CBSE · Class 10 Mathematics · Chapter 1
Real Numbers Class 10 Worksheet with Solutions and MCQs
Practise Real Numbers with a worksheet that supports independent learning and revision. Use the solutions to check your method, then strengthen your understanding with the worked examples, MCQs and frequently asked questions below.
A Real Numbers Class 10 worksheet with solutions helps you move from knowing a rule to applying it correctly. Begin with prime factorisation, HCF and LCM, then practise explaining why a number is irrational. Write your working before checking an answer so that you can identify the exact step that needs improvement.
This page brings together a downloadable worksheet resource and additional Real Numbers questions and answers for revision. The examples below are extra practice on this page. They show how to organise calculations and explanations clearly for Class 10 Maths Chapter 1.
What are real numbers? Definition and examples
Real numbers are all the numbers represented by points on the number line. They include rational numbers and irrational numbers. Integers, fractions, terminating decimals and recurring decimals are rational; numbers such as √2, √3 and π are irrational.
The real numbers symbol is ℝ. For example, −8, 0, 3/5, 2.75 and √7 are real numbers. There is no first or smallest real number: for any real number, a smaller one can be found. Between any two different real numbers, there are infinitely many real numbers.
Real numbers chart
| Number group | Examples | Key point |
|---|---|---|
| Natural numbers | 1, 2, 3, 4, … | Counting numbers under the convention used here. |
| Whole numbers | 0, 1, 2, 3, … | Natural numbers together with zero. |
| Integers | …, −2, −1, 0, 1, 2, … | Whole numbers and their negatives. |
| Rational numbers | −4, 0, 3/7, 0.25 | Can be written as p/q, where p and q are integers and q ≠ 0. |
| Irrational numbers | √2, √5, π | Cannot be written as a ratio of two integers. |
Important Real Numbers Class 10 concepts
Fundamental Theorem of Arithmetic
Every composite number can be expressed as a product of primes, uniquely apart from the order of the factors. For example, 84 = 2² × 3 × 7. This prime factorisation is the basis for many HCF and LCM questions.
HCF and LCM
The HCF uses common prime factors with their smallest powers. The LCM uses every prime factor present, with its greatest power. Decide which is needed from the problem rather than guessing from a keyword.
Proofs of irrationality
To prove that √2, √3 or √5 is irrational, begin by assuming it is rational in lowest terms. Use the resulting divisibility statements to reach a contradiction.
Word problems
Equal grouping and cutting problems often involve HCF. Finding the earliest common repetition of events often involves LCM. Translate the situation into a mathematical requirement before calculating.
Real Numbers formulas for Class 10
For two positive integers a and b:
HCF(a, b) × LCM(a, b) = a × b
LCM(a, b) = (a × b) ÷ HCF(a, b)
HCF(a, b) = (a × b) ÷ LCM(a, b)
If a and b are co-prime positive integers, their HCF is 1 and their LCM is a × b. The product relationship above applies to two positive integers; do not automatically extend it to three numbers.
Real Numbers Class 10 questions with worked solutions
Example 1: Find the HCF and LCM of 72 and 120
72 = 2³ × 3²
120 = 2³ × 3 × 5
HCF = 2³ × 3
HCF = 24
LCM = 2³ × 3² × 5
LCM = 360
Check: 24 × 360 = 8640 and 72 × 120 = 8640. Both products agree.
Example 2: Use the HCF–LCM relationship
The HCF and LCM of two positive integers are 12 and 180. If one integer is 36, find the other.
12 × 180 = 36 × b
2160 = 36 × b
b = 2160 ÷ 36
b = 60
The HCF of 36 and 60 is 12, and their LCM is 180, so the result satisfies the given conditions.
Example 3: Prove that √3 is irrational
Assume √3 = a/b, where a and b are co-prime positive integers.
a² = 3b²
Therefore, 3 divides a².
Since 3 is prime, 3 divides a.
Let a = 3k.
9k² = 3b²
b² = 3k²
Therefore, 3 also divides b.
This contradicts the assumption that a and b are co-prime. Hence, √3 is irrational.
Real Numbers Class 10 MCQs with answers
Try these original objective questions before opening their answers. Explain your choice as well as selecting an option.
1. What is the HCF of two co-prime positive integers?
A. 0
B. 1
C. Their sum
D. Their product
Answer: B. 1. Co-prime numbers have no common positive factor other than 1.
2. If a = 2² × 3³ and b = 2³ × 3, what is their LCM?
A. 12
B. 36
C. 108
D. 216
Answer: D. 216. Take the greatest power of each prime: LCM = 2³ × 3³ = 216.
3. Which of the following is irrational?
A. √49
B. 0.125
C. √7
D. 5/8
Answer: C. √7. The other options are rational; √49 = 7.
4. What is the prime factorisation of 140?
A. 2 × 5 × 14
B. 2² × 5 × 7
C. 4 × 35
D. 2 × 70
Answer: B. 2² × 5 × 7. Every factor in a prime factorisation must be prime.
Real Numbers Class 10 case study questions
Case-based practice: Equal ribbon pieces
A school has ribbons of lengths 84 cm and 126 cm. Each ribbon must be cut into pieces of the same length, with no ribbon left over. The pieces should be as long as possible.
- Should you calculate the HCF or LCM?
- What is the greatest possible length of each piece?
- How many pieces will be obtained altogether?
View the solution
Use the HCF because the required length must divide both ribbon lengths exactly.
84 = 2² × 3 × 7
126 = 2 × 3² × 7
HCF = 2 × 3 × 7
HCF = 42 cm
Pieces from the first ribbon = 84 ÷ 42 = 2
Pieces from the second ribbon = 126 ÷ 42 = 3
Total pieces = 5
How to use the Real Numbers worksheet with answers
Use the worksheet download option on this page for practice. Attempt the questions independently, then check the provided solutions. For MCQs, write a short calculation or reason so you can distinguish understanding from guessing.
For Real Numbers word problems, first identify what must be shared, divided or repeated. In proof questions, state your assumption clearly and explain the contradiction. After checking your work, retry missed questions without referring to the solution.
Common mistakes to avoid: confusing the smallest and greatest prime powers, treating 1 as a prime number, assuming every square root is irrational, and giving a proof without explaining why it contradicts the starting assumption.
Related Class 10 Maths resources
Revise the chapter using the Real Numbers Class 10 notes and mind map. Use the summary to connect rational and irrational numbers before returning to the practice questions.
Explore the CBSE Class 10 Maths notes collection for related study resources. For practice with algebraic expressions and zeroes, try the Polynomials Class 10 worksheet with solutions and MCQs.
Frequently asked questions about Real Numbers Class 10
What are real numbers in simple words?
Real numbers are all the numbers that have a place on the number line. They include negative numbers, zero, positive numbers, fractions and irrational numbers such as √2.
Does this Real Numbers Class 10 worksheet include solutions?
This resource is presented as a worksheet with solutions and MCQs. Attempt the worksheet before checking its solutions. The additional worked examples and MCQ answers on this page also support self-revision.
Which Real Numbers questions should I prioritise for Class 10 revision?
Practise prime factorisation, HCF and LCM, their relationship, application problems and proofs of irrationality. Include both objective questions and answers that require a clear written explanation. Follow the topics prescribed in your current textbook and school syllabus.
How do I decide whether a word problem needs HCF or LCM?
Use HCF when seeking the greatest quantity that divides several quantities exactly, such as the longest equal ribbon pieces. Use LCM when seeking the smallest common multiple, such as when repeating events next occur together.
Are co-prime numbers always prime?
No. Co-prime numbers only need to have HCF 1. For example, 8 and 15 are both composite, but they are co-prime because their only common positive factor is 1.
Is zero a real number?
Yes. Zero is a real number, an integer and a rational number. It can be written as 0/1. Real numbers also include negative values, so the set does not start at zero or one.
Is the square root of every natural number irrational?
No. The square root of a perfect square is rational: √25 = 5. The square root of a positive integer that is not a perfect square is irrational, such as √6.
What is the difference between rational and irrational decimals?
A rational number has a terminating or non-terminating recurring decimal expansion. An irrational number has a non-terminating, non-recurring decimal expansion. “Non-terminating” alone is not enough to identify an irrational number.
When does a rational number have a terminating decimal expansion?
First write the fraction in lowest terms. Its decimal expansion terminates when the denominator has no prime factors other than 2 and 5. For example, 7/40 terminates because 40 = 2³ × 5. This is useful background for worksheets containing decimal-expansion questions.
Can the sum of two irrational numbers be rational?
Yes. For example, √2 + (−√2) = 0. Both numbers are irrational, but their sum is rational. Their product can also be rational: √2 × √2 = 2.
What are real numbers called in Hindi?
Real numbers are called वास्तविक संख्याएँ. संख्या रेखा पर दर्शाई जा सकने वाली सभी संख्याएँ वास्तविक संख्याएँ हैं। इनमें परिमेय और अपरिमेय दोनों प्रकार की संख्याएँ शामिल हैं।
Where can I revise Real Numbers before solving the worksheet?
Use the Real Numbers Class 10 mind map and notes, then return to the worksheet. For related chapters, browse the CBSE Class 10 Maths notes.
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