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What is the Distributive Property of Natural Numbers?
Grade Level:
Class 6
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The Distributive Property tells us how multiplication behaves with addition or subtraction. It means you can multiply a number by a sum (or difference) by multiplying that number by each part of the sum (or difference) separately, and then adding (or subtracting) the results.
Simple Example
Quick Example
Imagine you buy 2 packets of biscuits. Each packet has 5 chocolate biscuits and 3 plain biscuits. Instead of first adding 5+3=8 biscuits per packet and then multiplying by 2 (2 x 8 = 16), you can find the chocolate biscuits (2 x 5 = 10) and plain biscuits (2 x 3 = 6) separately and then add them (10 + 6 = 16). The total is the same!
Worked Example
Step-by-Step
Let's calculate 7 x (10 + 3) using the Distributive Property.
Step 1: Identify the number outside the bracket and the numbers inside.
Here, the number outside is 7, and inside are 10 and 3.
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Step 2: Distribute (multiply) the outside number by each number inside the bracket separately.
This means we do 7 x 10 and 7 x 3.
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Step 3: Calculate each multiplication.
7 x 10 = 70
7 x 3 = 21
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Step 4: Add the results from Step 3.
70 + 21 = 91
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Step 5: Compare with direct calculation (optional, for checking).
10 + 3 = 13
7 x 13 = 91
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Answer: 7 x (10 + 3) = 91
Why It Matters
Understanding the Distributive Property makes calculations easier and is a fundamental skill in algebra. It's used in computer programming to optimize calculations, in data science to process information efficiently, and even in economics to model how costs are spread. Engineers use it to design efficient systems.
Common Mistakes
MISTAKE: Multiplying the outside number by only the first number inside the bracket. For example, calculating 5 x (2 + 4) as (5 x 2) + 4 = 10 + 4 = 14. | CORRECTION: Remember to multiply the outside number by *every* number inside the bracket. So, 5 x (2 + 4) = (5 x 2) + (5 x 4) = 10 + 20 = 30.
MISTAKE: Applying the property incorrectly with subtraction, like 6 x (7 - 2) as (6 x 7) - (6 - 2). | CORRECTION: The outside number multiplies each term inside, and the operation inside (addition or subtraction) remains between the results. So, 6 x (7 - 2) = (6 x 7) - (6 x 2) = 42 - 12 = 30.
MISTAKE: Forgetting to put the operation sign (plus or minus) between the distributed terms. For example, writing 3 x (5 + 2) as (3 x 5)(3 x 2). | CORRECTION: Always place the original operation sign (addition or subtraction) between the products. So, 3 x (5 + 2) = (3 x 5) + (3 x 2) = 15 + 6 = 21.
Practice Questions
Try It Yourself
QUESTION: Calculate 4 x (8 + 2) using the Distributive Property. | ANSWER: (4 x 8) + (4 x 2) = 32 + 8 = 40
QUESTION: Use the Distributive Property to solve 9 x (15 - 5). | ANSWER: (9 x 15) - (9 x 5) = 135 - 45 = 90
QUESTION: A shopkeeper sells 6 boxes of pens. Each box has 10 blue pens and 7 black pens. How many pens are there in total using the Distributive Property? | ANSWER: 6 x (10 + 7) = (6 x 10) + (6 x 7) = 60 + 42 = 102 pens
MCQ
Quick Quiz
Which expression correctly shows the Distributive Property for 8 x (5 + 3)?
8 x 5 + 3
8 + 5 x 3
(8 x 5) + (8 x 3)
(8 + 5) x (8 + 3)
The Correct Answer Is:
C
Option C correctly shows that the number outside the bracket (8) is multiplied by each term inside the bracket (5 and 3) separately, and then the results are added. Options A, B, and D do not follow this rule.
Real World Connection
In the Real World
When calculating your monthly mobile data usage for a family plan, if 3 family members each use 5 GB for streaming and 2 GB for social media, you can use the Distributive Property. Total data is 3 x (5 + 2) GB, which is (3 x 5) + (3 x 2) = 15 + 6 = 21 GB. This helps telecom companies manage resources and bill customers.
Key Vocabulary
Key Terms
DISTRIBUTE: To spread something out or give a share to each of a group. | PROPERTY: A characteristic or rule in mathematics. | NATURAL NUMBERS: Counting numbers starting from 1 (1, 2, 3, ...). | EXPRESSION: A mathematical phrase that can contain numbers, variables, and operators.
What's Next
What to Learn Next
Great job learning the Distributive Property! Next, you can explore other properties of natural numbers like the Commutative and Associative Properties. These properties are building blocks for understanding more complex algebra and making calculations even faster!
