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Multiplication and division are linked to each other in the same way that addition and subtraction are. Multiplication and division are the two most essential arithmetic operations in mathematics. The result of multiplying two integers is known as the 'product.' A division is a process of dividing something into equal parts. The division is the inverse action of multiplication.
Key Terms: Addition, Subtraction, Multiplication, Division, Fraction, Decimal, Positive numbers, Negative numbers
Read More: Real Numbers Formula
What is Multiplication?
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Multiplication is the process of repeatedly adding a number. If we multiply m by n, it indicates that m is added to itself n times.
In other words, multiplication is a mathematical operation that is an abbreviated procedure of adding an integer to zero a defined number of times and that is extended to other numbers in line with integer-specific principles.
Multiplication is denoted by the cross sign “x”.
Example: 12 multiplied by 5 is equal to 60. How? Adding 12, 5 times to itself, we get;
12 + 12 + 12 + 12 +12 = 60
Therefore, we can write,
12 x 5 = 60.
Read More: Rationalize the Denominator
Properties of Multiplication
- The commutative property states that the order of two integers does not matter when multiplying them. For numbers a and b, a × b = b × a.
- Associative Property: For numbers a, b, and c, (a × b) × c = a × (b × c).
- Distributive Property: For numbers a, b, and c, a × (b + c) = (a × b) + (a × c).
What is Division?
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Division algorithm divides or distributes a number into equal parts. The division is a process of repeated subtraction. It is the inverse action of multiplication. It is characterised as the formation of equal groupings. When we divide numbers, we split them down into smaller amounts so that the multiplication of those smaller numbers equals the larger number taken.
The division is represented by a mathematical symbol consisting of a tiny horizontal line with a dot above and below it. They are two division symbols ÷ and /.
For example, if 60 is divided by 12, then 60 is divided into 5 equal parts. Therefore, the resultant value is 5.
Read More: Important Questions for Real Numbers
Parts of division
| Dividend ÷ Divisor = Quotient |
16 ÷ 4 = 4
In the above example, there are three parts for division.
16 is dividend
4 is divisor
4 is quotient (R.H.S)
The remainder is the remaining part of the dividend that remains after division. For example, when 86 is divided by 5, there remains a remainder of 1.
It means, 86 ÷ 5 = 17 and r =1,
Here, ‘r’ is the remainder.
Relation Between Multiplication And Division
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Multiplication and division are closely linked with each other. The division is the inverse process of multiplication. When we divide, we want to split into equal groups, but multiplication includes joining equal groups.
Division and multiplication are the inverse operations of each other. If we state that a multiplied by b equals c, then c divided by b equals a.
Mathematically, it can be represented as:
a × b = c
c ÷ b = a
Example:
10 x 10 = 100 [10 multiplied by 10 results in 100]
100 ÷ 10 = 10 [100 divided by 10 returns back 10]
Read More: Angle of Depression
Multiplication And Division Rule
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| Multiplication Rules | Division Rules |
|---|---|
| (+) × (+) = (+) (−) × (−) = (+) (+) × (−) = (-) (−) × (+) = (-) | (+) ÷ (+) = (+) (−) ÷ (−) = (+) (+) ÷ (−) = (-) (−) ÷ (+) = (-) |
Multiplication and Division of Decimal Numbers by 10, 100, and 1000
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| Arithmetic operations | Rules | Examples |
|---|---|---|
| Multiply by 10 ( 101) | the number will move one place value to the left | 8.74 x 10 =87.4 |
| Multiply by 100 ( 102) | the number will move two places value to the left | 8.74 x 100 =874 |
| Multiply by 1000 ( 103) | the number will move three places value to the left | 8.74 x 1000 =8740 |
| Divide by 10 (10) | the number will move one place value to the right | 87.4 ÷ 10 =8.74 |
| Divide by 100 ( 102) | the number will move two places value to the right | 87.4 ÷ 100 =0.874 |
| Divide by 1000 ( 103) | the number will move three places value to the right | 87.4 ÷ 1000 =0.0874 |
Multiplication and Division of Integers
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Integers are non-fractional quantities that can be negative, positive, or zero. On a number line, the integers are simply represented.
Example: Multiplying and dividing any integer by a whole number, a fraction, or an integer itself is shown below.
3 x 7 = 21 (Integer x Whole number)
2 x ½ = 1 (Integer x Fraction)
3 x -5 = -15 (Integer x Integer)
Read More: Relation Between HCF and LCM
Multiplication and Division of Fractions
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½ is a fraction that represents half of a whole number or any value, for example. The upper component is referred to as the numerator, while the lower part is referred to as the denominator. Let's look at an example of how to multiply and divide fractions.
¼ ÷ ½ = (1 x 2)/(1 x 4) = 2/4 = ½
Multiplication and Division of Decimals
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Multiplying decimals is the same as multiplying whole numbers. Decimals are numbers that contain a decimal point (Example: 5.67). They signify a fraction of something or a number, for example, 1/2 = 0.5.
When you multiply decimals, you arrange the decimal point in the product such that the number of decimal places in the product equals the sum of the decimal places in the factors.
Things to Remember
- When we multiply a number by one, the result is the same. Under multiplication, 1 is the identity element.
- When we multiply a number by zero, the result is zero.
- De Morgan invented the oblique bar as an indication in the division process in 1845.
- When you divide anything by one, the result is always the original number. It indicates that if the divisor is 1, the quotient will always equal the dividend.
- Division by 0 has no meaning.
- The division of the same dividend and divisor is always 1.
Read More: Pair of Linear Equations in Two Variables Formula
Sample Questions
Ques: Calculate the following: [3 marks]
(1) 678 x 45
(2) 1111 x 100
(3) 2345 x 67
Ans:
- 678 x 45 = 30,375
- 1111 x 100 = 111100
- 2345 x 67 = 1.57,115
Ques: Martha has 4 dogs. She bought 8 chewable bones to feed them both equally. How many bones will each dog get? [3 marks]
Ans: Given, the number of dogs is 4, and the number of bones = 8.
Thus, the number of bones for each dog = 8 ÷ 4 = 2.
Therefore, each dog will get 2 bones.
Ques: There are 16 rows of seats in a railway station. Each row has 20 benches. How many passengers can sit at the railway station? [3 marks]
Ans: The number of benches in one row= 20
No. of rows = 16
Multiplying the number with the number of rows = 20 x 16
Total number of seats available for students= 320.
Ques: The monthly salary of a man is Rs. 50000. Calculate the annual income by salary? [3 marks]
Ans: Number of months in a year= 12
Salary of 1 month= Rs 50,000
Annual salary= salary of 12 months= 50,000 × 12= Rs. 6,00,000
Therefore, annual income = Rs 6,00,000
Ques: There are 40 apples in a basket. A worker has to distribute them equally between 20 labors. How many apples each labor will get? [3 marks]
Ans: Total number of apples, i.e. dividend= 40
No. of labors, i.e. divisor = 20
No. of labors = 40 / 20
After division, the quotient comes out to be 2.
Therefore, each labor will get 2 apples.
Ques: Rs. 1,00,000 are distributed equally among 50 men. How much money will each person get? [3 marks]
Ans: Money received by 50 men= Rs 1,00,000
So, money received by 1 man= Rs 1,00,000 ÷ 50= Rs. 2000
Hence, each man will get Rs. 2000.
Ques: A carton holds 25 dozens of bananas. How many bananas can be packed in 45 cartons? [3 marks]
Ans: Number of dozens in 1 carton= 25
Number of banana in 1 dozen = 12
Number of banana in 45 carton = 25 x 12 x 45 = 13,500
Therefore, altogether there are 13,500 bananas in 45 cartons.
Ques: A soda factory produces 2000 liters of soda a day. How many liters will the factory produce in a year? [3 marks]
Ans: Number of liters soda factory produces in one day = 2000
Number of days = 365
Total number of liters produced in a year = 2000 x 365 = 7,30,000
Therefore, the soda company will produce 7,30,000 liters of soda in a year.
Ques: How many hours are there in 2000 minutes? [3 marks]
Ans: Number of minutes in 1 hr. = 60 minutes
This means that 60 minutes= 1 hour
By Unitary Method,
1 minute= 1/60 hour
Hence, 1500 minutes= 1 ÷ 60 × 2000
Or,
2000 ÷ 60= 33.33 hours
So there are 33.33 hours in 2000 minutes.
Ques: During a school trip there are 2420 students in the school. A bus can only seat 45 students. How many buses will he need to take all the students on a trip? [3 marks]
Ans: Number of students = 2420
Number of seats in a bus = 45
Number of buses required to carry all the students = 2420/45 = 53.77
Students Left Unseated = 2420 - (53 x 45)
=2420 - 2385
=35
So the school need 53 buses with 35 students unseated
Ques: A worker has 4500 loose sheets. He has to make notebooks with 120 pages. How many notebooks can he make? [3 marks]
Ans: Number of loose sheets = 4500
Number of pages in a notebook = 120
Number of notebooks = 4500 / 120 = 37.5
Total number of notebooks is 37.5
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