Mahesh

Students can go through AP Board 6th Class Maths Notes Chapter 5 Fractions and Decimals to understand and remember the concepts easily.

AP State Board Syllabus 6th Class Maths Notes Chapter 5 Fractions and Decimals

→ Fraction: A fraction is a numerical representation of apart of a whole. The whole may be a single object or a group of objects.
Eg: \(\frac{4}{7}\).
The fraction \(\frac{4}{7}\) represents four out of seven.
In the fraction \(\frac{4}{7}\), 4 is called the numerator and 7 is called the denominator.

→ Proper fraction: A fraction in which the numerator is less than its denominator is called a proper fraction.
Eg: \(\frac{1}{7}\), \(\frac{2}{5}\), \(\frac{3}{11}\), …..
All proper fractions are less than 1.

→ Improper fraction: A fraction in which the numerator is greater than its denominator is called an improper fraction.
Eg: \(\frac{5}{11}\), \(\frac{4}{7}\), \(\frac{2}{3}\),….
All improper fractions are greater than or equal to 1.

→ Mixed fraction: A mixed fraction is a combination of a whole number and a proper fraction.
Fraction in lowest terms: A fraction is said to be in its lowest terms if the numerator and the denominator have no factors in common other than 1.
Eg: \(\frac{2}{11}\), \(\frac{3}{7}\), \(\frac{18}{25}\),……
Equivalent fractions: Two fractions are said to be equivalent if they have same numerators and same denominators respectively when expressed in their lowest terms.
Eg : \(\frac{2}{5}\) & \(\frac{8}{20}\)
Equivalent fractions have the same value.

→ Like fractions:
Fractions with the same denominators are called like fractions
Eg: \(\frac{3}{13}\), \(\frac{4}{13}\), \(\frac{7}{13}\), \(\frac{21}{13}\), ….

→ Un-like fractions:
Fractions with different denominators are called like fractions.
Eg: \(\frac{7}{11}\), \(\frac{3}{5}\), \(\frac{9}{17}\), ….

→ Comparison of fractions:

• Out of two fractions with the same denominators (like fractions), the fraction with the smallest numerator is smaller than the other.
• Similarly out of two fractions with the same denominators (like fractions), the fraction with the greatest numerator is greater than the other.
  Eg: \(\frac{2}{11}\) < \(\frac{5}{11}\) \(\frac{9}{17}\) > \(\frac{4}{17}\)
• Out of two given fractions with the same numerator, the fraction with smaller denominator is greater than the other.
• Similarly out of two given fractions with the same numerator, the fraction with greater denominator is smaller than the other.
  Eg: \(\frac{11}{2}\) > \(\frac{11}{5}\) \(\frac{13}{8}\) < \(\frac{13}{11}\)
• To compare unlike fractions, convert them in to like fractions with L.C.M. as the same denominators, and then compare the like fractions.
  Eg: \(\frac{2}{3}\) and \(\frac{4}{5}\).
  LCM of 2, 5 is 15
  \(\frac{2}{3}\) = \(\frac{10}{15}\) \(\frac{4}{5}\) = \(\frac{12}{15}\)
  Now \(\frac{10}{15}\) < \(\frac{12}{15}\) there by \(\frac{2}{3}\) < \(\frac{4}{5}\)

→ Addition and subtraction of like fractions:

• To add like fractions we add their numerators while retaining the common denominator. Eg: \(\frac{5}{7}\) + \(\frac{2}{7}\) = 5 + \(\frac{2}{7}\) = \(\frac{7}{7}\)
• To subtract like fractions we subtract their numerators while retaining the common denominator.
  Eg: \(\frac{6}{13}\) – \(\frac{2}{13}\) = 6 – \(\frac{2}{13}\) = \(\frac{4}{13}\)

→ Addition and subtraction of un-like fractions:

• Convert the given unlike fractions in to like fractions, (denominator = LCM of given denominators)
• Now add or subtract as we do in case of like fractions.
• To multiply a fraction with a whole number, first multiply the numerator of the fraction by the whole number while keeping the denominator the same.
  Eg: \(\frac{3}{4}\) × 5 = 3 × \(\frac{5}{4}\) = \(\frac{15}{4}\)
  8 × \(\frac{2}{3}\) = 8 × \(\frac{2}{3}\) = \(\frac{16}{3}\)

→ Multiplication of two fractions = product of numerators/product of denominators
Eg: \(\frac{5}{6}\) × \(\frac{4}{9}\) = 5 × \(\frac{4}{6}\) × 9 = \(\frac{20}{54}\)

• The product of any two proper fractions is always less than each of its fraction.
  Eg: \(\frac{1}{5}\) × \(\frac{2}{7}\) = \(\frac{2}{35}\), \(\frac{2}{35}\) < \(\frac{1}{5}\) & \(\frac{2}{35}\) < \(\frac{2}{7}\)
• The product of any two improper fractions is always greater than each of its fraction.
  Eg: \(\frac{7}{3}\) × \(\frac{5}{2}\) = \(\frac{35}{6}\) \(\frac{7}{3}\) < \(\frac{35}{6}\) \(\frac{5}{2}\) < \(\frac{35}{6}\)
• The product of a proper fraction and an improper fraction is always greater than its proper fraction and less than its improper fraction.
  Eg: \(\frac{2}{5}\) × \(\frac{7}{4}\) = \(\frac{14}{20}\) \(\frac{2}{5}\) < \(\frac{14}{20}\) < \(\frac{7}{4}\)

→ Reciprocal of a fraction: A fraction obtained by interchanging the numerator and the denominator of a given fraction is called its reciprocal fraction.
Eg: Reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\)
\(\frac{1}{a}\) of b means \(\frac{1}{a}\) × b = \(\frac{b}{a}\)

→ Division of a whole number by a fraction: To divide a whole number by a fraction we multiply the given whole number by the reciprocal of the given fraction.
Eg: 5 ÷ \(\frac{3}{4}\) = 5 × \(\frac{20}{5}\) = \(\frac{4}{5}\)

• Any two non-zero numbers whose product is equal to 1 are called reciprocals to each other.
  Eg: \(\frac{3}{7}\) and \(\frac{7}{3}\) are reciprocals to each other.
• To divide a whole number by a mixed fraction, first convert the mixed fraction into improper fraction and then multiply the whole number with the reciprocal of the improper fraction.
  Eg: 7 ÷ 3\(\frac{2}{5}\) = 7 ÷ \(\frac{17}{5}\) = 7 × \(\frac{5}{17}\) = \(\frac{35}{17}\)

→ Division of a fraction by a whole number:

• To divide a fraction by a whole number we multiply the given fraction by the reciprocal of the given whole number.
  Eg: \(\frac{5}{4}\) ÷ 3 = \(\frac{5}{4}\) × \(\frac{1}{3}\) = \(\frac{5}{12}\)
• To divide a mixed fraction by a whole number, first convert the mixed fraction into an improper fraction and then multiply the improper fraction by the reciprocal of whole number.
  Eg : 4\(\frac{3}{4}\) ÷ 8 = \(\frac{19}{4}\) × \(\frac{1}{8}\) = \(\frac{19}{32}\)

→ Division of a fraction by another fraction: To divide a fraction by another fraction, we multiply the first fraction with the reciprocal of the second fraction.
E.g: \(\frac{3}{5}\) ÷ \(\frac{7}{11}\) = \(\frac{3}{5}\) × \(\frac{11}{7}\) = \(\frac{33}{35}\)

→ Decimal numbers or Decimal fractions: A decimal is a way of expressing a fraction.
The fraction \(\frac{1}{10}\) is written as 0.1 in decimal form.

Examples for decimal numbers: 0.7, 0.4, 0.23, ..etc The decimal number 0.7 is read as zero point seven.
The decimal number 5.8 is read as five point eight.
The dot or the point between the two digits is called the decimal point.
The number of digits after decimal point is called the number of decimal places. Decimal places of 5.247 is 3.
The decimal point separates a decimal number into two parts. The number on its left as integer part and the digits on its right as decimal part.
The decimal part of a decimal number is always less than 1. As we move from left to right each decimal place decreases by tenth of its previous value.
The decimal places after the decimal point are (\(\frac{1}{10}\)-tenths), (\(\frac{1}{100}\)-hundreths), (\(\frac{1}{1000}\)-thousandths) and so on.

These are also called the place values of the decimal part.
If we divide a whole number into ten equal parts each part of the whole represents tenths part. \(\frac{1}{10}\)
If we divide a whole number into hundred equal parts each part of the whole represents hundredths part. \(\frac{1}{100}\)
If we divide a whole number into thousand equal parts each part of the whole represents thousandths part. \(\frac{1}{1000}\)

→ Converting fractions into decimals and vice versa:
Fractions with denominators 10, 100, 1000 can be easily converted into decimals by placing decimal point in the numerator accordingly.

• If the denominator is 10, then we place the decimal point in the numerator after one digit from RHS. The number of decimal places is equal to 1.
  Eg: \(\frac{256}{10}\) = 25.6
• If the denominator is 100, then we place the decimal point in the numerator after two digits from RHS. The number of decimal places is equal to two.
  Eg: \(\frac{256}{100}\) = 2.56
• If the denominator is 1000, then we place the decimal point in the numerator after three digits from RHS, The number of decimal places is equal to three.
  Eg: \(\frac{256}{1000}\) = 0.256 and read as zero point two five six.

→ Decimals Can Also be converted into Conversion of simple fractions into decimal fractions:
To convert simple fractions into decimal numbers:
To convert simple fractions into decimal numbers first convert the denominators to 10/100/1000 accordingly and then place the decimal point in the numerator as required.
Eg: \(\frac{23}{2}\) = 23 × \(\frac{5}{10}\) = \(\frac{115}{10}\) = 11.5
\(\frac{7}{4}\) = 7 × \(\frac{25}{100}\) = \(\frac{175}{100}\) = 1.75
\(\frac{3}{5}\) = 3 × \(\frac{2}{10}\) = \(\frac{6}{10}\) = 0.6
Writing zeroes at the end of a decimal number does not change its value.
Eg: 5.2 = 5.20 = 5.200 = 5.2000 and so on
Like and unlike decimal fractions:
Decimals having the same number of decimal places are called like decimals.
Eg: 3.2,5.6,4.8.
Decimals having the different number of decimal places are called unlike decimals. Eg : 5.23, 8.3, 4.214
Unlike decimals can be converted into like decimals by converting them into equivalent decimals.
Eg: 2.7 & 6.54
2.7= 2.70 and now-2.70 and 6.54 are like decimals.

→ Comparing and ordering of decimals:
To compare the given decimals
a) First convert them to like decimals.
b) Now compare the integer / whole number part. The number with greater whole part is greater than the other.
c) If the whole / integer parts are equal, then compare the tenths digits. The number with greater tenths digit is greater than the other.
d) If the tenths digits are also equal, then compare the hundredths digits. The number with greater hundredths digit is greater than the other.
e) If the hundredths digits are also equal, then compare the thousandths digits. The number with greater thousandths digit is greater than the other.
Eg : 54.235 and 54.238
54.235 < 54.238

→ Addition and subtraction of decimal fractions:
To add or subtract the given decimal fractions first convert them into like decimals. Now add / subtract the digits in the corresponding place values.

→ Uses of decimals: Decimal fractions are used in expressing money, distance, weight and capacity.

Students can go through AP Board 7th Class Maths Notes Chapter 11 Exponents to understand and remember the concepts easily.

AP State Board Syllabus 7th Class Maths Notes Chapter 11 Exponents

→ Very large numbers are easier to read, write and understand when expressed in exponential form.
Eg : 10000 = 10⁴
8 × 8 × 8 × 8 × 8 × ….. × 8 (16 times) = 8¹⁶.

→ When a number is multiplied by itself for many number of times (repeated multiplication) then we write it in exponential form.
Eg : 2 × 2 × 2 × 2 = 2⁴ Here 2 is base 4 is exponent.
3 × 3 × 3 × 3 × 3 = 3⁵ Here 3 is base 5 is exponent.

→ a . a . a . a ….. a (m times) = a^m.

→ Here ‘a’ is called the base and ‘m’ is called the exponent.

→ Laws of exponents

i) a^m × aⁿ = a^m+n

ii)  (a^m)ⁿ = a^mn

iii) (ab)^m = a^m . b^m

iv) a^m = aⁿ ⇒ m = n

v) a^-n = \(\frac{1}{a^{n}}\)

vi) \(\frac{\mathrm{a}^{\mathrm{m}}}{\mathrm{a}^{\mathrm{n}}}\) = a^m-n  if (m > n)
= \(\frac{1}{a^{n-m}}\)  if (m < n)
= 1 if (m = n)

vii) \(\left(\frac{a}{b}\right)^{m}\) = \(\frac{a^{m}}{b^{m}}\)

viii) a⁰ = 1 where a ≠ 0

Students can go through AP Board 7th Class Maths Notes Chapter 10 Algebraic Expressions to understand and remember the concepts easily.

AP State Board Syllabus 7th Class Maths Notes Chapter 10 Algebraic Expressions

→ Variable: It takes different values.
Ex: x, y, z, a, b, c, m etc.
Constant: The value of constant is fixed.
Ex: 1, 2, \(\frac{-2}{3}\), \(\frac{4}{5}\) etc.

→ Algebraic Expression: An algebraic expression is a single term or a combination of terms connected by the symbols ‘+’ (plus) or(minus).
Ex: 2x + 3, \(\frac{2}{5}\)p, 3x – 1 etc.

→ Numerical Expression : If every term of an expression is a constant term, then the expression is called a Numerical expression.
Ex: 2 + 1, -5 × 3, (12 + 4) ÷ 3.
Note: In the expression 2x + 9, ‘2x’ is an algebraic term. ‘9’ is called numeric term.

→ Like terms are terms which contain the same variables with the same exponents.
Ex: 12x, 25x, -7x are like terms.
2xy², 3xy², 7xy² are like terms.

→ Coefficient: In a.xⁿ, ‘a’ is called the numerical coefficient and ‘x’ is called the literal coefficient.
Types of algebraic expressions.

→ Degree of a monomial: The sum of all exponents of the variables present in a monomial is called the degree of the term or degree of the monomial.
Ex: The degree of 9x²y² is 4 [∵ 2 + 2 = 4]
Note: Degree of constant term is zero.
The highest of the degrees of all the terms of an expression is called the degree of the expression.
Ex: The degree of the expression ax + bx² + cx³ + dx⁴ + ex⁵ is 5.

→ The difference between two like terms is a like term with a numerical coefficient equal to the difference between the numerical coefficients of the two like terms.
Note:

1. If no two terms of an expression are alike then it is said to be in the simplified form.
2. In an expression, if the terms are arranged in such a way that the degrees of the terms are in descending order then the expression is said to be in standard form.
3. Addition (or) subtraction of expressions should be done in two methods, they are
   i) Column or Vertical method.
   ii) Row or Horizontal method.

Students can go through AP Board 7th Class Maths Notes Chapter 9 Construction of Triangles to understand and remember the concepts easily.

AP State Board Syllabus 7th Class Maths Notes Chapter 9 Construction of Triangles

→ A triangle can be drawn of any three of its elements are known.

→ To construct a triangle, we need

1. Three sides
2. Any two sides and the angle included between them.
3. Two angles and the side included between them.
4. Hypotenuse and one adjacent side of a right angled triangle.
5. Two sides and a non-included angle.

Students can go through AP Board 7th Class Maths Notes Chapter 8 Congruency of Triangles to understand and remember the concepts easily.

AP State Board Syllabus 7th Class Maths Notes Chapter 8 Congruency of Triangles

→ Two figures are said to be identical if their shapes are same.
Eg: Any two squares, circles or equilateral triangles.

→ Two figures are said to be congruent if they are identical in shape and equal in size.

→ Two line segments are congruent if they have same lengths.

→ Two triangles are congruent if the corresponding angles are equal.

→ We establish the congruency of the triangles by following criteria.

→ S.S.S. criterion: If three sides of a triangle are equal to the corresponding three sides of another triangle, then the triangles are congruent.

FA = TI; AN = IN; FN = TN then △FAN ≅ △TIN

→ S.A.S. criterion: If two sides and the angle included between the two sides of a triangle are equal to the corresponding two sides and the included angle of another triangle, then the two triangles are congruent.

CA = PI; ∠C = ∠P; CT = PG
then, △CAT ≅ △PIG

→ A.S.A. criterion : If two angles and the included side of a triangle are equal to the corresponding two angles and included side of another triangle then the triangles are congruent.

∠A = ∠I; AT = IM; ∠T = ∠M then △MAT ≅ △DIM.

→ R.H.S. criterion: In two right angled triangles, if the hypotenuse and one corresponding side are equal then they are congruent.

∠B = ∠E = 90°
BC = EF
AC = DF
then △ABC ≅ △DEF

→ If by any criterion two triangles are congruent then all the corresponding parts are equal.

Students can go through AP Board 6th Class Maths Notes Chapter 4 Integers to understand and remember the concepts easily.

AP State Board Syllabus 6th Class Maths Notes Chapter 4 Integers

→ Positive numbers: All numbers which are greater than zero are called positive numbers {1, 2, 3, …..}

→ Negative numbers: In our real life we come across many situations where in we have to use numbers whose value is less than zero; such numbers are called negative numbers.
Example: Very low temperature, loss in a business, depth below a surface, etc. Negative numbers are represented by the minus symbol

→ Zero is neither a negative number nor a positive number.

→ Integers: The set of positive numbers, zero along with the set of negative numbers are called Integers. The set of integers is denoted by I or Z.
I = Z = {…. 4, -3, -2, -1, 0, 1, 2, 3,…..}

→ Historical Notes: Brahma Gupta (598 – 670 AD), Indian mathematician first used a special sign (-) for negative numbers and stated rules for operations on negative numbers.
The letter ”Z” was first used by the Germans because the word for Integers in the language is Zehlen which means NUMBER.

→ Representation of Integers on a number line:

On a number line all negative numbers lie on the left side of zero. All positive numbers lie on right side of zero.
A number line extends on either side endlessly.
All whole numbers are called non-negative integers.
The natural numbers are called positive integers.

On a number, of the given two numbers the number on LHS is always less than the number on the RHS.

On a number line as we move from left to right the value of numbers goes on increasing and vice versa.
A number line can be written in a vertical direction as given below.

Students can go through AP Board 6th Class Maths Notes Chapter 3 HCF and LCM to understand and remember the concepts easily.

AP State Board Syllabus 6th Class Maths Notes Chapter 3 HCF and LCM

→ Divisibility Rules:
A divisibility rule is a process by which we can determine whether a given number is completely divisible by other given number or not without performing actual division.
Reasons behind the rules:
Our number system is based on base 10 system. Every place value increases by 10-times as move from right to left.

→ Divisibility rule for 2: In the above place value table except ones place all other places namely 100/1000/10 000…. are completely divisible by 2. So for divisibility by 2 we need to check the unit digit only.
A number is divisible by 2 if it has any of the digits 0, 2, 4 or 8 in its units place.
In other words all even numbers are divisible by 2.

→ Divisibility rule for 3:
10/3 → not divisible
100/3 → not divisible
1000/3 → not divisible and it goes on..
But in all the cases, the remainder is 1. As such if the number 56817 is divided by 3 we get remainder, 5, 6, 8, 1 and 7 respectively. The sum of these remainders 5 + 6 + 8 + 1 + 7 = 27 is divisible by 3 as is the number is divisible by 3.
In other words if the sum of the digits of a given number is divisible by 3, then the given number is also divisible by 3.
The digital root of natural number is the single digit value obtained by repeated process of summing digits.

Example: The digital root of 325698 is 3 + 2 + 5 + 6 + 9 + 8 = 33 = 3 + 3 = 6
Note: While adding the digits of a number we can ignore 9’s or combinations of digits summing up to 9.

Example: The digital root of 87459634572 is
By dropping (4+5), (9), (6+3), (4+5), (7+2), the remaining digits are 8 & 7. From these digits eliminate 9 that is 8 + 7 – 9 = 15 – 9 = 6
Therefore the digital root of 87459634572 is 6. Hence divisible by 3.

Now add these digits to get the total remainder. If this remainder is completely divisible by 3, then the given number is also divisible by 3.

Divisibility rule of 3 will add the digits and then check if its divisible by 3. This is applicable for numbers which leaves remainder 1 when 10 is divided by that number.

→ Divisibility rule for 4: In the place value table starting from 100, all other higher places namely 1000/10 000/100 000, …etc. are all completely divisible by 4. So we need to check the digits in ten’s and unit’s place for divisibility by 4.

A number is divisible by 4 if the number formed by the digits in its ten’s place and unit’s place taken in the same order is divisible by 4 and also zeros on both places.

Example: Is the number 87534 divisible by 4?
Number formed by last two digits 34 is not divisible by 4 and hence the given number is also not divisible by 4.

Example : Is the number 779956 divisible by 4?
Number formed by last two digits 56 is divisible by 4 and hence the given number is also divisible by 4.

→ Divisibility rule for 5: In the place value table starting from 10, all other higher places namely 10/100/1000/10000/1 00 000, ..’etc. are all completely divisible by 5. So we need to check the digits in unit’s place for divisibility by 5.
A number is divisible by 5 if the number ends in either zero of 5.

Example: Is the number 779956 divisible 5?
The digit in unit’s place is 6, so it is not divisible by 5.

Example: Is the number 77995 divisible by 5?
The digit in unit’s place is 5, so it is divisible by 5.

Example: Is the number 779950 divisible by 5?
The digit in unit’s place is 0, so it is divisible by 5.

→ Divisibility rule for 6 : If a number is divisible by both 2 and 3, then it is also divisible by 6.
Example: Is the number 612432 divisible by 6?
As the given number is an even number it is divisible by 2.
Also the digital root of the number is 3, it is divisible by 6.
Hence it is divisible by 6.
In other words if a number is divisible by two relatively prime numbers, then their product also divides the given number.

→ Divisibility rule for 8: In the place value table starting from 1000, all other higher places namely 10 000/1 00 000…etc., are all completely divisible by 8. So we need to check the digits in hundred’s, ten’s and unit’s place for divisibility by 8.
A number is divisible by 8 if the number formed by the digits in its hundred’s place, ten’s place and unit’s place taken in the same order is divisible by 8, are also zeros on three places.
Example: Is the number 875344 divisible by 8?
Number formed by last three digits 344 is divisible by 8 and hence the given number is also divisible by 8.

→ Divisibility rule for 9: The rule is same as rule for 3
10/9 → not divisible
100/9 → not. divisible
1000/9 → not divisible and it goes on ..
But in all the cases, the remainder is 1. As such if the number 56817 is divided by 9 we get remainder, 5, 6, 8, 1 and 7 respectively. The sum of these remainders 5 + 6 + 8 + 1 + 7 = 27 is divisible by 9 as is the number is divisible by 3.

In other words if the sum of the digits of a given number is divisible by 9, then the given number is also divisible by 9.
The digital root of natural number is the single digit value obtained by repeated process of summing digits.

Example: The digital root of 325698 is 3 + 2 + 5 + 6 + 9 + 8 = 33 = 3 + 3 = 6
Note: While adding the digits of a number we can ignore 9’s or combinations of digits summing up to 9.

Example: Is the number 7854963 divisible by 9?
The digital root of the given number is 7 + 8 + 5 + 4 + 9 + 6 + 3 = 42 = 4 + 2 = 6, not divisible by 9.

→ Divisibility rule for 10: In the place value table starting from 10, all other higher places namely 10/100/1000/10000/100000…etc., are all completely divisible by 10. So we need to check the digits in unit’s place for divisibility by 10.
A number is divisible by 10 if the number ends in zero.
Example: Is the number 779956 divisible 10?
The digit in unit’s place is 6, so it is not divisible by 10.

Example: Is the number 779950 divisible 10?
The digit in unit’s place is 0, so it is divisible by 10.

→ Divisibility rule for 11: A number is divisible by 11, if the difference between the sum of digits at even places and the sum of digits at odd place is either zero or a multiple of 11.
Example: Is the number 52487 divisible 11?
Sum of the digits at odd places = 7 + 4 + 5 = 16 Sum of the digits at even places = 8 + 2 = 10 Difference = 16 – 10 = 6, not divisible by 11.
Note: If two numbers are divisible by a given number, then their sum, difference and the product are also divisible by that number.

→ Factor: A factor of a number is an exact divisor of that number.
Example: 15 = 5 × 3, here 5 divides 15 completely and 3 divides 15 completely. As such 1, 3, 5, 15 are factors of 15.
Also 15 = 1 × 15. It means 1 is a factor of every number and every number is a factor of itself.
Every factor of a number is less than or equal to the number.
Perfect number: A number for which the sum of all its factors is equal to twice the number is called a perfect number.
Example: 6 = 1 × 6
= 2 × 3, here 1, 2,3 and 6 are factors whose sum is (1 + 2 + 3 + 6 = 12) 12, twice the given number 6. So 6 is a perfect number.
6, 28, 496, 8128…… are perfect numbers. Euclid has given a formula to derive perfect
numbers.
If q is a prime of the form 2p – 1 where p is a prime, then q(q+1)/2 is an even perfect number.

→ Multiple: Multiples of a given number can be obtained by multiplying the given number with natural numbers i.e. 1, 2, 3, 4, …. etc.
Example: Multiple of 6 are:
= 6 × 1, 6 × 2, 6 × 3, 6 × 4, …..
= 6, 12, 18, 24, …..

→ Prime number: Numbers having only two factors namely one and itself are called prime numbers.
A prime number is a whole number that has exactly two factors, 1 and itself.
Example: 2, 3, 5, 7, 11,
All the above numbers have only two factors namely 1 and itself.
We can write infinitely many prime numbers.
2 is the only even prime number. Also 2 is the smallest prime number.

→ Composite number: Numbers having more than two factors are called composite numbers.
Example: 4, 6, 8, 9, ….
1 is neither a prime number nor a composite number.
The Sieve of Eratosthenes is an ancient algorithm that can help us find all prime numbers up to any given limit.

→ How does the Sieve of Eratosthenes work?
The following example illustrates how the Sieve of Eratosthenes, can be used to find all the prime numbers that are less than 100.
Step 1: Write the numbers from 1. to 100 in ten rows as shown below.
Step 2: Cross out 1 as 1 is neither a prime nor a composite number.
Step 3: Circle 2 and cross out all the multiples of 2. (2, 4, 6, 8, 10, 12, ….)
Step 4: Circle 3 and cross out all the multiples of 3. (3, 6, 9, 12, 15, 18, ….)
Step 5: Circle 5 and cross out all the multiples of 5. (5, 10. 15, 20, 25, ….)
Step 6: Circle 7 and cross out all multiples of 7. (7, 14, 21. 28, 35, ….)
Circle all the numbers that are not crossed out and they are the required prime numbers less than 100.

Alternate method:
Finding prime numbers upto 100

First arrange the numbers from 1 to 100 in a table as shown above.
Enter 6 numbers in each row until the last number 100 is reached.
First we select a number and we strike off all the multiples of it.
Start with 2 which is greater than 1.
Round off number 2 and strike off entire column until the end.
Similarly strike off 4th column and 6th column as they are divisible by 2.
Now round off next number 3 and strike off entire column until end.
The number 4 is already gone.
Now round off next number 5 and strike off numbers in inclined fashion as shown in the figure (they are all divisible by 5). When striking off ends in some row, start again striking off with number in another end which is divisible by 5. New striking off line should be parallel to previous strike off line as. shown in the figure.
The number 6 is already gone.
Now round off number 7 and strike off numbers as we did in case of number 5.
8,9,10 are also gone.
Stop at this point.
Count all remaining numbers. Answer will be 25.

→ Prime numbers
There are 25 prime numbers less than 100.
These are:

What if we go above 100? Around 400 BC the Greek mathematician. Euclid, proved that there are infinitely many prime numbers.

→ Co-primes: Two numbers are said to be co-prime if they have no factors in common. Example: (2, 9), (25, 28)
Any two consecutive numbers always form a pair of co-prime numbers.
Example: (7 & 8), (21 & 22), …..
Co-prime numbers are also called relatively prime number to one another.
Example: 3, 5, 8, 47 are relatively prime to one another/co-prime to each other.

→ Twin primes: Two prime numbers are said to be twin primes, if they differ by 2. Example: (3, 5), (5, 7), (11, 13), …etc.

→ Prime factorization: The process of expressing the given number as the product of prime numbers is called prime factorization.
Example: Prime factorization of 24 is
24 = 2 × 12 = 2 × 2 × 6
= 2 × 2 × 2 × 3, this way is unique.
Every number can be expressed as product of primes in a unique manner. We can factorize a given number in to product of primes in two methods. They are
a) Division method
b) Factor tree method

→ Common factors: The set of all factors which divides all the given numbers are called their common factors.
Example: Common factors to 24, 36 & 48 are 1, 2, 3, 4, 6 & 12
Factors of 24 = 1, 2, 3, 4, 6, 8, 12 & 24
Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18 & 36
Factors of 48 = 1, 2, 3, 4, 6, 8, 12, 16, 24 & 48
Common factors to 24, 36 & 48 are 1, 2, 3, 4, 6 & 12
We can see that among their common factors 12 is the highest common factor. It is called H.C.F. of the given numbers. So H.C.F. of 24, 36 & 48 is 12.

→ H.C.F./G.C.D : The highest common factor or the greatest common divisor of given numbers is the greatest of their common factors.
H.C.F. of given two or more numbers can be found in two ways.
a) By prime factorization
b) By continued division
H.C.F. of any two consecutive numbers is always 1.
H.C.F. of relatively prime/co-prime numbers is always 1.
H.C.F. of any two consecutive even numbers is always 2.
H.C.F. of any two consecutive odd numbers is always 1.

→ Common multiples:
Multiples of 8: 8, 16, 24, 32, 40, 48, ….
Multiples of 12: 12, 24, 36, 48, ….
Multiples.common to 8 & 12: 24, 48; 72, 96, ….
Least among the common multiple is 24. This is called L.C.M. of 8 & 12. The number of common multiples of given two or more numbers is infinite, as such greatest common multiple cannot be determined.

→ L.C.M.: The least common multiple of two or more numbers is the smallest natural number among their common multiples.
L.C.M. of given numbers can be found by the
a) Method of prime factorization.
b) Division method.
L.C.M. of any two consecutive numbers is always equal to their product.
L.C.M. of 8 & 9 is 8 × 9 = 72
L.C.M. of co-prime numbers is always equal to their product.
L.C.M. of 8 & 15 is 8 × 15 = 120

→ Relation between the L.C.M. & H.C.F:
For a given two numbers N₁ & N₂, the product of the numbers is equal to the product of their L.C.M.(L) & H.C.F.(H)
N₁ × N₂ = L × H

Students can go through AP Board 7th Class Maths Notes Chapter 7 Data Handling to understand and remember the concepts easily.

AP State Board Syllabus 7th Class Maths Notes Chapter 7 Data Handling

→ Data: Information which is in the form of numbers or words and helps in taking decisions or drawing conclusions is called data. Tables and graphs are the ways in which data is presented.
The numerical entries in the data are called ‘observations’.

→ The average or Arithmetic Mean or Mean
A.M = \(\frac{\text { Sum of all observations }}{\text { Number of observations }}\)

→ (i.e.,) A.M. is equal to sum of all the observations of a data set divided by the number of observations. It lies between the lowest and highest values of the data.

→ Mode: An observation of data that occurs most frequently is called the mode of the data. A data may have one or more modes and sometimes none.

→ Median: Median is simply the middle observation, when all observations are arranged in ascending or descending order. In case of even number of observations median is the average of middle observations.

→ Mean, mode, median are representative values for a data set.

→ When all values of data set are increased or decreased by a certain number, the mean also increases or decreases by the same number.

→ Data can be presented in bar graphs / double bar graphs or pie chart.

→ Bar graph: Bar graph are made up of bars of uniform width which can be drawn horizontally or vertically with equal spacing between them. The length of each bar tells us the frequency of the particular item. We take convenient scale for the length of bar graph.

→ Double bar graph: It presents two observations side by side.

→ Pie chart: A circle is divided into sectors to represent the given data.
Angle subtended by the sector at the centre of the circle is directly proportional to each observation.

Students can go through AP Board 7th Class Maths Notes Chapter 6 Ratio – Applications to understand and remember the concepts easily.

AP State Board Syllabus 7th Class Maths Notes Chapter 6 Ratio – Applications

→ Ratio: A ratio is an ordered comparison of quantities of the same units.
We use the symbol ‘:’ to represent a ratio. The ratio of two quantities ‘a’ and ‘b’ is a : b and we read it as “a is to b”. The two quantities ‘a’ and ‘b’ are called the terms of the ratio. The first quantity ‘a’ is called first term or antecedent and the second quantity ‘b’ is called consequent.

→ Proportion: If two ratios are equal, then the four terms of the ratios are said to be in proportion. We use the symbol : : (is as)
If two ratios a : b and c : d are equal, we write a : b :: c : d or a : b = c : d
Here ‘a’, ‘d’ are called extremes and b, c are called means.

→ Unitary Method: The method in which we first find the value of one unit and then the value of the required number of units is known as unitary method.
Eg: If the cost of 5 pens is Rs. 85; then the cost of 12 pens is ……… ?
Solution. Cost of 5 pens = Rs. 85
Cost of 1 pen = \(\frac{85}{5}\) = Rs. 17
∴ Cost of 12 pens = 12 × 17 = Rs. 204

→ Direct proportion: If in two quantities, when one quantity increases, the other also increase or vice-versa then the two quantities are said to be in direct proportion.
Eg: The number of books and their cost are in direct proportion.
As the number of books increases, the cost also increases.

→ Ratios also appear in the form of percentages.

→ The word percent means “per every hundred” or for a hundred. The symbol % is used to denote percentage.

→ To convert a quantity into its equivalent percentage

• express it as a fraction.
• multiply it with 100.
• assign % symbol.

Eg: A man purchased an article for Rs. 80 and sells it for Rs. 100. Find his gain percent.
Solution. Cost price = Rs. 80
Selling price = Rs. 100
gain = Rs. 20
gain as a fraction = \(\frac{20}{80}\)
gain as percent = \(\frac{20}{80}\) × 100 = 25%

→ When C.P > S.P there incurs loss.

→ When C.P < S.P there is gain.

→ When C.P = S.P neither loss nor gain.

→ Loss = C.P – S.P gain = S.P – C.P

→ Discount is always expressed as some percentage of marked price.

→ In general when P is principle; R% is rate of interest per annum and I is the interest, then
I = R% of P
I = R% of P for T years

Students can go through AP Board 7th Class Maths Notes Chapter 5 Triangle and Its Properties to understand and remember the concepts easily.

AP State Board Syllabus 7th Class Maths Notes Chapter 5 Triangle and Its Properties

→ Triangles can be classified according to properties of their sides and angles. Based on sides, triangles are of three types.

→ Equilateral triangle: A triangle in which all the three sides are equal is called an equilateral triangle. In △ABC
AB = BC = CA, also ∠A = ∠B = ∠C In an equilateral triangle each angle is equal to 60°.

→ Isosceles triangle: A triangle in which two sides are equal is called an isosceles triangle.
In △PQR
PQ = PR also ∠Q = ∠R

The non-equal side in an isosceles triangle may be taken as base of the triangle.

→ Scalene triangle: A triangle in which no two sides are equal is called a scalene triangle.

In △BAT
BA ≠ AT ≠ BT also ∠B ≠ ∠A ≠ ∠T.

→ Based on angles, triangles can be classified into three types.

→ Acute angled triangle: A triangle in which all the three angles are acute is called an acute-angled triangle.

In △TAP,
∠T, ∠A, ∠P are acute angles.

→ Obtuse angled triangle: A triangle in which one angle is obtuse is called an obtuse angled triangle.

In △FAN,
∠A is obtuse angle.
A triangle cannot have more than one obtuse angle,

→ Right angled triangle: A triangle in which one angle is a right angle is called a right angled triangle.

In △COT
ZO is right angle (i.e) 90°.
A triangle cannot have more than one right angle,

→ Right angled isosceles triangle: A triangle in which one angle is right angle and two sides are equal is called a right angled isosceles triangle.

In △POT,
PO = OT and ∠O = 90° also
∠P = ∠T = 45°

→ Family of triangles – Flow chart

→ Relation between sides of a triangle
In any triangle the sum of the lengths of any two sides is greater than the length of the third side.

In △TIN,
TI + IN > TN; TN + NI > TI; TI + TN > IN
Also the difference between lengths of any two sides of the triangle is less than the length of the third side.
In △TIN, TI > TN – NI; IN > TI – TN; TN > IN – TI

→ Altitutes of a triangle
The length of a line segment drawn from a vertex to its opposite side and is perpendicular to it is called an altitude or height of the triangle. An altitude can be drawn from each vertex.

Altitude of a triangle may be in its interior or exterior.

→ Medians of a triangle
A line segment joining a vertex and the mid-point of its opposite side is called a median.

A triangle has three medians.
The medians of a triangle are concurrent.
The point of concurrence of medians of a triangle is called the centroid of the triangle.
In △ABC, D, E and F are mid-points of the sides AB, BC and AC.
AE, CD and BF are mid-points.
G is the centroid.

→ Angle – sum property of a triangle: The sum of interior angles of a triangle is equal to 180° or two right angles.

In △BET,
∠B + ∠E + ∠T = 180°

→ Exterior angle of a triangle

→ When one side of a triangle is produced, the angle thus formed is called an exterior angle.

In △COL; the side OL is produced to D.
∠CLD is an exterior angle.
The exterior angle of a triangle is equal to the sum of the interior opposite angles.
∠COL + ∠OCL = ∠CLD

Students can go through AP Board 6th Class Maths Notes Chapter 2 Whole Numbers to understand and remember the concepts easily.

AP State Board Syllabus 6th Class Maths Notes Chapter 2 Whole Numbers

→ Natural numbers: The numbers which we use for counting are called Natural numbers N= {1, 2, 3, 4, 5, 6,…} .

→ Successor: Every natural number has a successor, which is one more than it. Example : Successor of 15 is 15 + 1 = 16

→ Predecessor: Every natural number has a predecessor except 0. Predecessor of a number is one less than it.
Example: Predecessor of 56 is 56 – 1 = 55

→ Whole numbers: The natural numbers along with zero forms the set of Whole numbers.
W = {0, 1, 2, 3, 4, 5, 6,…}

• Every whole number has a successor and every whole number has a predecessor except zero.
• Every natural number is a whole number and every whole number is a natural number except zero.
• The smallest natural number is 1.
• The smallest whole number is 0.
• Addition, subtraction and multiplication can be represented on a number line.

→ Closure property: Sum of any two whole numbers is always a whole number.
Example : 5 + 3 = 8, a whole number.
This is called closure property of whole numbers w.r.t. addition.

→ Closure property: Product of any two whole numbers is always a whole number. Example : 5 × 3 = 15, a whole number.
This is called closure property of whole numbers w.r.t. multiplication.
In other words whole numbers are closed under addition and multiplication.
But whole numbers are not closed under subtraction and division.
Example: 7 – 12, is not a whole number.
9 ÷ 14, is not a whole number.
Division by zero is not defined.
Example: 5 ÷ 0, is not defined

→ Commutative property: Sum of any two whole numbers taken in any order is always same.
Example: 5 + 3 = 3 + 5 = 8
This is called commutative property of whole numbers w.r.t. addition.

→ Commutative property: Product of any two whole numbers taken in any order is always same.
Example: 5 × 3 = 3 × 5 = 15
This is called commutative property of whole numbers w.r.t. addition.
In other words whole numbers are commutative w.r.t. addition and multiplication. But whole numbers are not commutative w.r.t. subtraction and division.
Example: 4 – 9 ≠ 9 – 4 and 8 ÷ 11 ≠ 11 ÷ 8

→ Associative property: The sum of any three whole numbers taken in any or der is always same.
Example : 5 + (8 + 3) = (5 + 8) + 3 =16
This is called associative property of whole numbers w.r.t. addition.

→ Associative property: The product of any three whole numbers taken in any order is always same.
Example : 5 × (8 × 3) = (5 × 8) × 3 = 120
This is called associative property of whole numbers w.r.t. multiplication.
In other words whole numbers are associative w.r.t. addition and multiplication.
But whole numbers are not associative w.r.t. subtraction and division.
Example : 5 – (8 – 3) ≠ (5 – 8) – 3
: 5 ÷ (8 ÷ 3) ≠ (5 ÷ 8) ÷ 3

→ Distributive property of multiplication over addition:
(Example: 5 × (8 + 3) = (5 × 8) + (5 × 3) = 55

→ Additive identity: If zero is added to any whole number, then the result is the number itself.- Here zero is called the additive identity.
Example: 5 + 0 = 5, 7 + 0 = 7, 0 + 9 = 9 and so on.

→ Multiplicative identity: If any whole number is multiplied by 1, then the result is the number itself. Here one is called the multiplicative identity.
Example: 5 × 1 = 5, 7 × 1 = 7, 1 × 9 = 9 and so on.
If we represent the number 1 as a (.) dot, then a whole number can be represented either as an array of a triangle or a square.
The triangular numbers are 3, 6, 10, 15, 21, 28, ……. etc.

The square numbers are 4, 9, 16, 25, 36, …… etc.

→ Multiplication by 9/99/999/9999….etc.
74 × 99 = (74 – 1)/(9 – 7)(10 – 4) = 73/26
256 × 999 = (256 – 1)/(9 – 2)(9 – 5)(10 – 6) = 255/744
4267 × 9999 = (4267 – 1)/(9 – 4)(9 – 2)(9 – 6)(10 – 7) = 4266/5733
Here the number of 9’s in the multiplier is equal to number of digits in the multiplicand.
Answer has two parts: LHS/RHS
LHS: (Multiplicand-1)
RHS: Subtract all digits from 9 but the last digit from 10

Students can go through AP Board 6th Class Maths Notes Chapter 10 Practical Geometry to understand and remember the concepts easily.

AP State Board Syllabus 6th Class Maths Notes Chapter 10 Practical Geometry

→ The above are the components we see in a geometry box. They are pair of set squares, protractor, graduated ruler, compasses and the divider.

→ Graduated ruler: A scale is used to draw straight edges of given length. It is also used to measure the lengths of the given line segments. A scale is also called a graduated ruler.

→ Compass: A compass is used to draw a circle of given radius. Sometimes we draw only a part of a circle which is called an arc. A compass is used in the construction of a line segment of given length, a perpendicular to a given line, given angle and in many more geometrical shapes.

→ Divider: A divider is used to measure the lengths of straight line segments and curved lines. It is also used to compare the lengths of two line segments.

→ Set squares: The pair of set squares is used to draw the pair of parallel lines.

→ Perpendicular bisector: The perpendicular bisector of a given line segment is the line which divides the given line segment into two equal parts at right angles.

→ Angle bisector: Angle bisector is a ray which divides the given angle into two equal parts.