VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Curriculum

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1 VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Curriculum Chapter As Per NCERT Text Book 1. Integers 2. Fractions and Decimals 3. Data Handling 4. Simple Equations 5. Lines and Angles 6. The Triangle and its Properties 7. Congruence of Triangles 8. Comparing Quantities 9. Rational Numbers 10. Practical Geometry 11. Perimeter and Area 12. Algebraic Expressions 13. Exponents and Powers 14. Symmetry 15. Visualizing Solid Shapes Topic Operations on Integers Properties of Integers Multiplication and Division on Fraction Multiplication and Division on Decimals Data Representation Data Value Introduction to Simple Equations Application of Simple Equations Angles Pairs of Lines Triangles Properties of Triangles Congruence of Plane Figures Criteria for Congruence of Triangles Ratios and Proportions Percentages Conversions Application of Percentages Introduction to Rational Numbers Comparison of Rational Numbers Operations on Rational Numbers Construction of Triangles Construction of Parallel Lines Plane Figures Circles Understanding Algebraic Expressions Operations on Algebraic Expressions Application of Algebraic Expressions Exponents and Powers Laws of Exponents Line Symmetry Rotational Symmetry Introduction to Solid Shapes Viewing the different section of Solids VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Curriculum Page 1

2 1. Integers Operations on Integers When two positive integers are added, we get a positive integer. When two positive integers are added, we get a positive integer. Eg: = 10 When two negative integers are added, we get a negative integer. Eg: -6 + (-3) = -9 When a positive and a negative integer are added, the sign of the sum is always the sign of the bigger number of the two, without considering their signs. Eg: = 20 and = -25 The additive inverse of any integer a is - a, and the additive inverse of (- a) is a. Eg: Additive inverse of (-12) = - (-12) = 12 Subtraction is the opposite of addition, and, therefore, we add the additive inverse of the integer that is being subtracted, to the other integer. Eg: = 23 + Additive inverse of 43 = 23 + (- 43) = - 20 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 is the inverse operation of multiplication. 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 VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 1

3 integer. For any integer p, p multiplied by zero is equal to zero multiplied by p, which is equal to zero. For any integer p, p divided by zero is not defined, and zero divided by p is equal to zero, where p is not equal to zero. Properties of Integers Integers are closed under addition, i.e. for any two integers,a and b, a+b is an integer. 1. Closure property: 2. Closure property under addition: Integers are closed under addition, i.e. for any two integers,a and b, a+b is an integer. Eg: 3+4=7;-9+7=2 3. Closure property under subtraction: Integers are closed under subtraction, i.e. for any two integers,a and b, a-b is an integer. Eg: -21-(-9)=-12;8-3=5 4. Closure property under multiplication: Integers are closed under multiplication, i.e. for any two integers,a and b, ab is an integer. Eg: 5 6=30; -9-3=27 5. Closure property under division: Integers are NOT closed under division, i.e. for any two integers, Eg: VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 2

4 6. Commutative property: 7. Commutative property under addition: Addition is commutative for integers. For any two integers, a and b, a+b=b+a Eg:5+(-6)=5-6=-1; (-6)+5=-6+5=-1 5+(-6)=(-6)+5 8. Commutative property under subtraction: Subtraction is NOT commutative for integers. For any two integers, a-b b-a Eg:8-(-6)=8+6=14; (-6)-8=-6-8=-14 8-(-6) Commutative property under multiplication: Multiplication is commutative for integers. For any two integers, a and b, ab=ba Eg:9 (-6)=-(9 6)=-54; (-6) 9=-(6 9)=-54 9 (-6)=(-6) Commutative property under division: Division is NOT commutative for integers. For any two integers, Eg:3/6=1/2; 11. Associative property: 12. Associative property under addition: Addition is associative for integers. For any three integers, a, b and c, a+(b+c)=(a+b)+c Eg:5+(-6+4)=5-2=3; (5-6)+4=-1+4=3 5+(-6+4)=(5-6)+4 VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 3

5 13. Associative property under subtraction: Subtraction is associative for integers. For any three integers, a-(b-c) (a-b)-c Eg:5 (6-4)=5-2=3; (5-6)-4=-1-4=-5 5 (6-4) (5-6) Associative property under multiplication: Multiplication is associative for integers. For any three integers, a, b and c, (a b) c=a (b c) Eg: [(-3) (-2)) 4]=(6 4)=24 [(-3) (-2 4) ]=(-3-8)=24 [(-3) (-2)) 4]=[(-3) (-2 4) ] 15. Associative property under division: Division is NOT associative for integers. 16. Distributive property: 17. Distributive property of multiplication over addition: For any three integers, a, b and c, a (b+c) = a b+a c Eg: -2 (4 + 3) = -2 (7) = -14-2(4+3)=(-2 4)+(-2 3) =(-8)+(-6) = Distributive property of multiplication over subtraction: For any three integers, a, b and c, a (b-c)= a b-a c Eg: -2 (4-3) = -2 (1) = -2-2(4-3)=(-2 4)-(-2 3) =(-8)-(-6) =-2 VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 4

6 The distributive property of multiplication over the operations of addition and subtraction is true in the case of integers. 19. Identity under addition: Integer 0 is the identity under addition. That is, for an integer a, a+0=0+a=a. Eg: 4+0=0+4=4 20. Identity under multiplication: The integer 1 is the identity under multiplication. That is, for an integer a, 1 a=a 1=a Eg: (-4) 1=1 (-4)=-4 VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 5

7 2. Fractions and Decimals Multiplication and Division on Fraction To multiply a fraction by a whole number, multiply the whole number by the numerator of the fraction. Multiplication of fractions: To multiply a fraction by a whole number, multiply the whole number by the numerator of the fraction. Eg: While multiplying a whole number by a mixed fraction, change the mixed fraction into an improper fraction. Eg: To multiply two fractions, multiply their numerators and denominators. Eg: 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. Division of fractions: To obtain the reciprocal of a fraction, interchange the numerator with the denominator. Eg:. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 6

8 To divide a whole number by a fraction, take the reciprocal of the fraction and then multiply it by the whole number. Eg: To divide a fraction by a whole number, multiply the fraction by the reciprocal of the whole number. Eg: To divide a fraction by a fraction, multiply the first fraction by the reciprocal of the second fraction. Multiplication and Division on Decimals Ignore the decimals and multiply the two numbers. Multiplication of decimals: 1. To multiply a whole number by a decimal number, follow these steps: 2. Ignore the decimals and multiply the two numbers. 3. Count the numbers of digits to the right of decimal point in the original decimal number. 4. Insert the decimal, from right to left, in the answer by the same count. Eg: (i) 3 0.2=0.6 (ii) 3 0.4= To multiply a decimal number by a decimal number, follow these steps: 2. Ignore the decimals and multiply the two numbers. 3. Count the number of digits to the right of decimal point in both the decimal numbers. 4. Add the number of digits counted and insert the decimal, from right to left, in the answer by the same count. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 7

9 Eg: (i) =0.14 (ii) = To multiply a decimal number by 10, 100 or 1000, follow these steps: 2. While multiplying a decimal number by 10, retain the original number and shift the decimal to the right by one place. 1. While multiplying a decimal number by 100, retain the original number and shift the decimal to the right by two places. 2. While multiplying a decimal number by 1000, retain the original number and shift the decimal to the right by three places. Division of decimals: 1. To divide a decimal number by a whole number, follow these steps: 2. Convert the decimal number into a fraction. 3. Take the reciprocal of the divisor. 4. Multiply the reciprocal by the fraction. Eg: 1. To divide a decimal number by another decimal number, follow these steps: 2. Convert both the decimal numbers into fractions. 3. Take the reciprocal of the divisor. 4. Multiply the reciprocal by the fraction. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 8

10 3. Data Handling Data Representation Representation of data using bars of uniform width. 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. Graphs are a visual representation of organised data. 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. 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. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 9

11 Data Value Arithmetic mean is a number that lies between the highest and the lowest value of data. Arithmetic mean is a number that lies between the highest and the lowest value of data. Note that we need not arrange the data in ascending or descending order to calculate arithmetic mean. Range = Highest observation - Lowest observation Mode refers to the observation that occurs most often in a given data. The following are the steps to calculate mode: Step - 1: Arrange the data in ascending order. Step - 2: Tabulate the data in a frequency distribution table. Step - 3: The most frequently occurring observation will be the mode. 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. The following are the steps to calculate median. Step - 1: Arrange the data in ascending order. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 10

12 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 representative values of a group of observations or data, and lie between the minimum and maximum values of the data. They are also called measures of the central tendency. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 11

13 4. Simple Equations Introduction to Simple Equations An equation is a condition of equality between two mathematical expressions. An equation is a condition of equality between two mathematical expressions. Eg:2x-3=5, 3x+9=11, 4y+2=12 If the left hand side of an equation is equal to its right hand side for any value of the variable, then that value is called the solution of that equation. Eg:For the equation, 5x+5=15, x=2 is a solution. When we add or subtract the same number to or from both the sides of an equation, the value of the left hand side remains equal to its value on the right hand side. Eg:5x+3=13 On subtracting 2 from both sides, we get 5x+3-2=13-2 5x+1=11 When we divide or multiply an equation 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. Eg: 1) 5x+1=1 3 On dividing both sides by 4, we get 2) 5x+1=13 On multiplying both sides by 4, we get 4(5x+1)=4(13) 20x+4=52 VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 12

14 Application of Simple Equations To find the solution of an equation, we have to perform identical mathematical operations To find the solution of an equation, we have to perform identical mathematical operations on the two sides of the equation so that only the variable remains on one side. Eg:3x+8=84 3x+8-8=84-8 3x=76 Transposing means moving a term of the equation to the other side. Transposing a number is the same as adding or subtracting the same number from both sides of the equation. Eg:To solve 2x+8=24 Given, 2x+8=24 Transposing 8 to the right hand side, we get 2x=24-8 2x=16 x=8 Hence, the value of x is 8. When a number is transposed from one side of the equation to the other, its sign changes. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 13

15 5. Lines and Angles Angles An angle is formed when lines or line segments meet. An angle is formed when lines 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. The angle formed by two lines or line segments can be an acute angle, where the measure of the angle is less than The angle formed by two lines or line segments can be a right angle, where the measure of the angle is90 0. The angle formed by two lines or line segments can be an obtuse angle, where the measure of the angle is greater than VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 14

16 The angle formed by two lines or line segments can be a straight angle, where the measure of the angle is equal to The angle formed by two lines or line segments can be a reflex angle, where the measure of the angle is greater than When the sum of the measures of two angles is 90 0, the angles are called complementary angles. When the sum of the measures of two angles is 180 0, the angles are called supplementary angles. Adjacent angles have a common vertex and a common arm, and the non-common arms are on either side of the common arm. A linear pair is pair of adjacent angles whose non-common sides are opposite rays. Vertically opposite angles are opposite to each other and are equal. Adjacent angles have no common interior points. The angles in a linear pair are supplementary. Angles may have a common vertex and a common arm. Two angles are said to be adjacent angles if they have a common side or arm. Two adjacent angles can be either supplementary or complementary. If two adjacent angles are supplementary, then they form a straight angle, and are also called a linear pair. Two angles that are not adjacent but have a common vertex are called vertically opposite angles. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 15

17 Vertically opposite angles are equal. Pairs of Lines Lines that meet at a point are called intersecting lines. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 16

18 Lines that meet at a point are called intersecting lines. Lines that always remain the same distance apart and never meet are called parallel lines. A line that intersects two or more lines at distinct points is called a transversal. When two lines are intersected by a transversal, pairs of corresponding angles, alternate angles and interior angles are formed. 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. Each pair of interior angles on the same side of the transversal are supplementary, each pair of corresponding angles are equal and each pair of alternate interior angles are equal. If a transversal cuts two lines such that the pairs of corresponding angles are equal, then the lines are parallel. Eight angles are formed when a transversal intersects two lines. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 17

19 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. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 18

20 6. The Triangle and its Properties Triangles A triangle is said to be equilateral if each one of its sides is of the same length and each of one its angles measures The Triangle and Its Properties 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. The sides and the angles of every triangle differ from one another; therefore, they do not look alike. Triangles can be classified based on their sides and angles. Based on their sides, there are equilateral, isosceles and scalene triangles. Based on their angles, there are acute, obtuse and right-angled triangles. 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 an Scalene triangle. Acute-angled triangle: A triangle with all its angles less than 90 is known as an acute-angled triangle. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 19

21 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 90 angle is called the hypotenuse, and is the longest side of the triangle. Mark the mid-point of the side of a triangle, and join it to its opposite vertex. This line segment is called a median. It is defined as a line segment drawn from a vertex to the mid-point of the opposite side. You can draw three medians to a given triangle. The medianspass through a common point. Hence, the mediansof a triangle are concurrent. This point of concurrence is called the centroid, and VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 20

22 is denoted by G. The centroidand medians of a triangle always lie inside the triangle. The centroid of a triangle divides the median in the ratio 2:1. Altitude: The altitude of a triangle is a line segment drawn from a vertex and is perpendicular to the opposite side. 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. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 21

23 Properties of Triangles The sum of the three angles in a triangle Angle sum property: The sum of the three angles in a triangle is equal to Eg: If A, B and C are the angles of a triangle, then Suppose a line XY is parallel to side BC. AB is a transversal that cuts line XY and AB, at A and B, respectively. As the alternate interior angles are equal,. Also,. form a linear pair, and their sum is Exterior angle property: An exterior angle of a triangle is equal to the sum of its opposite interior angles. Eg: In the figure here, 4 is called the exterior angle to triangle ABC, and 4 = The sum of the lengths of any two sides of a triangle is greater than the third side. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 22

24 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 legs. a 2 = b 2 + c 2 Converse: If the Pythagoras property holds, then the triangle must be right-angled. That is, if there is a triangle such that the sum of the squares on two of its sides is equal to the square of the third side, then it must be a right-angled triangle. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 23

25 7. Congruence of Triangles Congruence of Plane Figures If two objects are of exactly the same shape and size, they are said to be congruent. If two objects are of exactly the same shape and size, they are said to be congruent. The relation between two congruent objects is called congruence. 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. 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. Eg: Two line segments, say, PQ & RS are congruent if they have equal lengths. We write this as If two angles have the same measure, they are congruent. Also, if two angles are congruent, their measures are the same. Eg: Two angles, say, and, are congruent if their measures are equal. We write this as or as. However, commonly, we write VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 24

26 A plane figure is any shape that can be drawn in two dimensions. Congruence of Plane Figures A plane figure is any shape that can be drawn in two dimensions. A plane figure is any shape that can be drawn in two dimensions. Examples: Rectangle, square, triangle, rhombus, etc. If two objects are of exactly the same shape and size, they are said to be congruent, and the relation of the two objects being congruent is called congruence. Or we can state it as: "Two plane figures are congruent if each, when superimposed on the other, covers it exactly." Congruence is denoted by. However, how do you check if two figures drawn on a paper are congruent or not? One method is to 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 other. However, take care not to twist, bend or stretch the traced or cut image. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 25

27 If two angles have the same measure, then they are congruent. Also, if two angles are congruent, then their measures are the same. If two angles are congruent, then the lengths of their arms do not matter. Criteria for Congruence of Triangles Two triangles are congruent if all the sides and all the angles 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 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. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 26

28 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. We can tell if two triangles are congruent using 4 axioms: SAS axiom, ASA axiom, SSS axiom and RHS axiom. 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. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 27

29 We can tell if two triangles are congruent using 4 axioms: SAS axiom, ASA axiom, SSS axiom and RHS axiom. 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. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 28

30 VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 29

31 8. Comparing Quantities Ratios and Proportions Ratios are used to compare quantities. Ratios are used to compare quantities. 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 into like fractions. If these like fractions are equal, then we say that the given ratios are equivalent. Eg:6 pens cost Rs 90. What would be the cost 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=150 The ratio of two quantities in the same unit is a fraction that shows how many times one quantity is greater or smaller than the other. 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. Percentages: Percentage is another method used to compare quantities. Percentages are numerators of fractions with the denominator 100. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 30

32 Meaning of percentage: Per cent is derived from the Latin word 'per centum', which means per hundred. Per cent is represented by the symbol - %. Percentages Percentage is the numerator of a fraction, whose denominator is hundred. Percentage is the numerator of a fraction, whose denominator is hundred. Converting fractional numbers into percentage: To compare fractional numbers, we need a common denominator. To convert a fraction into a percentage, multiply it by hundred and then place the % symbol. For example, 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 100%. Conversion of percentages into fractions or decimals: Examples: VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 31

33 Percent 2% 45% Fraction 2/100 Decimal Conversions A given percentage can be converted into fractions and decimals. Also, a decimal can be converted as percentage. Let us see one by one. A given percentage can be converted into fractions and decimals. Also, a decimal can be converted as percentage. Let us see one by one. To convert a percentage into a fraction: Step1: Drop the percent sign, and then divide the number by hundred. 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. Eg: 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. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 32

34 Application of Percentages Percentages are helpful in comparison. Percentages are helpful in comparison. To calculate percentage increase/decrease, we use the formula: Percentage increase/decrease = Profit = Selling price -Cost price ; Loss = Cost price-selling price Profit percentage or loss percentage is always calculated on the cost price. 1. To find the profit percent, we use the following formula: 2. To find the loss percent, we use the following formula: Amount at the end of the year can be obtained by adding the sum borrowed (principal) and the interest. Thus, Amount = Principal + Interest 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 given by If you are given any three of these quantities, the fourth one can be calculated using this formula. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 33

35 9. Rational Numbers Introduction to Rational Numbers All numbers, including whole numbers, integers, fractions and decimal numbers, can be written in the numerator-denominator form. All numbers, including whole numbers, integers, fractions and decimal numbers, can be written in the numerator-denominator form. 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. A rational number is positive if its numerator and denominator are both either positive integers or negative integers.. If either the numerator or the denominator of a rational number is a negative integer, then the rational number is called a negative rational number. The rational number zero is neither negative nor positive. On the number line: o Positive rational numbers are represented to the right of 0. o Negative rational numbers are represented to the left of 0. By multiplying or dividing both the numerator and the denominator of a rational number by the same non-zero integer, we can get another rational number that is equivalent to the given rational number. 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. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 34

36 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 While comparing positive rational numbers with the same denominator, the number with the greatest numerator is the largest. While comparing positive rational numbers with the same denominator, the number with the greatest numerator is the largest. It is easy to compare these numbers if their denominators are the same. Eg: A positive rational number is always greater than a negative rational number. While comparing negative rational numbers with the same denominator, compare their numerators ignoring the minus sign. The number with the greatest numerator is the smallest. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 35

37 Positive rational numbers lie to the right of 0, while negative rational numbers lie to the left of 0 on the number line. To compare rational numbers with different denominators, convert them into equivalent rational numbers with the same denominator, which is equal to the LCM of their denominators. You can find infinite rational numbers between any two given rational numbers. Operations on Rational Numbers To add rational numbers with different denominators Addition of rational numbers: The following points are to be noted while finding the sum of rational numbers: 1. If the denominators of the given rational numbers are the same, then the denominator of their sum will also be the same. The numerator of the sum of two rational numbers with the same denominator is the sum of the numerators of the given numbers. 2. To add rational numbers with different denominators, we convert them into equivalent rational numbers with the same denominator. 3. Thus, the sum is. 1. Two rational numbers whose sum is zero are called additive inverses of each other. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 36

38 2. The denominator of the difference of two rational numbers with the same denominator is the same as the common denominator of the given numbers. Subtraction of rational numbers: 1. To subtract rational numbers with different denominators, we must convert them into equivalent rational numbers with the same denominator. Multiplication of rational numbers: 1. To multiply two rational numbers, we simply multiply their numerators and denominators with their correct signs. 2. Two rational numbers whose product is 1 are called reciprocals of each other. = 3. A rational number and its reciprocal will always have the same sign. Division of rational numbers: 1. To divide one rational number by another, we actually multiply the first number with the reciprocal of the second number. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 37

39 10. Practical Geometry Construction of Triangles The sum of the measures of the three angles... 1) The sum of the measures of the three angles of a triangle is 2) The sum of the lengths of any two sides of a triangle is always greater than the length of the third side. 3) The difference between the lengths of any two sides of a triangle is always less than the length of the third side. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 38

40 4) The measure of an exterior angle is equal to the sum of the two remote or opposite interior angles. Practical Geometry - Construction of Triangles The properties of triangles are as follows: To construct a triangle, we should know any one of the following: 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 Parallel Lines Two lines in a plane that never meet each other at any point are said to be parallel to each other. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 39

41 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: 1. All pairs of alternate interior angles formed by parallel lines and a transversal are equal. 2. All pairs of corresponding angles formed by parallel lines and a transversal are equal. 3. All pairs of alternate exterior angles formed by parallel lines and a transversal are equal. 4. The interior angles formed on the same side of the transversal are supplementary (the sum of their measures is ). Steps to construct parallel lines using the alternate interior angle property: 1. Draw a line, l. 2. Mark point A outside line l. 3. Mark point B on line l. 4. Draw line joining points A and B. 5. Draw an arc with B as the centre, such that it intersects lines l and n at points D and E, respectively. 6. Draw another arc with the same radius and A as the centre, such that it intersects line n at F. 7. Ensure that the arc drawn from A cuts line n between A and B. 8. Measure distance DE with the help of the compass. 9. Draw another arc with F as the centre and radius equal to DE. 10. Mark the point of intersection of this arc and the previous arc as G. 11. Draw line m passing through A and G. 12. Lines l and m are parallel. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 40

42 Verification of the construction: If the pair of alternate interior angles ABC and BAG are equal in measure, then line l //line m. Hence, the construction is verified. Steps to construct parallel lines using the corresponding angle property: 1. Draw line l and point P outside it. 2. Mark point Q on line l. 3. Draw line joining point P and point Q. 4. Draw an arc with Q as the centre, such that it intersects line l at R and line n at S. 5. Draw another arc with the same radius and P as the centre, such that it intersects line n at X. 6. Ensure that the arc drawn from P cuts line n outside QP. 7. Draw another arc with X as the centre and distance RS as the radius, such that it intersects the previous arc at Y. 8. Draw line m passing through points P and Y. 9. Lines l and m are parallel. Verification of the construction: If the pair of corresponding angles PQR and XPY are equal in measure, then line l II line m. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 41

43 11. Perimeter and Area Plane Figures The area of the parallelogram is given by base x height. Triangle: A triangle is a polygon with three vertices, and three sides or edges that are line segments. A triangle with vertices A, B, and C is denoted as ABC The perimeter of a triangle is the sum of the lengths of its sides. If the three sides are a, b, and c, then perimeter VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 42

44 The area of a triangle is the space enclosed by its three sides. It is given by the formula, b is the base and h is the altitude. where A simple closed figure bounded by four line segments is called a quadrilateral. Various types of quadrilateral are: Rectangle Square Parallelogram Rhombus A rectangle is a quadrilateral with opposite sides equal, and each angle of measure 90 o. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 43

45 The perimeter of a rectangle is twice the sum of the lengths of its adjacent sides. In the figure, the perimeter of rectangle ABCD = 2(AB + BC). The area of a rectangle is the product of its length and breadth. In the figure, the area of rectangle ABCD = AB x BC. A square is a quadrilateral with four equal sides, and each angle of measure 90 o. The perimeter of a square with side s units is 4s. In the figure, the perimeter of square ABCD = 4AB or 4BC or 4CD or 4DA. The area of a square with side s is s 2 In the figure, the perimeter of square ABCD = AB 2 or BC 2 or CD 2 or DA 2. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 44

46 A quadrilateral in which both the pairs of opposite sides are parallel is called a parallelogram. The perimeter of a parallelogram is twice the sum of the lengths of the adjacent sides. In the figure, the perimeter of parallelogram ABCD = 2(AB + BC) The area of a parallelogram is the product of its base and perpendicular height or altitude. Any side of a parallelogram can be taken as the base. The perpendicular dropped on that side from the opposite vertex is known as the height (altitude). In the figure, the 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. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 45

47 Circles A circle is defined as a collection of points on a plane that are at an equal distance... Circle: 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. Circumference: The distance around a circular region is known as its circumference. Diameter: Any straight line segment that passes through the centre of a circle and whose end points are on the circle is called its diameter. Radius: Any line segment from the centre of the circle to its circumference. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 46

48 Circumference of a circle =, where r is the radius of the circle or, where d is the diameter of the circle. Π is an irrational number, whose value is approximately equal to. Circumference = Diameter x 3.14 Diameter(d) is equal to twice radius(r). Circles with the same centre but different radii are called concentric circles. 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: if radius r is given if diameter D is given if circumference C is given VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 47

49 12. Algebraic Expressions Understanding Algebraic Expressions Expressions that contain only constants are called numeric or arithmetic expressions. Expressions that contain only constants are called numeric or arithmetic expressions. Expressions that contain constants and variables, or just variables, are called algebraic expressions. 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 joined together by plus (+) signs are called its terms. 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. 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. Algebraic expressions that contain only one term are called monomials. Algebraic expressions that contain only two unlike terms are called binomials. Algebraic expressions that contain only three unlike terms are called trinomials. All algebraic expressions that have one or more terms are called polynomials. Therefore, binomials and trinomials are also polynomials. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 48

50 Operations on Algebraic Expressions We can perform arithmetic operations on algebraic expressions. To add the like terms in an algebraic expression... We can perform arithmetic operations on algebraic expressions. To add the like terms in an algebraic expression, multiply the sum of their coefficients with their common algebraic factors.to subtract the like terms in an algebraic expression, multiply the difference of their coefficients with their common algebraic factors. For example, let us subtract 9ab from. Thus, to add or subtract 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 the terms, move them with the correct '+' or '-'sign before them. Let us see this example. Add and We know that, the terms are and are like items. Therefore, Unlike terms remain unchanged in the sum or difference of algebraic expressions. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 49

51 Application of Algebraic Expressions Algebraic expressions can be used to represent number patterns. Algebraic expressions can be used to represent number patterns. In the following example, we can find the relation between number of cones and the number of icecream scoops. Number of cones Number of icecream scoops (2n) (n) x 15 Thus, we can find the value of an algebraic expression if the values of all the variables in the expression are known. In a similar way, we can write formulas for the perimeter and area for different geometrical figures using simple, easy-to-remember algebraic expressions If s represents the side of a square, then its: o Perimeter is 4s o Area is s 2 If l represents the length and b represents the breadth of a rectangle, then its:perimeter is Area is l x b VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 50

52 13. Exponents and Powers Exponents and Powers An exponent or power is a mathematical representation that indicates the number of times that a number is multiplied by itself. 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 = a m Here, a is called the base, and m is called the exponent, 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 a n = b, where a and b are integers and n is a natural number, then a n is called the exponential form of b. The factors of a product can be expressed as the powers of the prime factors of 100. This form of expressing numbers using exponents is called the prime factor product form of exponents. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 51

53 Even if we interchange the order of the factors, the value remains the same. So a raised to the power of x multiplied by b raised to the power of y, is the same as b raised to the power of y multiplied by a raised to the power of x. The value of an exponential number with a negative base raised to the power of an even number is positive. 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. Laws of Exponents When numbers with the same base are multiplied, the power of the product... When numbers with the same base are multiplied, the power of the product is equal to the sum of the powers of the numbers. More precisely if are whole numbers then, 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. That is, if is a non-zero integer, and and are whole numbers then, The other laws of exponents are as follows: VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 52

54 1),where> are non - zero integers, and is a whole number. 2),where are non - zero integers, and is a whole number. 3),where> a is a non-zero integer, and m and n are whole numbers. 4) The value,where> a is a non-zero integer. 5),where> a is a non - zero integer, and m and n are whole numbers. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 53

55 14. Symmetry Line Symmetry A figure has line symmetry if a line can be drawn dividing the figure into two identical parts. The word symmetry comes from the Greek word symmetros, which means even. A figure has line symmetry if a line can be drawn dividing it into two identical parts. The line is called the line of symmetry or axis of symmetry. Line symmetry is also known as reflection symmetry because a mirror line resembles the line of symmetry, where one half is the mirror image of the other half. Remember, while looking at a mirror, an object placed on the right appears to be on the left, and vice versa. For a line segment, the perpendicular bisector is the line of symmetry. For an equilateral triangle, the bisectors of the internal angles are the lines of symmetry. For a square, the lines of symmetry are the diagonals and the lines joining the mid-points of the opposite sides. The lines of symmetry of a rectangle are the lines joining the mid-points of the opposite sides. The line of symmetry of an isosceles triangle is the perpendicular bisector of the non-equal side. A scalene triangle, in which all the sides are of different lengths, doesn't have any line of symmetry. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 54

56 Regular polygon: A polygon is said to be a regular polygon if all its sides are equal in length and all its angles are equal in measure. If a polygon is not a regular polygon, then it is said to be an irregular polygon. Regular and irregular polygons have lines of symmetry. An equilateral triangle is regular because each of its sides has the same length, and each of its angles measures sixty degrees. The number of lines of symmetry in a regular polygon is equal to the number of sides that it has. A pentagon has five lines of symmetry. Similarly, a regular octagon has eight sides, and therefore, it will have eight lines of symmetry, while a regular decagon has ten sides, so it will have ten lines of symmetry. Irregular polygon: Most irregular polygons do not have line symmetry. However, some of them do. Look at the rectangle and the isosceles triangle. A rectangle has two lines of symmetry, and an isosceles triangle has one line of symmetry. Some letters have line symmetry. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 55

57 The letters A, B, C, D, E, I, K, M, T, U, V, W and Y have one line of symmetry. The letter H overlaps perfectly both vertically and horizontally. So it has two lines of symmetry. Similarly, the letter X has two lines of symmetry. The letters F, G, J, L, N, P, Q, R, S and Z have no line of symmetry. Rotational Symmetry Any object or shape is said to have rotational symmetry if it looks exactly the same... Any object or shape is said to have rotational symmetry if it looks exactly the same at least once during a complete rotation through three hundred and sixty degrees. During the rotation, the object rotates around a fixed point. Its shape and size do not change. This fixed point is called the centre of rotation. Rotation may be clockwise or anti-clockwise. A full turn refers to a rotation of three hundred and sixty degrees. A half turn refers to a rotation of one hundred and eighty degrees. A quarter turn refers to a rotation of ninety degrees. The angle at which a shape or an object looks exactly the same during rotation is called the angle of rotation. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 56

58 The order of rotational symmetry can be defined as the number of times that a shape appears exactly the same during a full 360 o rotation. The centre of rotation of a square is its centre. The angle of rotation of a square is 90 degrees, and its order of rotational symmetry is 4. The centre of rotation of a circle is the centre of the circle. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 57

59 There are many shapes that have only line symmetry and no rotational symmetry at all. Some objects and shapes have both, line symmetry as well as rotational symmetry. The Ashok Chakra in the Indian national flag has both, line symmetry and rotational symmetry. Symmetry can be seen in the English alphabet as well. The letter H has both line symmetry and rotational symmetry. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 58

60 15. Visualizing Solid Shapes Introduction to Solid Shapes All two-dimensional figures have only length and breadth. All two-dimensional figures have only length and breadth. For example, paper has only length and breadth, and hence, it is classified as a plane or twodimensional figure. Three-dimensional solid shapes have length, breadth and height. For example, a biscuit tin is in three-dimensional shape. Faces: The flat surfaces that form the skin of solids are called faces. Edges: The line segments that form the skeleton of solids are called edges. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 59

61 Vertices: The points where three edges meet are called vertices. The table shows the number of the faces, edges and vertices of some shapes. Face (F) Edge (E) Vertex (V) Cylinder Cone Triangular pyramid The net of a three-dimensional solid is a two-dimensional skeleton outline, which, when folded, results in the three-dimensional shape. Net solid VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 60

62 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. They do not have exact dimensions, but still convey all the significant aspects of the appearance of a solid. 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. 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. You can draw them using a squared paper. The first step is to draw the front face of the cube. Then draw the opposite face of the cube. This face should also be of the same size as that of the first square. Use the number of squares as a reference for maintaining the size. Now, join the corresponding corners of the squares. And finally, draw the edges that cannot be seen, with dotted lines. You 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, draw a rectangle of length three VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 61

63 units and breadth two units. Then draw four parallel line segments, each of length two units, starting from the four corners of the rectangle. Finally, connect the matching corners with appropriate line segments. The isometric sketch of the cuboid is ready. Observe that the measurements are of the exact size, which is not the case with oblique sketches. Viewing the different section of Solids Visualizing solid shapes is a very useful skill. Visualizing 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. VII I.I.T. Foundation, N.T.S.E. & Mathematics Olympiad Chapter Notes Page 62