Operations on Rational Numbers - R.D. Sharma Class 7
Introduction
Rational numbers are a fundamental concept in mathematics that form the basis for understanding fractions, decimals, and more advanced mathematical concepts. This comprehensive guide, following R.D. Sharma's approach as presented in Class 7, will explain operations on rational numbers with clear explanations and examples.
Understanding Rational Numbers
A rational number is any number that can be expressed as a fraction p/q, where p and q are integers and q 0. Rational numbers include positive and negative whole numbers, fractions, and terminating or repeating decimals. For example, 1/2, -3/4, 5 (which can be written as 5/1), and 0.25 (which can be written as 1/4) are all rational numbers.
Addition of Rational Numbers
Adding Rational Numbers with Same Denominators
When adding rational numbers with the same denominators, we simply add the numerators and keep the denominator the same.
Adding Rational Numbers with Different Denominators
When adding rational numbers with different denominators, we first find the least common multiple (LCM) of the denominators, convert each fraction to an equivalent fraction with the LCM as the denominator, and then add the numerators.
Step 1: Find LCM of 5 and 4, which is 20
Step 2: Convert fractions:
2/5 = (24)/(54) = 8/20
3/4 = (35)/(45) = 15/20
Step 3: Add the numerators:
8/20 + 15/20 = (8+15)/20 = 23/20
Subtraction of Rational Numbers
Subtracting Rational Numbers with Same Denominators
Subtract the numerators while keeping the denominator the same.
Subtracting Rational Numbers with Different Denominators
Find the LCM of denominators, convert each fraction to an equivalent fraction with the LCM as the denominator, and then subtract the numerators.
LCM of 5 and 4 is 20
3/5 = 12/20
1/4 = 5/20
12/20 - 5/20 = (12-5)/20 = 7/20
Multiplication of Rational Numbers
Multiplying Rational Numbers
To multiply rational numbers, multiply the numerators together and multiply the denominators together.
Multiplication by Simplifying First
It's often easier to simplify before multiplying, especially with larger numbers.
First, simplify:
5/7 14/15 = 5/7 (27)/(35) = 5/7 (27)/(35)
Cancel common factors:
5/7 14/15 = 1/7 14/3 = 1/1 2/3 = 2/3
Division of Rational Numbers
Dividing Rational Numbers
To divide rational numbers, multiply the first fraction by the reciprocal of the second fraction.
Properties of Operations on Rational Numbers
- Closure Property: The sum, difference, and product of two rational numbers is always a rational number.
- Commutative Property: Addition and multiplication are commutative, meaning the order doesn't affect the result: a/b + c/d = c/d + a/b
- Associative Property: Addition and multiplication are associative: (a/b + c/d) + e/f = a/b + (c/d + e/f)
- Distributive Property: a/b (c/d + e/f) = a/b c/d + a/b e/f
Solved Examples
Example 1:
Perform the following operation: 3/5 + 2/3 + 1/2
LCM of 5, 3, and 2 is 30
3/5 = 18/30
2/3 = 20/30
1/2 = 15/30
18/30 + 20/30 + 15/30 = (18+20+15)/30 = 53/30
Example 2:
Find the product: 5/6 3/4 5/8
5/6 3/4 5/8 = 5/6 3/4 8/5
= (538)/(645)
= (134)/(641) [Simplifying 5s]
= (131)/(611) [Simplifying 4s]
= 3/6 = 1/2
Example 3:
The product of two rational numbers is 8/15. If one of them is 4/5, find the other.
Let the other number be x.
Then, 4/5 x = 8/15
x = 8/15 4/5
x = 8/15 5/4
x = (85)/(154)
x = 40/60
x = 2/3
Practice Problems
- Simplify: 2/5 + 3/7 - 1/2
- Find the product: 7/9 (-6/11) 33/14
- Divide: 5/6 10/9
- Verify: (-2/3) (3/4 + 5/6) = (-2/3) 3/4 + (-2/3) 5/6
- Find the number which when multiplied by -4/5 gives 8/15.
Summary
Operations on rational numbers form a critical foundation for advanced mathematical concepts. Mastering these operations requires understanding the rules for each operation and practicing with various examples. Remember to:
- Always simplify your answers to their lowest terms
- Convert mixed numbers to improper fractions before performing operations
- For division, multiply by the reciprocal
- Apply the properties of operations to simplify calculations
- Be careful with negative signs, especially when multiplying or dividing
With regular practice, you'll become proficient in performing operations on rational numbers and this skill will serve you well in higher mathematics.
```
R.D. Sharma Class 7 Operations On Rational Numbers and Reference File Download Link
R.D. Sharma Class 7 Integers Exercise Solutions and Reference File Download Link
R D Sharma Solutions For Class 10 Maths Chapter 7 Statistics Exercise 7.1 and Reference Fi...