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  • Direct And Inverse Proportions | Study

  • Factorisation | Study

  • Introduction to Graphs | Study

  • Exponents And Powers | Study

  • Data Handling | Study

  • Squares And Square Roots | Study

  • Cubes And Cube Roots | Study

  • Comparing Quantities | Study

  • Algebraic Expressions And Identities | Study

  • Linear Equations In One Variable | Study

  • Rational Numbers | Study

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  • Direct And Inverse Proportions | Study

    Direct And Inverse Proportions | Study

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    Direct and Inverse Proportions | Speed Notes

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    Variations: If the values of two quantities depend on each other in such a way that a change in one causes corresponding change in the other, then the two quantities are said to be in variation. (Scroll down to continue …)

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    Direct Variation or Direct Proportion:

    Extra:

    Two quantities x and y are said to be in direct proportion if they increase (decrease) together in such a manner that the ratio of their corresponding values remains

    constant. That is if

    =k [k is a positive number, then x and y are said to vary directly.

    In such a case if y1, y2 are the values of y corresponding to the values x1, x of x

    respectively then = .

    If the number of articles purchased increases, the total cost also increases. More than money deposited in a bank, more is the interest earned.

    Quantities increasing or decreasing together need not always be in direct proportion, same in the case of inverse proportion.

    When two quantities x and y are in direct proportion (or vary directly), they are

    written as

    . Symbol

    stands for ‘is proportion to’.

    Inverse Proportion: Two quantities x and y are said to be in inverse proportion if an increase in x causes a proportional decrease in y (and vice-versa) in such a manner that the product of their corresponding values remains constant. That is, if xy

    = k, then x and y are said to vary inversely. In this case if y1, y2 are the values of y

    corresponding to the values x1, x2 of x respectively then

    x1, Y1 = x2, y2 or

    =

    When two quantities x and y are in inverse proportion (or vary inversely), they are

    written as x

    . Example: If the number of workers increases, time taken to finish

    the job decreases. Or If the speed will increase the time required to cover a given distance decreases.

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    • Factorisation | Study

      Factorisation | Study

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      Factorisation: Representation of an algebraic expression as the product of two or more expressions is called factorization. Each such expression is called a factor of the given algebraic expression. (Scroll down to continue …)

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      When we factorise an expression, we write it as a product of factors. These factors may be numbers, algebraic variables or algebraic expressions.

      An irreducible factor is a factor which cannot be expressed further as a product of factors.

      A systematic way of factorising an expression is the common factor method. It consists of three steps:

      1. Write each term of the expression as a product of irreducible factors
      2. Look for and separate the common factors and
      3. Combine the remaining factors in each term in accordance with the distributive law.

      Sometimes, all the terms in a given expression do not have a common factor; but the terms can be grouped in such a way that all the terms in each group have a common factor. When we do this, there emerges a common factor across all the groups leading to the required factorisation of the expression. This is the method of regrouping.

      In factorisation by regrouping, we should remember that any regrouping (i.e., rearrangement) of the terms in the given expression may not lead to factorisation. We must observe the expression and come out with the desired regrouping by trial and error.

      A number of expressions to be factorised are of the form or can be put into the form: a2 + 2ab + b2, a2 – 2ab + b2, a2 – b2 and x2 + (a + b)x + ab. These expressions can be easily factorised using Identities I, II, III and IV

      a2 + 2ab + b2 = (a + b)2

      a2 – 2ab + b2 = (a – b)2

      a2 – b2 = (a + b) (a – b)

      Factorisation

      x2 + (a + b)x + ab = (x + a)(x + b)

      In expressions which have factors of the type (x + a) (x + b), remember the numerical term gives ab.

      Its factors, a and b, should be so chosen that their sum, with signs taken care of, is the coefficient of x.

      We know that in the case of numbers, division is the inverse of multiplication. This idea is applicable also to the division of algebraic expressions.

      In the case of division of a polynomial by a monomial, we may carry out the division either by dividing each term of the polynomial by the monomial or by the common factor method.

      In the case of division of a polynomial by a polynomial, we cannot proceed by dividing each term in the dividend polynomial by the divisor polynomial. Instead, we factorise both the polynomials and cancel their common factors.

      In the case of divisions of algebraic expressions that we studied in this chapter, we have Dividend = Divisor × Quotient.

      In general, however, the relation is Dividend = Divisor × Quotient + Remainder

      Thus, we have considered in the present chapter only those divisions in which the remainder is zero.

      There are many errors students commonly make when solving algebra exercises.

      You should avoid making such errors.

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      • Introduction to Graphs | Study

        Introduction to Graphs | Study

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        Graphical presentation of data is easier to understand.

        1. A bar graph is used to show comparison among categories.
        2. A pie graph is used to compare parts of a whole.
        3. A Histogram is a bar graph that shows data in intervals. (Scrol down to continue …)

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        Introduction to Graphs

        A line graph displays data that changes continuously over periods of time. A line graph which is a whole unbroken line is called a linear graph.

        For fixing a point on the graph sheet we need, x-coordinate and y-coordinate.

        The relation between dependent variable and

        through a graph.

        independent variable is shown

        A Bar Graph: A pictorial representation of numerical data in the form of bars (rectangles) of uniform width with equal spacing. The length (or height) of each bar

        represents the given number.

        A Pie Graph: A pie graph is used to compare parts of a whole. The various

        observations or components are represented by the sectors of the circle.

        A Histogram: Histogram is a type of bar diagram, where the class intervals are shown on the horizontal axis and the heights of the bars (rectangles) show the frequency of the class interval, but there is no gap between the bars as there is no gap between the

        class intervals.

        Linear Graph: A line graph in which all the line segments form a part of a single line. Coordinates: A point in Cartesian plane is represented by an ordered pair of numbers.

        Ordered Pair: A pair of numbers written in specified order.

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        • Exponents And Powers | Study

          Exponents And Powers | Study

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          Exponents and Powers | Speed Notes

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          Numbers with exponents obey the following laws of exponents. 

          Very small numbers can be expressed in standard form using negative exponents. (Scroll down to continue …)

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          Use of Exponents to Express Small Number in Standard form:

          1. Very large and very small numbers can be expressed in standard form.
          2. Standard form is also called scientific notation form.

          (iii) A number written as number such that is said to be in standard form if m is a decimal and n is either a positive or a negative integer.

          Examples: 150,000,000,000 = 1.5 x 1011.

          Exponential notation is a powerful way to express repeated multiplication of the same number.

          For any non-zero rational number ‘a’ and a natural number n, the product a x a x a x x a(n times) = an.

          It is known as the nth power of ‘a’ and is read as ‘a’ raised to the power n’.

          The rational number a is called the base and n is called exponent.

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          • Data Handling | Study

            Data Handling | Study

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            Data Handling | Speed Notes

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            Data Handling: Deals with the process of collecting data, presenting it and getting result.

            Data mostly available to us in an unorganised form is called raw data. (Scroll down to continue …)

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            Grouped data can be presented using histogram. Histogram is a type of bar diagram, where the class intervals are shown on the horizontal axis and the heights of the bars show the frequency of the class interval. Also, there is no gap between the bars as there is no gap between the class intervals.

            In order to draw meaningful inferences from any data, we need to organise the data systematically.

            Frequency gives the number of times that a particular entry occurs.

            Raw data can be ‘grouped’ and presented systematically through ‘grouped frequency distribution’.

            Statistics: The science which deals with the collection, presentation, analysis and interpretation of numerical data.

            Observation: Each entry (number) in raw data.

            Range: The difference between the lowest and the highest observation in a given data.

            Array: Arranging raw data in ascending or descending order of magnitude. Data can also presented using circle graph or pie chart. A circle graph shows the relationship between a whole and its part.

            There are certain experiments whose outcomes have an equal chance of occurring. A random experiment is one whose outcome cannot be predicted exactly in advance. Outcomes of an experiment are equally likely if each has the same chance of occurring.

            Frequency: The number of times a particular observation occurs in the given data.

            Class Interval: A group in which the raw data is condensed.

            (i) Continuous: The upper limit of a class interval coincides with the lower limit of the next class.

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            • Squares And Square Roots | Study

              Squares And Square Roots | Study

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              Squares and Square Roots | Speed Notes

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              Square: Number obtained when a number is multiplied by itself. It is the number raised to the power 2. 22 = 2 x 2=4(square of 2 is 4).

              If a natural number m can be expressed as n2, where n is also a natural number, then m is a square number. (Scroll down to continue …)

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              All square numbers end with 0, 1, 4, 5, 6 or 9 at unit’s place. Square numbers can only have even number of zeros at the end. Square root is the inverse operation of square.

              There are two integral square roots of a perfect square number.

              Positive square root of a number is denoted by the symbol For example, 32=9 gives

              Perfect Square or Square number: It is the square of some natural number. If m=n2, then m is a perfect square number where m and n are natural numbers. Example: 1=1 x 1=12, 4=2 x 2=22.

              Properties of Square number:

              1. A number ending in 2, 3, 7 or 8 is never a perfect square. Example: 152, 1028, 6593 etc.
              2. A number ending in 0, 1, 4, 5, 6 or 9 may not necessarily be a square number. Example: 20, 31, 24, etc.
              3. Square of even numbers are even. Example: 22 = 4, 42=16 etc.
              4. Square of odd numbers are odd. Example: 52 = 25, 92 = 81, etc.
              5. A number ending in an odd number of zeroes cannot be a perferct square. Example: 10, 1000, 900000, etc.
              6. The difference of squares of two consecutive natural number is equal to their sum. (n + 1)2– n2 = n+1+n. Example: 42 – 32 =4 + 3=7. 122– 112 =12+11 =23, etc.
              7. A triplet (m, n, p) of three natural numbers m, n and p is called Pythagorean

              triplet, if m2 + n2 = p2: 32 + 42 = 25 = 52

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              • Cubes And Cube Roots | Study

                Cubes And Cube Roots | Study

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                Cubes and Cube Root | Speed Notes

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                Cube number: Number obtained when a number is multiplied by itself three times. 23 = 2 x 2 x 2 = 8, 33 = 3 x 3 x 3=27, etc.

                Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They

                can be expressed as sum of two cubes in two different ways.

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                Numbers obtained when a number is multiplied by itself three times are known as cube numbers. For example 1, 8, 27, … etc.

                If in the prime factorisation of any number each factor appears three times, then the

                number is a perfect cube.

                The symbol

                denotes cube root. For example

                Perfect Cube: A natural number is said to be a perfect cube if it is the cube of some natural number. Example: 8 is perfect cube, because there is a natural number 2 such that 8 = 23, but 18 is not a perfect cube, because there is no natural number whose cube is 18.

                The cube of a negative number is always negative.

                Properties of Cube of Number:

                1. Cubes of even number are even.
                2. Cubes of odd numbers are odd.
                3. The sum of the cubes of first n natural numbers is equal to the square of their sum.
                4. Cubes of the numbers ending with the digits 0, 1, 4, 5, 6 and 9 end with digits 0, 1, 4, 5, 6 and 9 respectively.
                5. Cube of the number ending in 2 ends in 8 and cube of the number ending in 8 ends in 2.
                6. Cube of the number ending in 3 ends in 7 and cube of the number ending in 7

                ends in 3.

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                • Comparing Quantities | Study

                  Comparing Quantities | Study

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                  Comparing Quantities | Speed Notes

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                  Ratio: Comparing by division is called ratio. Quantities written in ratio have the sameunit. Ratio has no unit. (Scroll down to continue …)

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                  Equality of two ratios is called proportion. Product of extremes = Product of means
                  Percentage: Percentage means for every hundred. The result of any division in
                  whichthe divisor is 100 is a percentage. The divisor is denoted by a special
                  symbol %, read as percent. Profit and Loss:
                  (i) Cost Price (CP): The amount for which an article is bought. (ii) Selling Price (SP): The amount for which an article is sold. Additional expenses made after buying an article are included in the cost price
                  and are known as overhead expenses. These may include expenses like amount
                  spent onrepairs, labour charges, transportation, etc. Discount is a reduction given on marked price. Discount = Marked Price – Sale
                  Price. Discount can be calculated when discount percentage is given. Discount

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                  • Algebraic Expressions And Identities | Study

                    Algebraic Expressions And Identities | Study

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                    Algebraic Expressions and Identities | Speed Notes

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                    Expressions are formed from variables and constants.

                    Constant: A symbol having a fixed numerical value.

                    Example: 2,, 2.1, etc. (Scroll down to continue …)

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                    Variable: A symbol which takes various numerical values. Example: x, y, z, etc.

                    Algebric Expression: A combination of constants and variables connected by the sign

                    +, -, and is called algebraic expression.

                    Terms are added to form expressions.

                    Terms themselves are formed as product of factors.

                    Expressions that contain exactly one, two and three terms are called monomials, binomials and trinomials respectively.

                    In general, any expression containing one or more terms with non-zero coefficients (and with variables having non- negative exponents) is called a polynomial.

                    Like terms are formed from the same variables and the powers of these variables are the same, too.

                    Coefficients of like terms need not be the same.

                    While adding (or subtracting) polynomials, first look for like terms and add (or subtract) them; then handle the unlike terms.

                    There are number of situations in which we need to multiply algebraic expressions: for example, in finding area of a rectangle, the sides of which are given as expressions.

                    Monomial: An expression containing only one term. Example: -3, 4x, 3xy, etc.

                    Binomial: An expression containing two terms. Example: 2x-3, 4x+3y, xy-4, etc.,

                    Polynomial: In general, any expression containing one or more terms with non-zero coefficients (and with variables having non-negative exponents).

                    A polynomial may contain any number of terms, one or more than one.

                    A monomial multiplied by a monomial always gives a monomial.

                    Multiplication of a Polynomial and a monomial:

                    While multiplying a polynomial by a monomial, we multiply every term in the polynomial by the mononomial.

                    Trinomial: An expression containing three terms.

                    Example:

                    3x+2y+5z, etc.

                    In carrying out the multiplication of a polynomial by a binomial (or trinomial), we multiply term by term, i.e., every term of the polynomial is multiplied by every term in the binomial (or trinomial).

                    Note that in such multiplication, we may get terms in the product which are like and have to be combined.

                    An identity is an equality, which is true for all values of the variables in the equality.

                    On the other hand, an equation is true only for certain values of its variables.

                    An equation is not an identity.

                    The following are the standard identities:

                    (a + b)2 = a2 + 2ab + b2

                    (a – b)2 = a2 – 2ab +b2 

                    (a + b)(a – b) = a2 –  b2

                    (x + a) (x + b) = x2 + (a + b) x + ab

                    The above four identities are useful in carrying out squares and products of algebraic expressions.

                    They also allow easy alternative methods to calculate products of numbers and so on.

                    Coefficients: In the term of an expression any of the factors with the sign of the term is called the coefficient of the product of the other factors.

                    Terms: Various parts of an algebraic expression which are separated by + and – signs. Example: The expression 4x + 5 has two terms 4x and 5.

                    1. Constant Term: A term of expression having no lateral factor.
                    2. Like term: The term having the same literal factors. Example 2xy and -4xy are like terms.

                    (iii) Unlike term: The terms having different literal factors.

                    Example:

                    are unlike terms.

                    and 3xy

                    Factors: Each term in an algebraic expression is a product of one or more number (s) and/or literals. These number (s) and/or literal (s) are known as the factor of that term. A constant factor is called numerical factor, while a variable factor is known as

                    a literal factor. The term 4x is the product of its factors 4 and x.

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                    • Linear Equations In One Variable | Study

                      Linear Equations In One Variable | Study

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                      A statement of equality of two algebraic expressions involving one or more variables. Example: x + 2 = 3

                      Linear Equation in One variable: The expressions which form the equation that contain single variable and the highest power of the variable in the equation is one. (Scroll down to continue …)

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                      Linear Equations in One Variable

                      An algebraic equation is an equality involving variables. It says that the value of the expression on one side of the equality sign is equal to the value of the expression on the other side.

                      The equations we study in Classes VI, VII and VIII are linear equations in one variable. In such equations, the expressions which form the equation contain only one variable. Further, the equations are linear, i.e., the highest power of the variable appearing in the equation is 1.

                      A linear equation may have for its solution any rational number.

                      An equation may have linear expressions on both sides. Equations that we studied in Classes VI and VII had just a number on one side of the equation.

                      Just as numbers, variables can, also, be transposed from one side of the equation to the other.

                      Occasionally, the expressions forming equations have to be simplified before we can solve them by usual methods. Some equations may not even be linear to begin with, but they can be brought to a linear form by multiplying both sides of the equation by a suitable expression.

                      The utility of linear equations is in their diverse applications; different problems on numbers, ages, perimeters, combination of currency notes, and so on can be solved

                      using linear equations.

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                      • Rational Numbers | Study

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                        Rational numbers are closed under the operations of addition, subtraction and multiplication, But not in division. (Scroll down to continue …)

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                        The operations addition and multiplication are
                        (i) commutative for rational numbers.

                        (ii) associative for rational numbers.

                        The rational number 0 is the additive identity for rational numbers.

                        The additive inverse of the rational number a/b is -a/b and vice- versa.

                        The reciprocal or multiplicative inverse of the rational number
                        is if a/b is c/d if (a/b)(c/d) =1

                        Distributive property of rational numbers:

                        For all rational numbers a, b and c, a(b + c) = ab + ac
                        and a(b – c) = ab – ac.

                        Rational numbers can be represented on a number line.

                        Between any two given rational numbers there are countless rational numbers.

                        The idea of mean helps us to find rational numbers between two rational numbers.

                        Positive Rationals: Numerator and Denominator both are either positive or negative.

                        Example: 2/3, -4/-5

                        Positive Rationals: Numerator and Denominator both are either positive or negative.

                        Example: -2/3, 4/-5

                        .

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