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Tag: CBSE 8
Comparing Quantities | Study
Algebraic Expressions And Identities | Study
Rational Numbers | Study
Linear Equations In One Variable | Study
Class 8 | Mathematics | All In One
Comparing Quantities | Study
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English Version 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. DiscountHindi Version Dig Deep
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Algebraic Expressions And Identities | Study
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English Version 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.
- Constant Term: A term of expression having no lateral factor.
- 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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Rational Numbers | Study
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English Version Rational Numbers | Speed Notes
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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) =1Distributive 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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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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