(a) Let \(\frac { a }{ b }\) and c are two rational numbers, then \(\frac { a }{ b } \times c=\frac { ac }{ b }\) (b) When we multiply two rational numbers :
On multiplying two rational numbers, we get result as a rational number.
\(\frac { 1 }{ 3 } \times \frac { 1 }{ 4 } =\frac { 1 }{ 4 } \times \frac { 1 }{ 3 }\) (commutative i.e., on changing the order the result remains same)
If 0 is multiplied to any rational number, the result is always zero.
Multiplication of Rational Numbers on a Number Line
The product of two rational numbers on the number line can be calculated in the following way. This result reconfirms that the product of two rational numbers is rational number whose numerator is the product of the numerators of the given rational numbers and the denominator is the product of the denominators of the given numbers. ∴ Multiplication is closure (product is rational), commutative (ab = ba) and associative (a(bc) = (ab)c) for rational number.
Division of Rational Numbers
(a) Let \(\frac { a }{ b }\) be a rational number then its reciprocal will be \(\frac { b }{ a }\)
The product of a rational number with its reciprocal is always 1.
Zero has no reciprocal as reciprocal of 0 = \(\left( \frac { 0 }{ 1 } \right)\) is \(\frac { 1 }{ 0 }\) (which is not defined).
The reciprocal of a rational number is called the multiplicative inverse of rational number.
1 and –1 are the only rational numbers which are their own reciprocal.
Reciprocal of a (+ve) rational number is (+ve) and reciprocal of (–ve) rational number is (–ve). To divide one rational number by other rational numbers we multiply the rational number by the reciprocal of the other
Zero divided by any rational number is always equal to zero.
Note:
When a rational number (except zero) is divided by another rational number (except 0) the quotient is always a rational number. (closed under division)
Division of any rational number by itself gives the quotient 1.
When a rational number is divided by 1, the quotient is a rational number itself.
Multiplication and Division of Rational Numbers Problems with Solutions
1. Find the product: Solution: 2. Find the value of: Solution: 3. Multiply: Solution: 4. Simplify: Solution:
When we take 100 as the denominator of fractions, the numerators are called percentages. For convenience, the symbol % is used for percent. When we take 100 as the denominator of fractions, the numerators are called percentages. For convenience, the symbol % is used for percent. Or ‘‘A percentage is simply a ratio in which the second term is arranged to be 100’’. Also percent is an abbreviation of the Latin phrase per centum, meaning per hundred or hundredths.
A fraction may be converted into a percentage by multiplying that fraction by 100%. This does not change its value, since 100% is 1.
A decimal may be converted into a percentage by multiplying it by 100%.
Uses of Percentages
Interpreting percentages.
Converting percentage to ‘How many’.
Converting ratio to percentage.
Increase or decrease as percent.
Eg: Raju invests 10% of his pocket money in buying toffees means ₹10 out of ₹100 are invested by Raju in buying the toffees.
Eg: A local cricket team played 20 matches in one season. It won 25% of them. How many matches did they win ? Here, the total number of matches played are 20. Out of these 25% are won by the team.
I method (direct): Out of 100, 25 matches are won by the team. So, out of 20, number of matches won by the team = \(\frac { 25 }{ 100 } \times 20\) = 5 matches.
II method (using percentage):
https://www.youtube.com/watch?v=J0BhMpB6SWo
Percentages Problems with Solutions
1. Express \(\frac { 7 }{ 20 }\) as a percentage. Solution:
2. Express 0.625 as a percentage. Solution: 0.625 = 0.625 × 100% = 62.5%
3. Solution:
4. Out of 50 students in a class, 15 like to play cricket. What is percentage of students who like to play cricket ? Solution: Total students = 50 Students who like to play cricket = 15 So, % age of students who like to play cricket
5. Convert the given decimals to percent: (a) 0.6 (b) 0.75 (c) 0.08 (d) 0.56 Solution: We have (a) 0.6 = (0.6 × 100)% = 60% (b) 0.75 = (0.75 × 100)% = 75% (c) 0.08 = (0.08 × 100)% = 8% (d) 0.56 = (0.56 × 100)% = 56%
6. Convert a percentage into fraction (i) 45% (ii) 65% (iii) 42.5% Solution: We have
7. Convert each of the following into decimal fraction : (a) 53% (b) 0.38% (c) 4.7% Solution:
8. What percentage of the adjoining figure is shaded and what percentage is unshaded ? Find it. Solution: First we find the fraction of the figure that is shaded or unshaded. From this fraction we will find the percentage of the shaded and unshaded regions.
9. Convert each of the following ratios into a percentage :Convert each of the following ratios into a percentage : (i) 15 : 45 (ii) 3 : 5 Solution: We have,
10. Estimate what region of the following figures is shaded and hence find percentage of that shaded region. Solution: We have,
11. Convert given percents to decimal fractions and also to fractions in simplest form :Convert given percents to decimal fractions and also to fractions in simplest form : (i) 25% (ii) 150% (iii) 20% (iv) 5% Solution:
12. The population of a city decreased from 25,000 to 24,500. Find the percentage decrease. The population of a city decreased from 25,000 to 24,500. Find the percentage decrease. Solution:
13. The population of India is 113 crore. If it increases by 1.7% every year, Find India’s population after one year. Solution:
If two rational numbers are to be added, we first express each one of them as rational number with positive denominator. There are two possibilities : (1) Either they have same denominators, or (2) They have different denominators.
Adding Rational Numbers with Same Denominator:
Let us add \(\frac { 8 }{ 5 }\) and \(\frac { -6 }{ 5 }\) Represent the numbers on the number line. Here, the distance between two consecutive points is \(\frac { -6 }{ 5 }\). For \(\frac { 8 }{ 5 }\), move 6 steps to the left of \(\frac { 8 }{ 5 }\) and we reach at \(\frac { 2 }{ 5 }\). Example 1: Add : \(\frac { -5 }{ 9 }\) and \(\frac { -17 }{ 9 }\). Solution: Given rational numbers are \(\frac { -5 }{ 9 }\) and \(\frac { -17 }{ 9 }\). Adding these two numbers, we have Which is the required answer.
Example 2: Add : \(\frac { -23 }{ 28 }\) and \(\frac { 5 }{ -28 }\). Solution: We first express \(\frac { 5 }{ -28 }\) as a rational number with positive denominator.
Addition of Rational numbers with Different Denominators: In this case, we convert the given rational numbers to a common denominator and then add.
Examples:
1. Add \(\frac { 8 }{ -5 }\) and \(\frac { 4 }{ -3 }\). Solution: The given rational numbers are \(\frac { 8 }{ -5 }\) and \(\frac { 4 }{ -3 }\). Clearly, they have different denominators. Here, first we express the given rational numbers into standard forms. Which is the required answer.
4. Add \(\frac { 7 }{ 9 }\) and \(\frac { -5 }{ 9 }\). Solution: In case, if denominator of the rational number is negative, first we make it (denominator) Positive and then add.
6. Find the sum of \(\frac { -8 }{ 5 }\) and \(\frac { -5 }{ 3 }\) Solution: Note : Addition of rational numbers is closure (the sum is also rational) commutative (a + b = b + a) and associative(a + (b + c)) = ((a + b) + c).
Additive inverse: The negative of a rational number is called additive inverse of the given number. Note: Zero is the only rational no. which is its even negative or inverse.
Subtraction of Rational Numbers
If we add the additive inverse of a rational number and other rational number then this is called subtraction of two rational numbers. So the subtraction is inverse process of addition and the term add the negative of use for subtraction.
Subtraction of Rational Numbers Problems with Solutions
1. Find value of \(\frac { 2 }{ 3 } -\frac { 4 }{ 5 }\). Solution:
2. Find value of \(\frac { 2 }{ 7 } -\left( \frac { -5 }{ 3 } \right)\). Solution:
Mr Sharma wanted to withdraw Rs. 1000 from his bank account to purchase books for his children. The cashier gave him 10 hundred-rupee notes, i.e., Rs. 10 × 100 = Rs. 1000 Mr Sharma got the required amount. But the cashier could also give the same amount in the following ways: Here, we observe that in each case Mr Sharma got the same amount of Rs. 1000. These numbers 1, 2, 5, 10, 20, 50, 100, 200, 500, and 1000 are factors of 1000. Hence, 1000 is a multiple of these numbers. Here we will discuss only the natural numbers, that is positive integers. If a = b × c, we say b and c are factors of a and a is a multiple of c and b.
Factors
Factor: A number which divides a given number exactly (without leaving any remainder) is called a factor of the given number. Example: Factors of 12 12 = 1 × 12 12 = 2 × 6 12 = 3 × 4 Here, 1, 2, 3, 4, 6, and 12 are factors of 12.
Properties of Factors
Every non-zero number is a factor of itself. Examples: 5 is a factor of 5. (5 ÷ 5 = 1) 12 is a factor of 12. (12 ÷ 12 = 1)
1 is a factor of every number. Examples: 1 is a factor of 5. (5 ÷ 1 = 5) 1 is a factor of 12. (12 ÷ 1 = 12)
Every non-zero number is a factor of 0. Example: 5 and 12 are factors of 0 because 0 ÷ 5 = 0 and 0 ÷ 12 = 0
The factors of a number are finite.
Multiples
Multiple: A multiple of any natural number is a number formed by multiplying it by another natural number. Example: Multiples of 6 are 6 × 1 = 6; 6 × 2 = 12; 6 × 3 = 18; 6 × 4 = 24 Here, 6,12,18,24 are multiples of 6. Example: Let us find the LCM and HCF of 24 and 36. Factors of 24 = 1, 2, 3, 4, 6, 8, 12, 24 Factors of 36 = 1, 2, 3, 4, 6, 9, 12, 18, 36 Here, the highest common factor is 12. ∴ HCF = 12 Multiples of 24 = 24, 48, 72, 96,… Multiples of 36 = 36, 72, 108,… Here, the lowest common multiple is 72. ∴ LCM = 72
Properties of Multiples
Every number is a multiple of itself. Examples (a) 3 × 1 = 3; 3 is the multiple of 3 (b) 7 × 1 = 7; 7 is the multiple of 7
Every number is the multiple of 1. Examples (a) 1 × 3 = 3; 3 is the multiple of 1 (b) 1 × 7 = 7; 7 is the multiple of 1
The multiples of a number are infinite (unlimited).
Even numbers: A number which is a multiple of 2 is called an even number. Example: 2, 4, 6, 8, 10,…
Odd numbers: A number which is not a multiple of 2 is called an odd number. Example: 1, 3, 5, 7, 9, 11,…
Prime numbers: A number which is greater than 1, and has exactly two factors (1 and the number itself) is called a prime number. Example: Factors of 2 = 1, 2 Factors of 3 = 1, 3 Factors of 5 = 1, 5 Factors of 7 = 1, 7 Factors of 11 = 1, 11 Here, 2, 3, 5, 7, 11 etc. are all prime numbers.
Composite numbers: A number, which is greater than 1 and has more than two factors is called a composite number. Examples: Here, Factors of 4 = 1, 2, 4 Factors of 6 = 1, 2, 3, 6 Factors of 8 = 1, 2, 4, 8 Factors of 9 = 1, 3, 9 Factors of 10 = 1, 2, 5
FINDING PRIME NUMBERS FROM 1 TO 100 We can find the prime numbers from 1 to 100 by following these steps (given by the Greek mathematician Eratosthenes). Step 1: Prepare a list of numbers from 1 to 100. Step 2: As 1 is neither prime nor composite number, cross it out. Step 3: Encircle ‘2’ as a prime number and cross out all its other multiples. Step 4: Encircle ‘3’ as a prime number and cross out all its other multiples. Step 5: Encircle ‘5’ as a prime number and cross out all its other multiples. Step 6: Continue this process till all the numbers are either encircled or crossed out. All the encircled numbers are prime numbers and the crossed out numbers (except 1) are composite numbers. Numbers 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97 are the prime numbers between 1 and 100. This is called the ‘Sieve of Eratosthenes’.
Twin primes: Two prime numbers having a difference of 2 are known as twin primes. Example: ( 3, 5 ), ( 5,7 ), (11,13 ), (17,19 ), etc are twin primes.
Co-primes: Two numbers are said to be co-primes if they have no common factor other than 1. In other words, two natural numbers are co-primes if their HCF is 1. Example: ( 2, 3 ), ( 3, 4 ), ( 5, 6 ), ( 7, 8 ), and so on.
Example 1: Is 16380 a multiple of 28? Solution: To check whether 16380 is a multiple of 28 or not, we have to divide 16380 by 28. If the remainder becomes zero, then it is a multiple of the number. So, 16380 = 28 × 585, hence 16380 is a multiple of 28.
Example 2: Express 29 as the sum of three odd prime numbers. Solution: 29 = 19 + 7 + 3 All 19, 7, and 3 are odd prime numbers.
DIVISIBILITY TESTS FOR 2, 3, 4, 5, 6, 7, 8, 9, 10, AND 11
If we want to know that a number is divisible by another number, we generally perform the actual division and see whether the remainder is zero or not. This process is time-consuming for division of large numbers. Therefore, to cut short our efforts, some divisibility tests of different numbers are given below.
Test of Divisibility by
Condition
Example
2
A number is divisible by 2, if its ones digit is 0, 2, 4, 6 or 8.
1372, 468, 500, 966 are divisible by 2, since their ones digit is 2, 8, 0 and 6 respectively.
3
A number is divisible by 3, if the sum of its digits is divisible by 3.
In 1881, the sum of digits is 1 + 8 + 8 + 1 = 18 which is divisible by 3. So 1881 is divisible by 3.
4
A number is divisible by 4, if the number formed by the last two digits is divisible by 4.
30776, 63784, 864 are all divisible by 4. Since last two digits of the numbers, i.e., 76, 84, and 64 are divisible by 4.
5
A number is divisible by 5, if its ones digit is either 5 or 0.
675, 4320, 145 all are divisible by 5 because their ones digit is 5 or 0.
6
A number is divisible by 6, if the number is divisible by 2 and 3.
In 5922, ones digit is 2, so it is divisible by 2. The sum of digits in 5922 is 5 + 9 + 2 + 2 = 18, which is divisible by 3. So, 5922 is divisible by 6.
7
A number is divisible by 7, if the difference between twice the last digit and the number formed by other digits is either 0 or a multiple of 7.
In number 2975, it is observed that the last digit in 2975 is 5. So, 297 – (2 x 5) = 287, which is a multiple of 7. Hence, 2975 is divisible by 7.
8
A number is divisible by 8, if the number formed by its last three digits is divisible by 8.
In 213456, the last three digits are 456 which is divisible by 8. So, the number 213456 is divisible by 8.
9
A number is divisible by 9, if the sum of its digits is divisible by 9.
In 538425, the sum of the digits are (5 + 3 + 8 + 4 + 2 + 5) = 27 which is divisible by 9. So, 538425 is divisible by 9.
10
A number is divisible by 10, if the digit at ones place of the number is 0.
The numbers 980, 63990 are all divisible by 10 because their ones digit is 0.
11
A number is divisible by 11, if the difference between the sum of digits at odd places and the sum of digits at even places is either 0 or a multiple of 11.
In number 27896, the sum of the digits at odd places are (2 + 8 + 6) = 16. The sum of the digits at even places are (7 + 9 ) = 16. Their difference is 16 – 16 = 0. So, the number 27896 is divisible by 11.
Example 3: Test whether 72148 is divisible by 8 or not? Solution: Here, the number formed by the last three digits is 148, which is not divisible by 8. So, 72148 is not divisible by 8.
Example 4: Test whether 8050314052 is divisible by 11 or not? Solution: The sum of the digits at even places = 8 + 5 + 3 + 4 + 5 = 25 The sum of digits at the odd places = 0 + 0 + 1 + 0 + 2 = 3 Difference = 25 – 3 = 22 22 is divisible by 11. So, the number 8050314052 is divisible by 11.
While some sequences are simply random values, other sequences have a definite pattern that is used to arrive at the sequence’s terms. Two such sequences are the arithmetic and geometric sequences. Let’s investigate the arithmetic sequence.
If a sequence of values follows a pattern of adding a fixed amount from one term to the next, it is referred to as an arithmetic sequence. The number added to each term is constant (always the same).
The fixed amount is called the common difference, d, referring to the fact that the difference between two successive terms yields the constant value that was added. To find the common difference, subtract the first term from the second term. Notice the linear nature of the scatter plot of the terms of an arithmetic sequence. The domain consists of the counting numbers 1, 2, 3, 4, … and the range consists of the terms of the sequence. While the x value increases by a constant value of one, the y value increases by a constant value of 3 (for this graph).
Examples:
Formulas used with arithmetic sequences and arithmetic series:
