GK: Class 12 General Q&A · Page 45
3,658 free answers on GK (General) for Class 12 students, across CBSE, ICSE and State boards.
Question & Answer
Questions
- GeneralClass 12All
What is the multiplicative inverse (reciprocal) of a rational number?
The multiplicative inverse of p/q is q/p (provided p ≠ 0). Property: A number multiplied by its reciprocal equals 1. (p/q) × (q/p) = 1 Examples: Reciprocal of 3/4 is 4/3 Reciprocal of -5/7 is -7/5 Reciprocal of 2 (or 2/1) is 1/2 Reciprocal of -1 is -1 Note: Zero (0) has no reciprocal because 1/0 is undefined.
- GeneralClass 12All
How do you divide rational numbers?
Multiply by the reciprocal (multiplicative inverse) of the divisor. Formula: (a/b) ÷ (c/d) = (a/b) × (d/c) = (a × d)/(b × c) Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 Another example: (-5/6) ÷ (3/7) = (-5/6) × (7/3) = -35/18
- GeneralClass 12All
How do you multiply two rational numbers?
Multiply the numerators together and the denominators together. Formula: (a/b) × (c/d) = (a × c)/(b × d) Example: 2/3 × 5/7 = (2 × 5)/(3 × 7) = 10/21 With negatives: (-3/4) × (2/5) = (-3 × 2)/(4 × 5) = -6/20 = -3/10 Rule for signs: Positive × Positive = Positive Negative
- GeneralClass 12All
What property is shown by: 2/3 + 5/7 = 5/7 + 2/3?
This demonstrates the Commutative Property of Addition. Definition: For any two rational numbers a and b: a + b = b + a The order of addition doesn't matter; the result is the same. Other examples: 1/4 + 3/5 = 3/5 + 1/4 -2/9 + 7/8 = 7/8 + (-2/9)
- GeneralClass 12All
Simplify: 1/2 + 1/3 + 1/6
Step 1: Find LCM(2, 3, 6) = 6 Step 2: Convert all fractions 1/2 = 3/6 1/3 = 2/6 1/6 = 1/6 Step 3: Add 3/6 + 2/6 + 1/6 = 6/6 = 1 Answer: 1
- GeneralClass 12All
Add: -3/7 + 5/7 - 2/7
Since all denominators are the same (7), simply add/subtract numerators: -3/7 + 5/7 - 2/7 = (-3 + 5 - 2)/7 = 0/7 = 0 Answer: 0
- GeneralClass 12All
What is the additive identity for rational numbers?
The additive identity for rational numbers is 0 (or 0/1). Property: For any rational number p/q: p/q + 0 = p/q Examples: 3/4 + 0 = 3/4 -5/7 + 0 = -5/7 0 + 2/3 = 2/3 Zero is called the additive identity because adding it to any number gives back the same number.
- GeneralClass 12All
Simplify: (2/5 + 3/10) - 1/4
Step 1: Add 2/5 + 3/10 LCM(5, 10) = 10 2/5 = 4/10 4/10 + 3/10 = 7/10 Step 2: Subtract 1/4 7/10 - 1/4 LCM(10, 4) = 20 7/10 = 14/20 1/4 = 5/20 14/20 - 5/20 = 9/20 Answer: 9/20
- GeneralClass 12All
What is the additive inverse of a rational number?
The additive inverse of p/q is -p/q (or equivalently, -(p/q)). Property: A number plus its additive inverse equals zero. p/q + (-p/q) = 0 Examples: Additive inverse of 3/5 is -3/5 Additive inverse of -7/9 is 7/9 Additive inverse of 0 is 0 Note: Also called the "opposite" or "negative" of a number.
- GeneralClass 12All
How do you subtract rational numbers?
Subtraction follows the same rules as addition, but subtract the numerators. Same denominator: p/q - r/q = (p - r)/q Different denominators: p/q - r/s = (p × s - r × q)/(q × s) Example: Subtract 5/6 - 1/4 Step 1: LCM(6, 4) = 12 Step 2: Convert 5/6 = 10/12 1/4 = 3/12 Step 3: Subtract 10/12 - 3
- GeneralClass 12All
What is the LCM method for adding rational numbers?
The LCM (Least Common Multiple) method finds the smallest common denominator. Steps: Find LCM of both denominators Convert each fraction to equivalent fraction with LCM as denominator Add numerators Simplify if needed Example: Add 1/4 + 1/6 Step 1: LCM(4, 6) = 12 Step 2: Convert 1/4 = 3/12 1/6 = 2/12 Step 3:<
- GeneralClass 12All
How do you add rational numbers with different denominators?
Use the cross-multiplication method or LCM method. Formula: p/q + r/s = (p × s + r × q)/(q × s) Example: Add 2/3 + 5/6 Step 1: Find LCM(3, 6) = 6 Step 2: Convert to equivalent fractions 2/3 = 4/6 5/6 = 5/6 Step 3: Add numerators 4/6 + 5/6 = 9/6 = 3/2 Answer: 3/2
- GeneralClass 12All
What is meant by "simplest form" or "lowest terms"?
A fraction is in simplest form (lowest terms) when the numerator and denominator have no common factors other than 1. Examples: 2/3 is in simplest form (GCD = 1) 8/12 is NOT in simplest form (GCD = 4), simplifies to 2/3 15/20 is NOT in simplest form (GCD = 5), simplifies to 3/4 Same as standard form: When denominator is also positive. Addition and Subtraction (Questions 21-30) 21. How do you
- GeneralClass 12All
What is the standard form of 0/5?
Step 1: Simplify 0/5 = 0 Step 2: Express as rational 0 = 0/1 Standard form: 0/1 or simply 0 Note: 0/n = 0 for any non-zero integer n. 19. Can -6/-8 be written in standard form? Yes. Step 1: Simplify the signs -6/-8 = 6/8 (negative ÷ negative = positive) Step 2: Find GCD(6, 8) GCD
- GeneralClass 12All
Express 36/-48 in standard form.
Step 1: Make denominator positive 36/-48 = -36/48 Step 2: Find GCD(36, 48) GCD = 12 Step 3: Divide both by GCD -36 ÷ 12 = -3 48 ÷ 12 = 4 Standard form: -3/4
- GeneralClass 12All
How do you check if two rational numbers are equal?
There are two methods: Method 1: Standard Form Convert both to standard form If they have the same standard form, they're equal Method 2: Cross Multiplication For a/b and c/d: if a × d = b × c, then a/b = c/d Example: Are 6/9 and 4/6 equal? Method 1: 6/9 = 2/3 (standard form) 4/6 = 2/3 (standard form) They'r
- GeneralClass 12All
What is GCD and why is it important for rational numbers?
GCD (Greatest Common Divisor) is the largest positive integer that divides both numbers without remainder. Importance: Used to reduce fractions to standard form Helps identify if two fractions are equivalent Essential for simplifying rational number operations Example: Find GCD of 24 and 36 Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24 Factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, 36 Comm
- GeneralClass 12All
How do you find equivalent rational numbers?
To find equivalent rational numbers of p/q: Method 1: Multiply both numerator and denominator by the same non-zero integer Example for 2/3: 2/3 = 4/6 (multiply by 2) 2/3 = 6/9 (multiply by 3) 2/3 = 10/15 (multiply by 5) Method 2: Divide both numerator and denominator by their common factor (to simplify) Example for 12/18: 12/18 = 6/9 (divide by 2)<
- GeneralClass 12All
What are equivalent rational numbers?
Equivalent rational numbers are different representations of the same rational number. They have the same value but different numerators and denominators. Property: If p/q is a rational number and m is a non-zero integer, then: p/q = (p × m)/(q × m) Example: These are all equivalent to 3/4: 6/8 (multiply by 2) 9/12 (multiply by 3) 15/20 (multiply by 5) -21/-28 (multiply by -7)<
- GeneralClass 12All
How do you convert a rational number to standard form?
Follow these steps: Step 1: If the denominator is negative, multiply both numerator and denominator by -1 Step 2: Find the GCD (Greatest Common Divisor) of the numerator and denominator Step 3: Divide both numerator and denominator by their GCD Example: Convert -8/28 to standard form GCD(8, 28) = 4 -8 ÷ 4 = -2 28 ÷ 4 = 7 Standard form: -