Multiplication and Division: Overview, Formulas, Solved Examples

The arithmetic operation of Mathematics includes addition, subtraction, multiplication and division on all the types of real numbers, including the integers. Division symbol is a form of the obelus as a horizontal line with a dot above and below the line, \( \div .\) It was first used as the sign for the division by the Swiss mathematician Johann Rahn in his book Teutsche Algebra in \(1659.\)

In mathematics, the term multiplication is one of the basic operations, and it means adding a number concerning another number repeatedly. The symbol for multiplication is \(×.\) In this article, we will provide detailed information on Multiplication and Division, Continue reading to learn more!

Define Multiplication and Division

Multiplication: Multiplication is used to find the product of two or more numbers. Multiplication is also known as repeated addition.
Example: When you want to multiply the numbers \(4 \times 12 = 48\) or \(12 + 12 + 12 + 12 = 48.\)
Division: The division is the opposite operation of multiplication. It is how one tries to determine how many times a number is contained into another.
We know that dividing \(20\) by \(5\) means finding the number that, when multiplied by \(5\) gives us \(20.\) Such a number is \(4.\)
Therefore, we write \(20 \div 5 = 4\) or \(\frac{{20}}{5} = 4.\)
Similarly, dividing \(36\) by \( – 9\) means finding the number which, when multiplied with \( – 9\) gives \(\left( {36} \right).\) Such a number is \( – 4.\)
Therefore, we write \(36 \div \left( { – 9} \right) = – 4\) or \(\frac{{36}}{{ – 9}} = – 4\)
Dividing \( – 35\) by \(\left( { – 7} \right)\) means what number should be multiplied by \(\left( { – 7} \right)\) to get \(\left( { – 35} \right).\)
Such a number is \(5.\)
Therefore, \(\left( { – 35} \right) \div \left( { – 7} \right) = 5\) or \(\frac{{ – 35}}{{ – 7}} = 5.\)
Dividend: The number to be divided is known as a dividend.
Divisor: The number which divides is known as the divisor.
Quotient: The result of division is known as the quotient.
Remainder: The number which is leftover after the division is known as remainder.
Here, \(r\) is the remainder, clearly, \(r = a – bq.\)
By using these terms, the division algorithm can be restated as:
\({\rm{Dividend}} = {\rm{Divisor}} \times {\rm{Quotient}} + {\rm{Remainder}}\)
Example: If we divide \(26\) by the number \(6,\) the dividend is \(26,\) the divisor is \(6,\) the quotient is \(26,\) and the remainder is \(2.\)

Rules of Multiplication and Division

The rules of multiplication and division are as follows:

Multiplication

To multiply the numbers, we follow the given rules:
Rule 1: The product of two numbers of opposite signs is equal to the additive inverse of the product of their absolute values.
Example: \(7 \times \left( { – 4} \right) = – \left( {7 \times 4} \right) = – 28\)
\(\left( { – 8} \right) \times 5 = – \left( {8 \times 5} \right) = – 40\)
Rule 2: The product of two numbers with like signs is equal to the product of their absolute values.
Example: \(7 \times 12 = 84\)
\(\left( { – 8} \right) \times \left( { – 13} \right) = 8 \times 13 = 104\)

Division

You know that when a dividend is negative and the divisor is negative, the quotient is positive. When a dividend is a negative number, and a divisor is a positive number, then the quotient is a negative number.
Thus, we have the following rules for the division of numbers:
Rule \(1:\) The quotient of the two numbers, both positive or negative, is a positive number equal to the quotient of the corresponding fundamental values of the numbers.
Thus, we split their values regardless of their sign and give plus sign to the quotient for dividing two numbers with like symbols.
Rule \(2:\) The quotient of a positive and a negative number is a negative number. The absolute value is equal to the quotient of the corresponding fundamental values of the numbers.
Thus, we divide their values regardless of their sign and give minus sign to the quotient for dividing numbers with unlike characters.

Properties of Multiplication and Division

The properties of multiplication and division are as follows:

Multiplication: The properties of multiplication are given below:
1. Commutativity: The multiplication of a whole number is commutative. In other words,
if \(a\) and \(b\) are any two whole numbers, then \(a \times b = b \times a\)
2. Multiplication by zero: If \(a\) is any whole number, then \(a \times 0 = 0 \times a = 0.\)
In other words, the product of any whole number and zero is always zero.
3. Existence of Multiplication identity: If \(a\) is a whole number, then \(a \times 1 = a = 1 \times a.\)
In other words, the product of any whole number and \(1\) is the number itself.
The number \(1\) is known as multiplication identify or the identifying element for multiplication of whole numbers because it does not change the identity (value) of the numbers during the operation of multiplication.
4. Associativity: if \(a,\,c\) are whole numbers, then
\(\left( {a \times b} \right) \times c = \left( {b \times c} \right)\)
The multiplication of whole numbers is associative; that is, the product of three real numbers does not change by changing their arrangements.
5. Distributivity of multiplication over addition: If \(a,\,b,\,c\) are any three whole numbers, then
\(a \times \left( {b + c} \right) = a \times b + a \times c\)
\(\left( {b + c} \right) \times a = b \times a + c \times a\)
The multiplication of whole numbers distributes over their addition.

Division: There are some of the properties of a division of numbers which are given below:
1. If \(a\) and \(b\) are integers, then \(a \div b\) is not necessarily an integer.
Example: \(14 \div 2 = 7.\) Here, the quotient is an integer.
But, in \(15 \div 4,\) we observe that the quotient is not an interger. Here, the result is
\(\frac{{15}}{4} = 3\frac{3}{4}.\) the quotient is \(3;\) the remainder is \(3\)
2. If \(a\) is an integer different from \(0,\) then \(a \div a = 1.\)
3. For every integer \(a,\) we have \(a \div 1 = a.\)
4. If \(a\) is a non-zero integer, then \(0 \div a = 0\)
5. If \(a\) is an integer, then \(a \div 0\) is not meaningful.
6. If \(a,\,b,\,c\) are integers, then
\(a > b \Rightarrow a \div c > b \div c,\) if \(c\) is positive.
\(a > b \Rightarrow a \div c > b \div c,\) if \(c\) is negative.

Formulas of Multiplication and Division

The formulas for multiplication and division are as follows:

Multiplication

The formulas of the multiplication of numbers are given below in the table:

In the case of the multiplication of numbers, you have to multiply the numbers without the sign. Once the product is acquired, then mark the symbol according to the rule of multiplication.

Division

The formulas of the division of numbers are given below in the table:

Same as the multiplication, you have to divide the numbers without the sign, then give the symbol according to the rule as given in the table.
The division of two numbers with the like signs gives a positive quotient, and the division of two numbers with an unlike symbol gives a negative quotient.

Multiplication and Division Facts

Multiplication: When you multiply the even numbers of negative integers, the result is always positive.
\(\left( – \right) \times \left( – \right)\left( – \right) \times \left( – \right) = \left( + \right)\)

Division: For every multiplication fact, we have two division facts.
Example: For the number 5 table, the division facts are \(10 \div 5 = 2,\,25 \div 5 = 5\) and \(50 \div 5 = 10\) and \(5 \times 2 = 10,\,2 \times 5 = 10.\)

Solved Examples on Multiplication and Division

Q.1. Multiply \(475\) by \(64\) by using the distributivity property.
Ans: We have, \(475 \times 64\)
\( = \left( {400 + 70 + 5} \right) \times 64\)
\( = 400 \times 64 + 70 \times 64 + 5 \times 64\) [Using distributivity]
\( = 25600 + 4480 + 320 = 30400\)

Q.2. Find the number which, when divided by \(46\) gives a quotient \(11\) and remainder \(18.\)
Ans: We have,
Divisor \( = 46,\) Quotient \( = 11\) and Remainder \( = 18.\)
We have to find the dividend. By division algorithm we have,
\({\rm{Dividend}} = {\rm{Divisor}} \times {\rm{Quotient}} + {\rm{Remainder}}\)
\( \Rightarrow {\rm{Dividend}} = 46 \times 11 + 18\)
\( = 506 + 18 = 524.\)
Hence the required answer is \(524.\)

Q.3. Find the product \(4 \times 4957 \times 25.\)
Ans: We observe that,
\(4 \times 25 = 100\)
So, we can arrange the numbers to find the desired product
\(4 \times 4925 \times 25 = \left( {4 \times 25} \right) \times 4957 = 100 \times 4957 = 495700\)
Hence, the required answer is \(495700.\)

Q.4. Find the value of: \(\left[ {32 + 2 \times 17 + \left( { – 6} \right)} \right] \div 15\)
Ans: We have,
\(\left[ {32 + 2 \times 17 + \left( { – 6} \right)} \right] \div 15\)
\( = \left[ {32 + 34 \div \left( { – 6} \right)} \right] \div 15 = \left( {66 – 6} \right) \div 15 = 60 \div 15 = \frac{{60}}{{15}} = 4\)
Hence, the required answer is \(4.\)

Q.5. Determine the product of the greatest number of four-digit and the greatest number of three-digit.
Ans: We know that the greatest four-digit number is \(9999\) and the greatest three-digit number is \(999.\)
\(\therefore \) Required product \( = 9999 \times 999\)
\( = 9999 \times \left( {1000 – 1} \right)\)
\( = 9999 \times 1000 – 9999 \times 1\,\,\,\left[ {\,a \times \left( {b – c} \right) = a \times b – a \times c} \right]\)
\( = \left( {1000 – 1} \right) \times 1000 – \left( {1000 – 1} \right) \times 1\)
\( = 1000 \times 1000 – 1000 \times 1 – \left( {1000 \times 1 – 1 \times 1} \right)\)
\(\left[ {\,\left( {a – b} \right) \times c = a \times c – b \times c} \right]\)
\( = 1000000 – 1000 – 10000 + 1\)
\( = 1000000 – 11000 = 9989001\)

Summary

Multiplication helps us in finding the product of two or more numbers. It is also known as repeated addition. The division helps students to determine how many times a number is contained into another. The division is also known as repeated subtraction. The symbol used for division is \( \div .\) There are four major terms used in division. The basic terms used in the division are dividend, divisor, quotient and remainder.

FAQs on Multiplication and Division

Q.1. How do you do multiplication and division easily?
Ans: Multiplication
For example: When you want to multiply the number \(5\) by any even number: \(5 \times 4 = \)
You have to take the number which is being multiplied by \(5\) and cut it in half, which means the number \(4\) will become \(2.\)
Add the number zero beside the number \(2,\) which means you got the number \(20,\) i.e., \(5 \times 4 = 20.\)
When you want to multiply the number \(5 \times 4 = 20.\) by any odd number:
For example: \(5 \times 3 = \)
You have to subtract one from the number multiplied by \(5,\) which means the number \(3 – 1 = 2.\)
Now, again you have to half the number \(2\) that means \(2 – 1 = 1,\) and add the digit \(5\) beside the digit \(1,\) which makes \(5 \times 3 = 15\)
Division
For example: Division by \(5:\) Here, you have to simple-merely multiply the number by \(2\) and then divide the product you get by the number \(10.\)
If you are dividing the number \(65432\) by \(5\) then,
You will write as \(65432 \div 5 = \left( {65432 \times 2} \right) \div 10 = 130864 \div 10 = 13086.4\)

Q.2. Explain Multiplication and Division with example?
Ans: In the multiplication of the numbers, we find the product of the given numbers by multiplying them.
Example: \(3 \times 10 = 30\) or \(10 + 10 + 10 = 30\)
In division, we divide the numbers to get the missing factor when the other two factors are given. The division is also known as repeated subtraction.
Example: \(56 \div 7 = 8,\,56 \div 8 = 7\) or \(56 – 8 – 8 – 8 – 8 – 8 – 8 – 8.\)

Q.3. What are \(4\) ways to show multiplication?
Ans: 1. Multiply the numbers using repeated addition
2. Multiply the numbers using the long multiplication method.
3. Multiply the numbers using the grid method.
4. Multiply the numbers by splitting the numbers into ones, tens, hundreds (according to the place value).

Q.4. What are the symbols for multiplication?
Ans: The symbol we use for representing the multiplication is a cross sign \(\left( \times \right),\) and sometimes we also use the dot \(\left( * \right)\) to represent the product of numbers.

Q.5. What are the symbols for multiplication and division?
Ans: The symbol we use for the multiplication is a cross sign \(\left( \times \right),\) and sometimes we also use the dot \(\left( \cdot \right)\) or \(\left( * \right)\) to represent the product of numbers. The symbol which is used for the division of numbers is \( \div .\)

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