What is the Distributive Property for Integers?

S3-SA4-0192

Grade Level:

Class 7

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The Distributive Property for Integers helps us multiply a single number by a sum or difference of two or more numbers. It means we can 'distribute' the multiplication to each number inside the bracket separately and then add or subtract the results. This property simplifies calculations by breaking down complex multiplications.

Simple Example

Quick Example

Imagine you buy 2 packets of biscuits, and 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), you can multiply 2 x 5 (chocolate biscuits) and 2 x 3 (plain biscuits) separately, then add the results. This gives you 10 + 6 = 16 biscuits in total.

Worked Example

Step-by-Step

Let's solve: 7 x (5 + (-3))

Step 1: Identify the numbers. We have 7 multiplying the sum of 5 and -3.
---Step 2: Apply the Distributive Property. This means 7 will multiply 5, AND 7 will multiply -3. So, it becomes (7 x 5) + (7 x (-3)).
---Step 3: Perform the first multiplication: 7 x 5 = 35.
---Step 4: Perform the second multiplication: 7 x (-3) = -21 (Remember, a positive integer multiplied by a negative integer gives a negative integer).
---Step 5: Now, add the results from Step 3 and Step 4: 35 + (-21).
---Step 6: Adding a negative number is the same as subtracting the positive number: 35 - 21 = 14.
---Step 7: The answer is 14.

Alternatively, solving inside the bracket first: 7 x (5 + (-3)) = 7 x (5 - 3) = 7 x 2 = 14. Both methods give the same answer!

Why It Matters

The Distributive Property is a fundamental building block for algebra and helps simplify complex equations in Data Science and Computer Science. Engineers use it to design efficient systems, and economists might use it to calculate total costs or revenues. Understanding this property is key to solving problems in higher mathematics and real-world applications.

Common Mistakes

MISTAKE: Only multiplying the first number inside the bracket. For example, in 4 x (2 + 3), students might write (4 x 2) + 3 instead of (4 x 2) + (4 x 3). | CORRECTION: Remember to multiply the number outside the bracket by EVERY term inside the bracket.

MISTAKE: Forgetting integer rules when distributing. For example, in 5 x (2 - 7), writing (5 x 2) - 7 instead of (5 x 2) + (5 x (-7)). | CORRECTION: Treat subtraction as adding a negative number (e.g., 2 - 7 is 2 + (-7)) to ensure correct sign multiplication.

MISTAKE: Applying the property incorrectly when there are more than two terms. For example, in 3 x (a + b + c), only doing (3 x a) + (3 x b) + c. | CORRECTION: The property applies to all terms inside the bracket. So it should be (3 x a) + (3 x b) + (3 x c).

Practice Questions

Try It Yourself

QUESTION: Use the Distributive Property to solve: 6 x (4 + (-2)) | ANSWER: 12

QUESTION: Simplify using the Distributive Property: -5 x (8 - 3) | ANSWER: -25

QUESTION: A shopkeeper sells 3 types of pens. He sold 10 blue pens, 5 red pens, and 7 black pens. If each pen costs Rs. 8, write an expression using the Distributive Property to find the total cost and then calculate it. | ANSWER: 8 x (10 + 5 + 7) = (8 x 10) + (8 x 5) + (8 x 7) = 80 + 40 + 56 = 176. Total cost is Rs. 176.

MCQ

Quick Quiz

Which of the following expressions correctly uses the Distributive Property for 9 x (7 - 4)?

(9 x 7) + 4

(9 x 7) - 4

(9 x 7) - (9 x 4)

9 x (7 + 4)

The Correct Answer Is:

Option C correctly distributes the multiplication of 9 to both 7 and 4, maintaining the subtraction operation between them. Options A and B only multiply the first term or apply the operation incorrectly. Option D changes the original operation.

Real World Connection

In the Real World

When a small business owner in India calculates monthly expenses, they might use the Distributive Property. For example, if they have 3 employees, and each gets a salary of Rs. 15,000 and a travel allowance of Rs. 2,000, the total expense can be calculated as 3 x (15,000 + 2,000) = (3 x 15,000) + (3 x 2,000). This helps in budgeting and financial planning.

Key Vocabulary

Key Terms

DISTRIBUTE: To spread out or give out to each part | INTEGER: A whole number (positive, negative, or zero) | PROPERTY: A rule or characteristic in mathematics | EXPRESSION: A mathematical phrase that can contain numbers, variables, and operators

What's Next

What to Learn Next

Great job mastering the Distributive Property for Integers! Next, you can explore the Distributive Property for Rational Numbers. This will build on what you've learned and help you apply this powerful tool to fractions and decimals, opening up even more problem-solving possibilities!