A Story of Numbers — Class 8 Mathematics NCERT Solutions (Free)
Free step-by-step NCERT solutions for Class 8 Mathematics chapter "A Story of Numbers" — 8 important questions with detailed answers for CBSE board exam preparation.
TL;DR: Free step-by-step NCERT solutions for Class 8 Mathematics chapter "A Story of Numbers" — 8 important questions with detailed answers for CBSE board ex…
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NCERT Solutions for Class 8 Mathematics — A Story of Numbers. Step-by-step answers to all 8 textbook questions from this chapter, written for CBSE board preparation and free to use.
A Story of Numbers — All 8 Questions Solved
Q1. Which of the following is a rational number? 0, √2, π, 1/3
A rational number is a number that can be expressed in the form p/q where p and q are integers and q ≠ 0.
Step 1: Check 0
0 can be written as 0/1
✓ This is a rational number
Step 2: Check √2
√2 = 1.414213... (non-terminating, non-repeating decimal)
It cannot be expressed as p/q where p and q are integers
✗ This is irrational
Step 3: Check π
π = 3.14159... (non-terminating, non-repeating decimal)
It cannot be expressed as p/q
✗ This is irrational
Step 4: Check 1/3
1/3 is already in the form p/q where p = 1 and q = 3 (both integers, q ≠ 0)
✓ This is a rational number
Answer: 0 and 1/3 are rational numbers
Q2. Find two rational numbers between 1/4 and 1/2.
We need to find two rational numbers that lie between 1/4 and 1/2.
Step 1: Convert to a common form
1/4 = 0.25
1/2 = 0.50
Step 2: Method 1 - Using simple fractions
We can find the average of the two numbers:
(1/4 + 1/2)/2 = (1/4 + 2/4)/2 = (3/4)/2 = 3/8
Check: 1/4 = 2/8, 3/8, 4/8 = 1/2
✓ 3/8 is between 1/4 and 1/2
Step 3: Find another rational number
Let's find the average of 1/4 and 3/8:
(1/4 + 3/8)/2 = (2/8 + 3/8)/2 = (5/8)/2 = 5/16
Check: 1/4 = 4/16, 5/16, 8/16 = 1/2
✓ 5/16 is between 1/4 and 1/2
Step 4: Verify
1/4 < 5/16 < 3/8 < 1/2
0.25 < 0.3125 < 0.375 < 0.50
Answer: Two rational numbers are 3/8 and 5/16
Q3. Add: 3/7 + 2/5
We need to add two fractions with different denominators.
Step 1: Find the LCM of denominators 7 and 5
Since 7 and 5 are both prime and different:
LCM(7, 5) = 7 × 5 = 35
Step 2: Convert to equivalent fractions with denominator 35
3/7 = (3 × 5)/(7 × 5) = 15/35
2/5 = (2 × 7)/(5 × 7) = 14/35
Step 3: Add the fractions
3/7 + 2/5 = 15/35 + 14/35 = (15 + 14)/35 = 29/35
Step 4: Check if the answer can be simplified
GCD(29, 35) = 1 (29 is prime, 35 = 5 × 7)
29/35 is already in simplest form
Answer: 3/7 + 2/5 = 29/35
Q4. Subtract: 7/8 - 1/6
We need to subtract two fractions with different denominators.
Step 1: Find the LCM of denominators 8 and 6
8 = 2³
6 = 2 × 3
LCM(8, 6) = 2³ × 3 = 24
Step 2: Convert to equivalent fractions with denominator 24
7/8 = (7 × 3)/(8 × 3) = 21/24
1/6 = (1 × 4)/(6 × 4) = 4/24
Step 3: Subtract the fractions
7/8 - 1/6 = 21/24 - 4/24 = (21 - 4)/24 = 17/24
Step 4: Check if the answer can be simplified
GCD(17, 24) = 1 (17 is prime, 24 = 2³ × 3)
17/24 is already in simplest form
Answer: 7/8 - 1/6 = 17/24
Q5. Simplify: (3/4) × (8/9)
We need to multiply two fractions.
Step 1: Multiply numerators and denominators
(3/4) × (8/9) = (3 × 8)/(4 × 9) = 24/36
Step 2: Simplify by canceling common factors
Before multiplying, we can cancel:
3 and 9 have GCD = 3: 3/9 = 1/3
8 and 4 have GCD = 4: 8/4 = 2/1
Or: (3/4) × (8/9) = (3 × 8)/(4 × 9)
Cancel 3 with 9: (1 × 8)/(4 × 3) = 8/12
Cancel 4 with 8: (2)/(3) = 2/3
Step 3: Verify
24/36 = (24÷12)/(36÷12) = 2/3 ✓
Answer: (3/4) × (8/9) = 2/3
Q6. Divide: (5/6) ÷ (10/12)
We need to divide two fractions.
Step 1: Use the rule a/b ÷ c/d = a/b × d/c
(5/6) ÷ (10/12) = (5/6) × (12/10)
Step 2: Multiply the fractions
(5/6) × (12/10) = (5 × 12)/(6 × 10) = 60/60
Step 3: Simplify
60/60 = 1
Step 4: Alternative method - cancel before multiplying
(5/6) × (12/10)
Cancel 5 and 10: (1/6) × (12/2)
Cancel 6 and 12: (1/1) × (2/2) = 1
Answer: (5/6) ÷ (10/12) = 1
Q7. What is the property demonstrated by: (1/2 + 1/3) + 1/4 = 1/2 + (1/3 + 1/4)?
This equation demonstrates a fundamental property of rational numbers.
Step 1: Identify the property
The grouping of numbers in addition is different on both sides, but the result is the same.
This is the Associative Property of Addition.
Step 2: Verify by calculating the left side
(1/2 + 1/3) + 1/4
First: 1/2 + 1/3 = (3 + 2)/6 = 5/6
Then: 5/6 + 1/4 = (10 + 3)/12 = 13/12
Step 3: Verify by calculating the right side
1/2 + (1/3 + 1/4)
First: 1/3 + 1/4 = (4 + 3)/12 = 7/12
Then: 1/2 + 7/12 = (6 + 7)/12 = 13/12
Step 4: Conclusion
Both sides equal 13/12
This shows that for rational numbers: (a + b) + c = a + (b + c)
Answer: This demonstrates the Associative Property of Addition for rational numbers
Q8. Which rational number is greater: 5/9 or 4/7?
We need to compare two fractions with different denominators.
Step 1: Find a common denominator
LCM(9, 7) = 9 × 7 = 63 (since 9 and 7 are coprime)
Step 2: Convert to equivalent fractions
5/9 = (5 × 7)/(9 × 7) = 35/63
4/7 = (4 × 9)/(7 × 9) = 36/63
Step 3: Compare the numerators
35/63 vs 36/63
Since 35 < 36, we have 35/63 < 36/63
Step 4: Alternative method - decimal form
5/9 = 0.555...
4/7 = 0.571...
0.555... < 0.571...
Answer: 4/7 is greater than 5/9
A Story of Numbers — Practice MCQs with Answers
Attempt these 20 multiple-choice questions after working through the solutions above. Each carries the correct option and the reasoning behind it, so a wrong answer tells you which idea to revisit.
MCQ 1. What is the smallest natural number?
- A. 0
- B. 1
- C. -1
- D. 1/2
Answer: B. 1 — Natural numbers are the counting numbers that start from 1 and go up infinitely.
MCQ 2. Which of the following is NOT an integer?
- A. -5
- B. 0
- C. 3/4
- D. 7
Answer: C. 3/4 — Integers include positive and negative whole numbers along with zero, but do not include fractions or decimals that are not whole numbers.
MCQ 3. Is every integer a rational number?
- A. Yes
- B. No
- C. Sometimes
- D. Only positive integers
Answer: A. Yes — Every integer 'a' can be expressed in the form a/1, which fits the definition of a rational number p/q where q is not zero.
MCQ 4. What is the additive inverse of -7/3?
- A. 7/3
- B. -7/3
- C. 3/7
- D. -3/7
Answer: A. 7/3 — The additive inverse of a number 'a' is '-a' such that when added together, they sum to zero.
MCQ 5. Which number is the multiplicative identity for rational numbers?
- A. 0
- B. 1
- C. -1
- D. 1/2
Answer: B. 1 — The multiplicative identity is the number that, when multiplied by any rational number, results in the same rational number.
MCQ 6. Which property is illustrated by the equation: (2/3) × (1/2 + 1/4) = (2/3) × (1/2) + (2/3) × (1/4)?
- A. Commutative property
- B. Associative property
- C. Distributive property
- D. Closure property
Answer: C. Distributive property — This equation demonstrates that multiplication distributes over addition.
MCQ 7. If x + 5/2 = -3/2, what is the value of x?
- A. -4
- B. -1
- C. 1
- D. 4
Answer: A. -4 — To solve for x, subtract 5/2 from both sides of the equation.
MCQ 8. How many rational numbers are there between any two given rational numbers?
- A. One
- B. Ten
- C. Hundred
- D. Infinitely many
Answer: D. Infinitely many — Between any two distinct rational numbers, there exist an infinite number of other rational numbers.
MCQ 9. What is the reciprocal (multiplicative inverse) of -4/5?
- A. 4/5
- B. -5/4
- C. 5/4
- D. -4/5
Answer: B. -5/4 — The reciprocal of a non-zero rational number p/q is q/p. The sign of the number remains unchanged.
MCQ 10. Which of the following operations is NOT closed for natural numbers?
- A. Addition
- B. Subtraction
- C. Multiplication
- D. Division
Answer: B. Subtraction — The subtraction of two natural numbers does not always result in a natural number (e.g., 3 - 5 = -2, which is not a natural number).
MCQ 11. The expression (a + b) + c = a + (b + c) represents which property for rational numbers?
- A. Commutative property
- B. Associative property
- C. Distributive property
- D. Identity property
Answer: B. Associative property — The associative property states that the way numbers are grouped in an addition (or multiplication) operation does not affect the result.
MCQ 12. Find three rational numbers between 1/3 and 1/2.
- A. (5/12, 11/24, 23/48)
- B. (2/6, 3/6, 4/6)
- C. (1/4, 1/5, 1/6)
- D. (1/3, 1/2, 2/3)
Answer: A. (5/12, 11/24, 23/48) — To find rational numbers between two fractions, express them with a common and sufficiently large denominator, then pick numerators in between.
MCQ 13. The sum of two rational numbers is -5/6. If one of the numbers is 1/3, what is the other number?
- A. -7/6
- B. -1/2
- C. -1
- D. 7/6
Answer: A. -7/6 — Set up an equation with the given information and solve for the unknown rational number.
MCQ 14. Simplify: (2/5) × (-3/7) - (1/14) - (3/7) × (3/5)
- A. -1
- B. 0
- C. 1/2
- D. -1/2
Answer: D. -1/2 — Apply the order of operations and properties of rational numbers to simplify the expression efficiently.
MCQ 15. What rational number is represented by point P on the number line if the distance between 0 and 1 is divided into 4 equal parts, and P is at the 3rd mark to the left of 0?
- A. 3/4
- B. -3/4
- C. 4/3
- D. -4/3
Answer: B. -3/4 — Points to the left of zero represent negative numbers. Each division represents 1/4 of a unit.
MCQ 16. Is the set of integers closed under division?
- A. Yes
- B. No
- C. Only for positive integers
- D. Only for negative integers
Answer: B. No — Dividing two integers does not always yield an integer (e.g., 3 ÷ 2 = 1.5, which is not an integer).
MCQ 17. Which of the following is the additive identity for rational numbers?
- A. 0
- B. 1
- C. -1
- D. 1/2
Answer: A. 0 — The additive identity is the number that, when added to any rational number, leaves the rational number unchanged.
MCQ 18. Which of the following numbers is a rational number?
- A. √2
- B. π
- C. 0.333...
- D. 0.1010010001...
Answer: C. 0.333... — Rational numbers include terminating or repeating decimals. 0.333... is a repeating decimal, equivalent to 1/3.
MCQ 19. The product of two rational numbers is -12/25. If one of the numbers is 3/5, what is the other number?
- A. -4/5
- B. -1/5
- C. -12/5
- D. 4/5
Answer: A. -4/5 — Set up an equation and solve for the unknown rational number by dividing the product by the known number (or multiplying by its reciprocal).
MCQ 20. Evaluate: (5/4 - 1/2) × (3/2 + 1/4)
- A. 21/16
- B. 15/16
- C. 7/8
- D. 5/4
Answer: A. 21/16 — First, evaluate the expressions inside each parenthesis, then multiply the results.
How to Use These Solutions
Attempt each question yourself first and only then compare with the worked answer. Marks in the CBSE board exam are awarded for the METHOD as much as the final result, so reproduce the steps rather than memorising the last line. Where a solution states a law, a formula or a definition, learn that wording — examiners look for it.
Related practice for this chapter: chapter MCQs, previous-year questions, revision notes and sample papers.