Rational Numbers: Class 7 Mathematics Q&A
36 free answers on Rational Numbers (Mathematics) for Class 7 students, across CBSE, ICSE and State boards. Includes questions such as “Main differences between rational and irrational numbers.”, “What is the product of a rational number and its multiplicative inverse?”, “What is the sum of a rational number and its additive inverse?”
Question & Answer
Questions
- MathematicsClass 7All
Main differences between rational and irrational numbers.
Rational and irrational numbers are two distinct types of real numbers. Here are five main differences between them: Definition: Rational Numbers: Rational numbers are numbers that can be expressed as the quotient or fraction b is not equal to zero. In other words, rational numbers are those that can be written as fractions of integers.Irrational Numbers: Irrational numbers cannot be expressed as fractions of integers. They cannot be represented as b are integers. Instead, the
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What is the product of a rational number and its multiplicative inverse?
The product is always one (1). Formula: (p/q) × (q/p) = 1 (where p ≠ 0) Examples: 3/4 × 4/3 = 12/12 = 1 7/5 × 5/7 = 35/35 = 1 (-2/9) × (-9/2) = 18/18 = 1 Explanation: This is the defining property of multiplicative inverse (reciprocal). The reciprocal is specifically the number that, when multiplied by the original, gives one. <strong
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What is the sum of a rational number and its additive inverse?
The sum is always zero (0). Formula: p/q + (-p/q) = 0 Examples: 3/7 + (-3/7) = 0 -5/9 + 5/9 = 0 2/3 + (-2/3) = 0 Explanation: This is the defining property of additive inverse. The additive inverse is specifically the number that, when added to the original, gives zero.
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Arrange in ascending order: 2/3, 5/6, 1/2
Step 1: Find LCM(3, 6, 2) = 6 Step 2: Convert all to denominator 6 2/3 = 4/6 5/6 = 5/6 1/2 = 3/6 Step 3: Compare numerators 3/6 < 4/6 < 5/6 Answer in ascending order: 1/2 < 2/3 < 5/6
- MathematicsClass 7All
How do you compare two rational numbers?
There are three methods: Method 1: Cross Multiplication For a/b and c/d: Calculate a × d and b × c If a × d > b × c, then a/b > c/d If a × d < b × c, then a/b < c/d If a × d = b × c, then a/b = c/d Method 2: Same Denominator Convert both to the same denominator (LCM) Compare numerators Method 3: D
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Convert 0.454545... (0.4̄5̄) to a rational number.
Step 1: Let x = 0.454545... Step 2: Since 2 digits repeat, multiply by 100 100x = 45.454545... Step 3: Subtract 100x - x = 45.454545... - 0.454545... 99x = 45 Step 4: Solve x = 45/99 Step 5: Simplify (GCD = 9) 45/99 = 5/11 Answer: 0.4̄5̄ = 5/11
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Convert 0.666... (repeating) to a rational number.
Step 1: Let x = 0.666... Step 2: Multiply by 10 (one repeating digit) 10x = 6.666... Step 3: Subtract original from multiplied 10x - x = 6.666... - 0.666... 9x = 6 Step 4: Solve for x x = 6/9 = 2/3 Answer: 0.666... = 2/3 Verification: 2 ÷ 3 = 0.666... ✓
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How do you convert a terminating decimal to a rational number?
Count the decimal places and use that as the power of 10 in the denominator. Examples: (a) 0.5 1 decimal place 0.5 = 5/10 = 1/2 (b) 0.75 2 decimal places 0.75 = 75/100 = 3/4 (c) 2.125 3 decimal places 2.125 = 2125/1000 = 17/8 (d) 0.008 3 decimal places 0.008 = 8/1000
- MathematicsClass 7All
How do you represent -2/5 on a number line?
Step 1: Identify position -2/5 is negative, so it lies to the LEFT of 0 It's between -1 and 0 Step 2: Divide the segment Divide the space between -1 and 0 into 5 equal parts Each part represents 1/5 Step 3: Mark the point Count 2 parts from 0 towards the left Mark this point as -2/5 Visual: -1 -4/5 -3/5 -
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Find five rational numbers between 1/3 and 1/2
Method 1: Common Denominator Convert to same denominator with enough space: 1/3 = 10/30 1/2 = 15/30 Five numbers between 10/30 and 15/30: 11/30, 12/30, 13/30, 14/30, 15/30 Simplified: 11/30, 2/5, 13/30, 7/15, 1/2 Wait, we need numbers BETWEEN 1/3 and 1/2, so: 11/30, 2/5, 13/30, 7/15 (not including 15/30 which equals 1/2) Better answer with more space:</st
- MathematicsClass 7All
How many rational numbers lie between two rational numbers?
Infinitely many rational numbers exist between any two rational numbers. This is called the dense property of rational numbers. Example: Between 1/2 and 1: Using the mean method: (1/2 + 1)/2 = 3/4 (1/2 + 3/4)/2 = 5/8 (3/4 + 1)/2 = 7/8 So: 1/2 < 5/8 < 3/4 < 7/8 < 1 This process can continue forever, finding more numbers each time. General method:</stro
- MathematicsClass 7All
What is the distributive property for rational numbers?
Multiplication distributes over addition. Formula: a × (b + c) = (a × b) + (a × c) Example: 2/3 × (1/4 + 1/5) Method 1 (add first): = 2/3 × (5/20 + 4/20) = 2/3 × 9/20 = 18/60 = 3/10 Method 2 (distribute): = (2/3 × 1/4) + (2/3 × 1/5) = 2/12 + 2/15 = 1/6 + 2/15 = 5/30 + 4/30 = 9/30 = 3/10 Both methods give 3/10 ✓
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Simplify: [(2/5) ÷ (3/7)] × (4/9)
Step 1: Solve division first (2/5) ÷ (3/7) = (2/5) × (7/3) = 14/15 Step 2: Multiply by 4/9 (14/15) × (4/9) = (14 × 4)/(15 × 9) = 56/135 Check for simplification: GCD(56, 135) = 1 Already in simplest form. Answer: 56/135
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Find the reciprocal of: (a) 7/9 (b) -3 (c) 1
(a) Reciprocal of 7/9: Flip the fraction: 9/7 Verification: 7/9 × 9/7 = 63/63 = 1 ✓ (b) Reciprocal of -3: Write as fraction: -3 = -3/1 Flip: -1/3 Verification: -3 × (-1/3) = 3/3 = 1 ✓ (c) Reciprocal of 1: 1 = 1/1 Flip: 1/1 = 1 Verification: 1 × 1 = 1 ✓
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What property is shown by: (2/3 × 4/5) × 1/2 = 2/3 × (4/5 × 1/2)?
This demonstrates the Associative Property of Multiplication. Definition: For any three rational numbers a, b, and c: (a × b) × c = a × (b × c) The grouping of multiplication doesn't matter; the result is the same. Verification: Left side: (2/3 × 4/5) × 1/2 = (8/15) × 1/2 = 8/30 = 4/15 Right side: 2/3 × (4/5 × 1/2) = 2/3 × (4/10) = 2/3 × 2/5
- MathematicsClass 7All
Divide: (5/6) ÷ (10/9)
Step 1: Convert to multiplication by reciprocal (5/6) ÷ (10/9) = (5/6) × (9/10) Step 2: Multiply (5 × 9)/(6 × 10) = 45/60 Step 3: Simplify (GCD = 15) 45/60 = 3/4 Answer: 3/4
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Multiply: (2/3) × (9/4) × (1/6)
Method: Multiply all numerators and all denominators (2 × 9 × 1)/(3 × 4 × 6) = 18/72 Simplify: GCD(18, 72) = 18 18/72 = 1/4 Answer: 1/4 Smart approach: Cancel before multiplying: (2/3) × (9/4) × (1/6) = (2 × 9 × 1)/(3 × 4 × 6) = (1 × 3 × 1)/(3 × 2 × 3) [cancel
- MathematicsClass 7All
What is the multiplicative identity for rational numbers?
The multiplicative identity is 1 (or 1/1). Property: For any rational number p/q: (p/q) × 1 = p/q Examples: 3/5 × 1 = 3/5 -7/9 × 1 = -7/9 1 × 4/11 = 4/11 One is called the multiplicative identity because multiplying any number by 1 gives back the same number.
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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.
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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