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Fraction and Decimals
Fraction
Fraction is a method for representing the parts of a whole number. An orange is divided into two equal parts and so the first part of orange is half of the whole orange and represented by \[\frac{1}{2}\] of the orange.
Types of Fractions
Proper Fractions
A fraction whose numerator is less than denominator is called a proper fraction.
\[\frac{3}{5}\]’ \[\frac{1}{2}\]’ \[\frac{7}{9}\] are Proper fractions.
Improper Fractions
A fraction is called improper fraction even if:
LCM and HCF
LCM (Least Common Multiple)
LCM of two or more numbers is their least common multiple. LCM of 4 and 6 is 12, it means, 12 is the least common multiple of 4 and 6, therefore, 12 is exactly divisible by each of 4 and 6.
LCM by Prime Factorization Method
The following steps are used to determine the LCM of two or more numbers by prime factorisation method:
Step 1: Find the prime factors of each number.
Step 2: Product of highest power of prime factors is their LCM.
LCM by Division Method
The following steps are used to determine the LCM of two or more numbers by division method:
Step 1: Numbers are arranged or separated in a row by commas.
Step 2: Find the number which divides exactly atleast two of the given numbers.
Step 3: Follow step 2 till there are no numbers (atleast two) divisible by any number.
Step 4: LCM is the product of all divisors and indivisible numbers.
Example:
Find the least number which is exactly divisible by each of 28 and 42.
(a) 64 (b) 84
(c) 52 (d) All of these
(e) None of these
Answer (b)
Explanation: \[28=2\times 2\times 7,\,\,42=2\times 3\times 7\]
LCM =\[2\times 2\times 3\times 7=84\]
HCF (Highest Common Factor)
Highest Common Factor is also called Greatest Common Measure (GCM) or Greatest Common Divisor (GCD). H.C.F of two or more numbers is the greatest number which exactly divides each of the numbers.
HCF by Prime Factorization Method
The HCF of two or more numbers is obtained by the following steps:
Step 1: Find the prime factors of each of the given number.
Step 2: Find the common prime factors from prime factors of all the given numbers.
Step 3: The product of the common prime factors is their HCF.
HCF by Continued Division Method
The HCF of two or more numbers can also be obtained by continued division method. The greatest number is considered as dividend and smallest number as divisor.
Follow the following steps to perform the HCF of the given numbers:
Step 1: Divide the greatest number by smallest.
Step 2: If remainder is zero, then divisor is the HCF of the given number.
Step 3: If remainder is not zero then, divide again by considering divisor as new dividend and remainder as new divisor till remainder becomes zero.
Step 4: The HCF of the numbers is last divisor which gives zero remainder.
HCF of more than two Numbers
The HCF of more than two numbers is the HCF of resulting HCF of two numbers with third number. Therefore, HCF of more than two numbers is obtained by finding the HCF of two numbers with third, fourth and so on.
HCF of Larger Numbers
The HCF of smaller number (One or two digit numbers) is simply obtained by division but division of larger numbers take more more...
Ratio and Proportion
Ratio
Ratio of two quantities is the comparison of the given quantities. Ratio is widely used for comparison of two quantities in such a way that one quantity is how much increased or decreased by the other quantity.
For example, Peter has 20 litres of milk but John has 5 litres, the comparison of the quantities is said to be, Peter has 15 litres more milk than John, but by division of both the quantity, it is said that Peter has, \[\frac{20}{5}=4\] times of milk than John. It can be expressed in the ratio form as\[4:1\]
Note: In the ratio\[a:b\]\[(b\ne 0)\], the quantities a and b are called the terms of the ratio and the first term (ie. a) is called antecedent and the second term (ie. b) is called consequent.
Simplest form of a Ratio
If the common factor of antecedent and consequent in a ratio is 1 then it is called in its simplest form.
Comparison of Ratio
Comparison of the given ratios are compared by first converting them into like fractions, for example to compare \[5:6,\text{ }8:13\text{ }and\text{ }9:16\]first convert them into the fractional form
i.e. \[\frac{5}{6},\frac{8}{13},\frac{9}{16}\].
The LCM of denominators of the fractions\[=2\times 3\times 13\times 8=624\]
Now, make denominators of every fraction to 624 by multiplying with the same number to both numerator and denominator of each fraction.
Hence,\[\frac{5}{6}\times \frac{104}{104}=\]\[\frac{520}{624},\frac{8}{13}\times \frac{48}{48}\]\[=\frac{384}{624}\]and\[\frac{9}{16}\times \frac{39}{39}\]\[=\frac{351}{624}\].Equivalent fractions of the given fractions are \[\frac{520}{624},\frac{384}{624},\frac{351}{624}\]. We know that the greater fraction has greater numerator, therefore the ascending order of the fractions are, \[\frac{351}{624}<\frac{384}{624}<\frac{520}{624}\] or \[\frac{9}{16}<\frac{8}{13}<\frac{5}{6}\] or \[9:16<8:13<5:6\], thus the smallest ratio among the given ratio is \[9:16\]and greatest ratio is\[5:6\].
Equivalent Ratio
The equivalent ratio of a given ratio is obtained by multiplying or dividing the antecedent and consequent of the ratio by the same number. The equivalent ratio of \[\text{a}\,\,\text{:}\,\,\text{b}\] is \[\text{a}\,\times \,\text{q}\,\,\text{:}\,\,\text{b}\,\times \,\text{q}\]whereas, a, b, q are natural numbers and q is greater than 1.
Hence, the equivalent ratios of \[5:8\]are, \[\frac{5}{8}\times \frac{2}{2}=\frac{10}{16}\] or\[10:16\], \[\frac{5}{8}\times \frac{3}{3}=\frac{15}{24}\] or\[15:24\], \[\frac{5}{8}\times \frac{12}{12}=\frac{60}{96}\] or\[60:96\].
Example: Mapped distance between two points on a map is 9 cm. Find the ratio of actual as well as mapped distance if 1 cm = 100 m.
(a) \[10000:1\] (b) \[375:1\]
(c) \[23:56\] (d) \[200:1\]
(e) None of these
Answer (a)
Explanation: Required ratio =\[900\times 100:9=\]
\[90000:9=10000:1\]
Example: Consumption of milk in a day is 6 litre. Find the ratio of Consumption of milk in month of April and quantity of milk in a day?
(a) \[99:2\] (b) \[30:1\]
(c) \[123:3\] (d) \[47:3\]
(e) None of these
Answer (b)
Explanation: Required ratio \[=30\times 6:6=30:1\]
Proportion
The equality of two ratios is called proportion. If a cake is distributed among eight boys and each boy gets equal part of the cake then cake is said to be distributed in proportion. The simplest form of ratio \[12:96\]is \[1:8\]and\[19:152\]is \[1:8\]therefore, \[12:96\]and \[19:152\] are in proportion and written as \[12:96::19:152\]or more...
Geometry and Symmetry
Basic Geometrical Shapes
Lines and angles are the main geometrical concept and every geometrical figure is made up of lines and angles. Triangles are also constructed by using lines and angles.
Point
A geometrical figure which indicates position but not the dimension is called a point. A point does not have length, breadth and height. A point is a fine dot. P is a point on a plane of paper as shown below.
Line
A set of points which can be extended infinitely in both directions is called a line.
Line Segment
A line of fix length is called a line segment.
In the above figure RS is a line segment and the length of RS is fixed.
Ray
A ray is defined as the line that can be extended infinitely in one direction.
In the above figure AB can be extended towards the direction of B. Hence, called a ray.
Note: A line segment has two end points, a ray has only one end point and a line has no end points.
Angle
Angle is formed between two rays which have a common point.
Vertex or common end point is O.
OA and OB are the arms of \[\angle AOB\]
The name of the above angle can be given as \[\angle AOB\]or \[\angle BOA\]
The unit of measurement of an angle is degree\[({}^\circ )\]
Types of Angles
Acute Angle
The angle between \[0{}^\circ \] and \[90{}^\circ \] is called an acute angle.
For example, \[10{}^\circ ,\,\,30{}^\circ ,\,\,60{}^\circ ,\,\,80{}^\circ \]are acute angles.
Right Angle
An angle of measure \[90{}^\circ \] is called a right angle.
Obtuse Angle
An angle whose measure is between \[90{}^\circ \] and \[180{}^\circ \] is called an obtuse angle.
Straight Angle
An angle whose measure is \[180{}^\circ \,\,\text{is}\]called a straight angle.
Reflex Angle
An angle whose measure is more than \[180{}^\circ \] and less than \[360{}^\circ \]is called a reflex angle.
Complementary Angle
Two angles whose sum is \[90{}^\circ \,\,\text{is}\] called the complimentary angle.
Complementary angle of any angle \[\theta \] is\[90{}^\circ \theta \].
Supplementary Angle
Two angles whose sum is \[180{}^\circ \,\,\text{is}\] called supplementary angles.
Supplementary angle of any angle \[\theta \] is\[180{}^\circ \theta \].
Adjacent Angle
Two angles are said to be adjacent if they have a common vertex and one common arm. In the following figure \[\angle AOC\] and \[\angle COB\]are adjacent angles.
Vertically more...
Mensuration
Perimeter and Area of Plane Figures
Perimeter of geometrical figure is the sum of its sides. There are different types of geometrical figures. Figures are classified by their shapes and sizes. Area of a geometrical figure is its total surface area.
Perimeter and Area of a Triangle
Data Handling
In this chapter we will learn about pictograph and bar graph.
Data
Data is a collection of facts, such as numbers, observations, words or even description things.
Observation
Each numerical figure in a data is called observation.
Frequency
The number of times a particular observation occurs is called its frequency.
Statistical Graph
The information provided by a numerical frequency distribution is easy to understand when we represent it in terms of diagrams or graphs.
To represent statistical data, we use different types of diagrams or graphs. Some of them are:
(i) Pictograph
(ii) Bar graph
Pictograph
A pictograph represents the given data through pictures of objects. It helps to answer the questions on the data at a glance.
Example:
The following pictograph, shows the number of cakes sold at a bakery over five days.
