Multiplication And Distributive Property

Student can exploit the distributive property to simplify large multiplications mentally.

6 lessons

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Multiplication Shortcuts

Base properties of multiplication.

Before diving into complex calculations, let's review four simple rules that make multiplying whole numbers much easier to do in your head.

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Diagram

The Distributive Property

Visual branching of distributive property.

$A polished process flowchart showing the distributive property of multiplication. The top node displays the equation 5 times (3 + 4). This splits into two separate downward branches: one calculating (5 times 3) and the other calculating (5 times 4). These branches converge at the bottom into a single node showing 15 + 20, leading to a final result of 35. polished process flowchart, rounded card-style nodes, pastel gradient fills, elegant sans-serif typography, generous whitespace, subtle connecting arrows, light neutral background$

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Distributing the multiplier across the terms inside the addition bracket.

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Worked Example

Expanding to Multiply

Using distribution over subtraction for 97.

Problem

Multiply $172 \times 97$

Notice that $97$ is very close to $100$ . Multiplying large numbers mentally becomes much easier if we expand one of them to a convenient base using the Distributive Property.

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Worked Example

Taking Out Common Factors

Using distributive property in reverse.

Problem

Solve $569 \times 45 + 569 \times 55$

Instead of doing two separate large multiplications, look for a common factor in both products. This is the reverse of expanding a bracket.

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reading

Complex Factoring

Three-term factoring using the property.

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Let us solve the numerical expression $361 \times 162 - 361 \times 60 - 2 \times 361$ using the distributive property.

First, we identify and take out the common factor from all three products. The isolated common factor placed outside the bracket is .

By putting the rest of the terms inside a bracket, the expression becomes $361 \times (162 - 60 - 2)$ .

Next, we simplify the subtraction inside the parentheses. Evaluating $162 - 60 - 2$ gives us a simple multiplier of .

Finally, we multiply our common factor by this new value to easily find the solution. The final evaluated product is , which is much faster to calculate mentally than evaluating each term separately.

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Word Problem: Distributive Property

Real-world application requiring distribution.

Use the Distributive Property to solve this real-world purchasing problem.

Problem. Rohan buys $12$ computers and $12$ printers. If the cost of one computer is ₹ $56,233$ and one printer is ₹ $7,867$ , find the total cost incurred by Rohan.

Hint: Use the distributive property: $a \times b + a \times c = a \times (b + c)$ .

Step 1: Write the expression*

Write the total cost using multiplication for computers and printers.

Step 2: Take out the common factor*

Both terms have the same number of items. Take that number outside the bracket.

Step 3: Add inside the bracket*

Add the cost of one computer and one printer.

Step 4: Multiply*

Multiply the bracket value by 12.

Final Answer*

Write the total cost in rupees.

← previousProperties Of Addition And Subtraction next →Division Principles And Algorithm

Worked Example

Expanding to Multiply

Using distribution over subtraction for 97.

Problem

Multiply $172 \times 97$

Notice that $97$ is very close to $100$ . Multiplying large numbers mentally becomes much easier if we expand one of them to a convenient base using the Distributive Property.

1 / 3

Worked Example

Taking Out Common Factors

Using distributive property in reverse.

Problem

Solve $569 \times 45 + 569 \times 55$

Instead of doing two separate large multiplications, look for a common factor in both products. This is the reverse of expanding a bracket.

1 / 3

reading

Complex Factoring

Three-term factoring using the property.

0 of 3 blanks filled

Let us solve the numerical expression $361 \times 162 - 361 \times 60 - 2 \times 361$ using the distributive property.

First, we identify and take out the common factor from all three products. The isolated common factor placed outside the bracket is .

By putting the rest of the terms inside a bracket, the expression becomes $361 \times (162 - 60 - 2)$ .

Next, we simplify the subtraction inside the parentheses. Evaluating $162 - 60 - 2$ gives us a simple multiplier of .

Finally, we multiply our common factor by this new value to easily find the solution. The final evaluated product is , which is much faster to calculate mentally than evaluating each term separately.

feedback

Word Problem: Distributive Property

Real-world application requiring distribution.

Use the Distributive Property to solve this real-world purchasing problem.

Problem. Rohan buys $12$ computers and $12$ printers. If the cost of one computer is ₹ $56,233$ and one printer is ₹ $7,867$ , find the total cost incurred by Rohan.

Hint: Use the distributive property: $a \times b + a \times c = a \times (b + c)$ .

Step 1: Write the expression*

Write the total cost using multiplication for computers and printers.

Step 2: Take out the common factor*

Both terms have the same number of items. Take that number outside the bracket.

Step 3: Add inside the bracket*

Add the cost of one computer and one printer.

Step 4: Multiply*

Multiply the bracket value by 12.

Final Answer*

Write the total cost in rupees.