers raised to the power of three are called cube numbers. Cube numbers are also read as two-cube, three-cube, four-cube, five-cube, and so on. Negative numbers can also be written using exponents. If an = b, where a b and n are integers, then an is called the exponential form of b. When numbers are expressed as the product of the powers of their prime factors, we get the prime factor product form. The order of factors in the prime factor product form of a number can be interchanged without changing the value of the number. e.g. ax × by = by × ax. When numbers are expressed using powers of 10, we get the expanded form of numbers. The value of an exponential number with a negative base raised to the power of an even number is positive. The value of an exponential number with a negative base raised to the power of an odd number is negative. (–1)odd number = –1 (–1)even number = + 1 If the base of two exponential numbers is the same, then the number with the greater exponent is greater than the number with the smaller exponent. A number can be expressed as a decimal number between 1.0 and 10.0, including 1.0, multiplied by a power of 10. Such a form of a number is known as its standard form. Large numbers when written in the standard form is much easier to read, understand and compare. Laws of Exponents Multiplication of Powers with the Same Base When numbers with the same base are multiplied, the power of the product is equal to the sum of the powers of the numbers. If 'a' is a non-zero integer, and 'm' and 'n' are whole numbers then, am × an = am+n. Division of Powers with the Same Base When numbers with the same base are divided, then the power of the quotient is equal to the difference between the powers of the dividend and the divisor. If 'a' is a non-zero integer, and 'm' and 'n' are whole numbers then, am ÷ an = am-n. Power of a Power If 'a' is any non-zero integer, and ‘m’and ‘n’ are whole numbers then, (am)n = amn. Multiplication of Powers with the Same Exponent If 'a' is any non-zero integer, and ‘m’ is a whole number then, am × bm = (ab)m. Division of Powers with the Same Exponent If a and b are any non-zero integers and m is a whole number then, am ÷ bm = (abab)m. Numbers with an Exponent of Zero For any non-zero integer a, a0 = 1. Introduction to Solid Shapes All two-dimensional figures have only length and breadth. e.g. A paper has only length and breadth, and hence, it is classified as a plane or two-dimensional figure. Three-dimensional solid shapes have length, breadth and height. e.g. a box is in three-dimensional shape. Faces The flat surface of a three dimensional shape is called a Face. Edges Two faces of a solid shape meet to form an Edge. Vertices The points where three edges meet are called vertices. The table shows the number of the faces, edges and vertices of some shapes. Solid shape Face (F) Edge (E) Vertices (V) 3 2 0 2 1 1 4 6 4 Nets of three-dimensional shapes The net of a three-dimensional solid is a two-dimensional skeleton outline, which, when folded, results in the three-dimensional shape. Drawing solid shapes on a flat surface Solid shapes can be drawn on a flat surface, which is known as the two-dimensional representation of a three-dimensional solid. Sketches of solids are of two types: oblique and isometric. Oblique sketches are drawn on squared paper. The oblique sketch gives the visual representation of a solid shaped object but does not the represent the actual dimensions. Isometric sketches are drawn on dotted or isometric sheets and have the exact measurements of solids. The two-dimensional surface on which we draw an image is usually flat. So, when we try drawing a solid shape on a paper or a board, the image appears a little distorted. However, this is just an optical illusion. When look at the picture of the cube. Not all the lines forming the cube are of equal length. Also, we are unable to see all the faces of the cube. In spite of this illusion, we can make out that the image is of a cube. Such skeletons of solids are called oblique sketches. They can be drawn using a squared paper. The first step is to draw the front face of the cube. Then the opposite face of the cube are drawn. This face should also be of the same size as that of the first square. The number of squares is used as a reference for maintaining the size. Now, the corresponding corners of the squares are joined. And finally, the edges that cannot be seen are drawn, with dotted lines. We can see that the front face and the opposite face of the cube are of the same size. Also, the edges appear equal, though we do not draw them of the same length. All of us must have solved puzzles on isometric sheets at some point of time. An isometric sheet divides a screen into small equilateral triangles made of dots. Using this sheet, we can draw sketches with measurements that agree with that of a given solid. Let’s try to draw the sketch of a cuboid of length three units, breadth two units and height two units. First, a rectangle of length three units and breadth two units is drawn. Then four parallel line segments, each of length two units are drawn starting from the four corners of the rectangle. Finally, the matching corners with appropriate line segments are connected. The isometric sketch of the cuboid is formed. Observe that the measurements are of the exact size, which is not the case with oblique sketches. Viewing the Different Section of Solids Visualising solid shapes is a very useful skill. We can see the hidden parts of a solid shape. For example, when a cuboid with a square face is cut vertically, then each face is a square. The face is a cross section of the cuboid. Three-dimensional objects or solids have length, breadth and height, and look different from different points of view. Sections of solid shapes can be viewed using three ways: • Viewing the cross sections • Using shadows • Viewing at certain angles A solid can be viewed from different angles. Viewing a solid from the front, side and top are the three most common ways of viewing it. Cutting or slicing a solid will result in its cross-section, which is also one way of viewing it. Observing the two-dimensional shadow of a three-dimensional solid is another method of viewing a solid. Shadows of solids are of different sizes depending on the position of the solid and the position of the source of light.