The Distributive Property Formula is applied when multiplying a given number by the sum of two numbers. In such cases, first, a number is added, then multiply them by the provided number, following the order of operations rule.

When you have mastered numbers, you will, in fact, no longer be reading numbers any more than you read words when reading books. You will be reading meanings. There are three main properties: associative property, distributive property, and commutative property, with which we interact again and again while learning the properties of numbers.

In this article, we will cover about distributive property formula. As the name itself says, distributive means distributing or giving a part or share of something to all. The other name of the distributive property formula is the distributive law of multiplication.

Properties of Rational Numbers

Rational numbers are those numbers that can be represented in \(\frac{p}{q}\) form, where \(q \ne 0.\) Natural numbers, whole numbers, and integers are all rational numbers as all of them can be written in \(\frac{p}{q}\) form with denominator \(1.\) There are some properties involved in the basic mathematical operations done on rational numbers. The main properties are closure property, commutative property, associative property, distributive property, and the existence of identity and inverse. Let us understand these properties in brief.

1. Closure property: If a mathematical operation is closed for a number system, the result will also belong to the same number system when we do the operation.
If \(x\) and \(y\) are two rational numbers, then if \(x + y\) is also a rational number, then the addition is closed under rational numbers.
For example, if \(x = 6\) and \(y = 7,\) then \(x + y = 6 + 7 = 13,\) which is a rational number.
The addition is closed for natural numbers, whole numbers, integers, and rational numbers.
The subtraction is closed for integers and rational numbers but not for whole numbers and natural numbers.
For example, If \(x = 20\) and \(y = 17,\,x – y = 20 – 17 = 3,\) which is a rational number but, if \(x = 1\) and \(y = 20,\,x – y = 1 – 20 = \, – 19,\) which is not a whole number.
Multiplication is also closed under all number systems, but the division is not. As division by zero is not defined.

2. Commutative property: If the order of the operands doesn’t influence the answer of operation, then that operation is said to be commutative.
If \(x\) and \(y\) are two rational numbers, then \(x + y = y + x.\)
For example, if, \(x = 6\) and \(y = 7\) then \(x + y = 6 + 7 = 13\) and \(y + x = 7 + 6 = 13,\) and thus \(x + y = y + x.\)
The addition and multiplication are commutative for all number systems. The subtraction and division are not commutative.
If \(x\) and \(y\) are two whole numbers, then \(x – y \ne y – x.\) For example,
If \(x=20\) and \(y = 17,\,x – y = 20 – 17 = 3\) and \(y – x = 17 – 20 = \, – 3.\)
Therefore, \(x – y \ne y – x.\)

3. Associative property: If the grouping of operands doesn’t change the result, then the operation is said to be associative.
For any three rational numbers \(x,\,y\) and \(z,\,x + \left( {y + z} \right) = \left( {x + y} \right) + z\)
For example, consider \(x = 8,\,y = 9\) and \(z = 10\)
\(x + \left( {y + z} \right) = 8 + \left( {9 + 10} \right) = 8 + 18 = 27\)
\(\left( {x + y} \right) + z = \left( {8 + 9} \right) + 10 = 17 + 10 = 27\)
And, thus, \(x + \left( {y + z} \right) = \left( {x + y} \right) + z.\)
The addition and multiplication are associative, whereas subtraction and division are not.
For any three whole numbers \(x,\,y\) and \(z,\,x – \left( {y – z} \right) \ne \left( {x – y} \right) – z,\) that is, the subtraction of whole numbers does not satisfy associativity.
For example, if \(x = 20,\,y = 15\) and \(z = 12\)
\(x – \left( {y – z} \right) = 20 – \left( {15 – 12} \right) = 20 – 3 = 17\) and \(\left( {x – y} \right) – z = \left( {20 – 15} \right) – 12 = 5 – 12 = \, – 7\)
Therefore, \(x – \left( {y – z} \right) \ne \left( {x – y} \right) – z.\)

Existence of Identity and Inverse

Additive Identity: An additive identity is a number that gives the sum as the number itself when added to any number, for any rational number \(x + 0 = 0 + x = x.\)
\(0\) is the additive identity.

Additive Inverse: The additive inverse is any number that can bring the final result to zero. For any rational number \(x,\) its additive inverse will be \(-x,\) as \(x + \left( { – x} \right) = 0\)
Multiplicative Identity: The multiplication of any whole number with \(1\) is the number itself. For any whole number \(x,\,x \times 1 = x,\,1 \times x = x,\) i.e., \(x \times 1 = 1 \times x.\)
Multiplicative Inverse: If \(x\) is any whole number \(\left( {x \ne 0} \right),\) then its multiplicative inverse will be \(\frac{1}{x}.\) So that, \(x \times \frac{1}{x} = 1,\) but \(\frac{1}{x}\) is a whole number only when \(x = 1.\)
Now, let us learn in detail about the distributive property.

Distributive Property

The distributive property states that multiplying the sum of two or more addends by a number will multiply each addend individually by adding the products together. The distributive property holds true for multiplication over addition and subtraction.

The distributive property formula states that for any expression of the form \(x\left( {y + z} \right),\) which means \(x \times \left( {y + z} \right),\) operand \(‘x’\) can be distributed among operands \(‘y’\) and \(‘z’\) as \(\left( {x \times y + y \times z} \right)\) i.e., \(x\left( {y + z} \right) = x \times y + y \times z\) or, \(\left( {p + q} \right)\left( {r + s} \right) = pr + ps + qr + qs\)
The distributive property formula of a given value can thus be expressed as,

Distributive Property of Multiplication Over Addition & Subtraction

For any three whole numbers \(x,\,y\) and \(z\)
\(x \times \left( {y + z} \right) = x \times y + x \times z\)
In other words, the multiplication of rational numbers is distributive over their addition and subtraction.
Consider the following examples,
Let \(x = 5,\,y = 3\) and \(z = 4\)

\(x \times \left( {y + z} \right)\) \( = 5 \times \left( {3 + 4} \right)\) \( = 5 \times 7\) \( = 35\)

\(x \times y + x \times z\) \( = 5 \times 3 + 54\) \( = 15 + 20\) \( = 35\)

In the same way, if we take more sets of values of \(x,\,y\) and \(z,\) every time, the results verify the distributivity of multiplication over addition.

The multiplication of rational numbers is also distributive over their subtraction, i.e., \(a \times \left( {b – c} \right) = a \times b – a \times c\) and \(\left( {b – c} \right) \times a = bc – ba,\) if \(b\) is greater than \(c.\)
For three ratioanl numbers \(x = 10,\,y = 15\) and \(z = 5\)
\(x \times \left( {y – z} \right) = 10 \times \left( {15 – 5} \right) = 10 \times 10 = 100\)
\(\left( {x \times y} \right) – \left( {x \times z} \right) = \left( {10 \times 15} \right) – \left( {10 \times 5} \right) = 150 – 50 = 100\)
Therefore, \(x \times \left( {y – z} \right) = x \times y – x \times z.\)
Consider any three integers, say, \(10,\, – 6\) and \(13.\) Multiply one of them by the sum of the other two in different ways.

\(10 \times \left[ { – 6 + 13} \right]\) \( = 11 \times \left( 7 \right)\) \( = 70\)

\(10 \times \left[ { – 6 + 13} \right]\) \( = 10 \times \left( { – 6} \right) + 10 \times \left( {13} \right)\) \( = \, – 60 + 130 = 70\)

As we can see, in both cases, we get the same result.
Therefore \(10 \times \left[ { – 6 + 13} \right] = 10 \times \left( { – 6} \right) + 10 \times \left( {13} \right)\)
Let us take another group of three integers, \( – 7,\,14\) and \( – 5,\) and repeat the process.

\( – 7 \times \left[ {14 + \left( { – 5} \right)} \right]\) \( = \, – 7 \times \left( 9 \right)\) \( = \, – 63\)

\( – 7 \times \left[ {14 + \left( { – 5} \right)} \right]\) \( = \, – 7 \times 14 + \left( { – 7} \right) \times \left( { – 5} \right)\) \( = \, – 98 + 35 = \, – 63\)

Here also we get the same result in both cases.
Therefore, \( – 7 \times \left[ {14 + \left( { – 5} \right)} \right] = \, – 7 \times 14 + \left( { – 7} \right) \times \left( { – 5} \right)\)
Hence, we can summarise in general, if \(x, y\) and \(z\) are ant three integers, then \(x \times \left( {y + z} \right) = \left( {x \times y} \right) + \left( {y \times z} \right).\) This is called the distributive property of integers over addition.
Similarly, with the help of three integers, we can show that,
\(x \times \left( {y – z} \right) = \left( {x \times y} \right) – \left( {y \times z} \right)\)
Thus, multiplication is distributive over addition and subtraction.

Applications of Distributive Property

The distributive property is one of the most important and most used properties in mathematics. We use the distributive property to simplify the mathematical equation effortlessly. The distributive property helps us easily doing complex multiplications.

Let us have a look at some examples for a better understanding.

Example 1: \(15 \times 235\)
Solution: \(15 \times 235 = 15 \times \left( {200 + 30 + 5} \right)\)
\( = 15 \times 200 + 15 \times 30 + 15 \times 5\)
\( = 3000 + 450 + 75\)
\( = 3525\)
Hence, the required answer is \(3525.\)

Example 2: \(27 \times 98\)
Solution: \(27 \times 98 = 27 \times \left( {100 – 2} \right)\)
\( = 27 \times 100 – 27 \times 2\)
\( = 2700 – 54\)
\( = 2646\)
Hence, the required answer is \(2646.\)

Example 3: \(35672 \times 20\)
Solution: \(35672 \times 20 = \left( {30000 + 5000 + 600 + 70 + 2} \right)\)
\( = 30000 \times 20 + 5000 \times 20 + 600 \times 20 + 70 \times 20 + 2 \times 20\)
\( = 600000 + 10000 + 12000 + 1400 + 40\)
\( = 713440\)
Hence, the required answer is \(713440.\)

Solved Examples

Q.1. Use distributive law to evaluate: \(342 \times 197\)
Ans: \(342 \times 197 = 342 \times \left( {200 – 3} \right)\)
Now, the multiplication of rational numbers is also distributive over their subtraction, i.e., \(a \times \left( {b – c} \right) = a \times b – a \times c\)
Thus,
\( = 342 \times 200 – 342 \times 3\)
\( = 68400 – 1026\)
\( = 67374\)
Hence, the required answer is \(67374.\)

Q.2. Use the distributive property to evaluate the value of \(479 \times 87 + 479 \times 13.\)
Ans: For any three whole numbers, \(x,\,y\) and \(z\)
\(x \times y + x \times z\) can be written as \(x \times \left( {y + z} \right)\)
Therefore, \(479 \times 87 + 479 \times 13 = 479 \times \left( {87 + 13} \right)\)
\( = 479 \times 100\)
\( = 47900\)
Hence, the required answer is \(47900.\)

Q.3. Find the value of \(p\) in \(p \times \left[ {15 + \left( { – 7} \right)} \right] = 4 \times 15 + 4 \times – 7.\)
Ans: Using the distributive property, we can write
\(4 \times 15 + 4 \times \left( { – 7} \right)\) as \(4 \times \left[ {15 + \left( { – 7} \right)} \right]\)
Thus, \(p \times \left[ {15 + \left( { – 7} \right)} \right] = 4 \times 15 + 4 \times \left( { – 7} \right)\)
Or \(p \times \left[ {15 + \left( { – 7} \right)} \right] = 4 \times \left[ {15 + \left( { – 7} \right)} \right]\)
By comparing LHS and RHS, we get,
\(p = 4\)
Hence, the value of \(p\) is equal to \(4.\)

Q.4. Verify and name the property used in the following. \( – 18 \times \left[ {\left( { – 5} \right) + \left( { – 3} \right)} \right] = \left( { – 18} \right) \times \left( { – 5} \right) + \left( { – 18} \right) \times \left( { – 3} \right)\)
Ans: LHS \( = \, – 18 \times \left[ {\left( { – 5} \right) + \left( { – 3} \right)} \right]\)
\( = \left( { – 18} \right) \times \left( { – 8} \right) = 144\)
RHS \( = \left( { – 18} \right) \times \left( { – 5} \right) + \left( { – 18} \right) \times \left( { – 3} \right)\)
\( = 90 + 54\)
\( = 144\)
LHS \(=\) RHS
Here, we have used the distributive property of multiplication over addition.

Q.5. Find the product of the greatest number of \(3\) digits and the greatest number of \(2\) digits.
Ans: The greatest number of \(3\) digits \( = 999\) and the greatest number of \(2\) digits \( = 99\)
Therefore, \(999 \times 99\)
\( = 999 \times \left( {100 – 1} \right)\)
Using distributive property, we get,
\( = 999 \times 100 – 999 \times 1\)
\( = \left( {1000 – 1} \right) \times 100 – 1 \times 100 – 1000 + 1\)
\( = 100000 – 100 – 1000 + 1\)
\(100001 – 1100 = 98901\)
Hence, the required answer is \(98901.\)

Q.6. Evaluate the expression \(3333 \times 987 + 13 \times 3333,\) using the distributive property.
Ans: For any three whole numbers \(x,\,y\) and \(z,\,x \times y + x \times z = x \times \left( {y + z} \right)\)
Now, the given expression is, \(3333 \times 987 + 13 \times 3333\)
Thus, \(3333 \times 987 + 13 \times 3333\) can be written as \(3333 \times \left( {987 + 13} \right)\)
Therefore, \(3333 \times \left( {987 + 13} \right) = 3333 \times 1000\)
\( = 3333000\)
Hence, the required answer is \(3333000.\)

Summary

In this article, we had a quick view of the properties of the numbers systems. We learned the closure property, commutative property, associative property and identity, and inverse properties for natural numbers, whole numbers, integers, and rational numbers.

Then we had a detailed discussion about the distributive property formula. We now know the application and usage of distributive property formulas in all the number systems. And finally, with the help of examples, we mastered the distributive property formula.

Learn About Properties of Real Numbers

FAQs

Q.1. What is the distributive property formula?
Ans: The distributive property formula states that for any expression of the form \(x\left( {y + z} \right),\) which means \(x \times \left( {y + z} \right),\) operand \(‘x’\) can be distributed among operands \(‘y’\) and \(‘z’\) as \(\left( {x \times y + y \times z} \right).\) That is,
\(x\left( {y + z} \right) = x \times y + y \times z\)
or, \(\left( {p + q} \right)\left( {r + s} \right) = pr + ps + qr + qs\)

Q.2. Does the distributive property come before division?
Ans: No, the distributive property does not come before division.

Q.3. When to use the distributive property?
Ans: Whenever there is a need to multiply a common number to a sum or difference of numbers, we use the distributive property.
That is,
\(x\left( {y + z} \right) = x \times y + y \times z\)
\(x\left( {y + z} \right) = x \times y – y \times z\)

Q.4. What is the distributive property of addition?
Ans: According to the distributive property, multiplying the sum of two or more addends by a number will multiply each addend individually by the number and then add the products together. The distributive property holds true for multiplication over addition and subtraction.

Q.5. If there is multiplication or division inside the parentheses, can the distributive property work?
Ans: No, the distributive property cannot work if there is multiplication or division inside the parentheses. The distributive property of multiplication lets us simplify expressions in which we multiply a number by a sum or difference.

We hope this detailed article on the distributive property formula helped you in your studies. If you have any doubts or queries regarding this topic, feel to ask us in the comment section. Happy learning!