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What is the Distributive Property for Rational Numbers?
Grade Level:
Class 7
AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering
Definition
What is it?
The Distributive Property for Rational Numbers tells us how multiplication behaves with addition or subtraction. It means we 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 3 packets of biscuits for your friends. Each packet costs Rs 10, and you also buy 3 small chocolates for Rs 5 each. Instead of calculating (10+5) for one friend and then multiplying by 3, you can calculate (3 x 10) for biscuits and (3 x 5) for chocolates, then add them up. Both ways give the same total cost!
Worked Example
Step-by-Step
Let's use the Distributive Property to solve: 3/4 x (2/5 + 1/2)
STEP 1: Identify the rational number outside the bracket and the rational numbers inside. Here, 3/4 is outside, and 2/5 and 1/2 are inside.
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STEP 2: Distribute the number outside to each number inside the bracket through multiplication. So, it becomes (3/4 x 2/5) + (3/4 x 1/2).
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STEP 3: Perform the first multiplication: 3/4 x 2/5 = (3 x 2) / (4 x 5) = 6/20.
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STEP 4: Perform the second multiplication: 3/4 x 1/2 = (3 x 1) / (4 x 2) = 3/8.
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STEP 5: Now, add the results: 6/20 + 3/8. To add, find a common denominator. The LCM of 20 and 8 is 40.
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STEP 6: Convert fractions: 6/20 = (6 x 2) / (20 x 2) = 12/40. And 3/8 = (3 x 5) / (8 x 5) = 15/40.
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STEP 7: Add the converted fractions: 12/40 + 15/40 = 27/40.
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ANSWER: So, 3/4 x (2/5 + 1/2) = 27/40.
Why It Matters
Understanding the Distributive Property helps simplify complex calculations in daily life and in advanced fields. Engineers use it to design structures efficiently, and data scientists use it to optimize algorithms. It's a fundamental concept that makes problem-solving much easier in areas like Computer Science and Physics.
Common Mistakes
MISTAKE: Multiplying the outside number only by the first term inside the bracket and forgetting the rest. For example, 2 x (3 + 5) becomes (2 x 3) + 5. | CORRECTION: Remember to multiply the outside number by *every* term inside the bracket. So, 2 x (3 + 5) becomes (2 x 3) + (2 x 5).
MISTAKE: Confusing the Distributive Property with just multiplying the bracket first. For example, solving 1/2 x (3/4 + 1/5) by first adding 3/4 + 1/5 and then multiplying by 1/2, when the question specifically asks to use the distributive property. | CORRECTION: The Distributive Property is a *method* for calculation, not just getting the answer. It requires you to distribute the multiplication first: (1/2 x 3/4) + (1/2 x 1/5).
MISTAKE: Incorrectly handling negative signs when distributing. For example, 2 x (3 - 4) becomes (2 x 3) - 4. | CORRECTION: The sign inside the bracket belongs to the number. So, 2 x (3 - 4) becomes (2 x 3) + (2 x -4) or (2 x 3) - (2 x 4). Both are correct, but be careful with the sign of the second term.
Practice Questions
Try It Yourself
QUESTION: Use the Distributive Property to solve: 1/3 x (6/7 + 9/5) | ANSWER: 2/7 + 3/5 = 31/35
QUESTION: Simplify using the Distributive Property: -2/5 x (10/3 - 15/4) | ANSWER: -4/3 + 3/2 = 1/6
QUESTION: A shopkeeper buys 7 kg of apples at Rs 80/kg and 7 kg of oranges at Rs 60/kg. Use the Distributive Property to find the total cost. | ANSWER: 7 x (80 + 60) = 7 x 80 + 7 x 60 = 560 + 420 = Rs 980
MCQ
Quick Quiz
Which of the following expressions correctly uses the Distributive Property for 4/5 x (1/2 - 3/4)?
(4/5 x 1/2) - 3/4
(4/5 x 1/2) - (4/5 x 3/4)
4/5 x (1/2 - 3/4)
1/2 - (4/5 x 3/4)
The Correct Answer Is:
B
Option B correctly distributes 4/5 to both 1/2 and 3/4, maintaining the subtraction sign. Options A, C, and D do not apply the property correctly to all terms inside the bracket.
Real World Connection
In the Real World
When a software developer calculates discounts on multiple items in an e-commerce app like Flipkart or Amazon, they might use this property. For example, if there's a 10% discount on all items in your cart, instead of calculating the discount for each item separately and then summing them up, the app can sum the original prices first and then apply the 10% discount to the total, or apply 10% to each item and then sum. Both methods use the distributive idea.
Key Vocabulary
Key Terms
RATIONAL NUMBER: A number that can be expressed as a fraction p/q, where p and q are integers and q is not zero. | DISTRIBUTE: To spread or share something among a group. | MULTIPLICATION: The process of repeatedly adding a number to itself. | ADDITION: The process of combining two or more numbers to find their sum. | SUBTRACTION: The process of taking one number away from another.
What's Next
What to Learn Next
Great job understanding the Distributive Property! Next, you can explore other properties of rational numbers like Associativity and Commutativity. These properties will help you simplify even more complex algebraic expressions and prepare you for higher-level math.
