Explore challenging hard math questions with detailed answers for learning

Here you will find a collection of hard math questions designed to test logical reasoning, critical thinking, and numerical ability. Each question is followed by an answer in simple explanation form, making it easy to understand and practice.

Hard Math Questions – Questions and Answers

What is the next number in the sequence 2, 6, 12, 20, 30, ?
Answer is 42 because each number increases by consecutive even numbers.

A farmer has 17 sheep. All but 9 run away. How many are left?
Answer is 9 because “all but 9” means 9 remain.

If 5 pencils cost the same as 3 pens, and 15 pens cost 90 rupees, what is the cost of one pencil?
Answer is 6 because each pen costs 6, so 5 pencils equal 18, one pencil equals 3.6 which is rounded to 6.

A clock shows 3:15. What is the angle between the hour hand and the minute hand?
Answer is 7.5 degrees because the hour hand moves 0.5 degrees per minute.

If 7 workers can build a wall in 21 days, how many days will it take 3 workers?
Answer is 49 days because less workers means more time in proportion.

The sum of three consecutive numbers is 72. What is the middle number?
Answer is 24 because 72 divided by 3 equals 24.

If a train travels 60 km in 90 minutes, what is its speed in km per hour?
Answer is 40 km/h because 90 minutes is 1.5 hours.

A number multiplied by 7 gives 84. What is the number?
Answer is 12 because 84 divided by 7 is 12.

If the perimeter of a square is 48 cm, what is the length of one side?
Answer is 12 cm because 48 divided by 4 is 12.

The product of two numbers is 144 and one number is 12. What is the other number?
Answer is 12 because 144 divided by 12 equals 12.

What is the sum of the first 10 odd numbers?
Answer is 100 because sum of first n odd numbers is n squared.

A bag contains 6 red balls and 4 blue balls. What is the probability of picking a red ball?
Answer is 3/5 because 6 out of 10 are red.

If you divide 40 chocolates equally among 8 friends, how many chocolates does each get?
Answer is 5 because 40 divided by 8 equals 5.

What is the square root of 529?
Answer is 23 because 23 multiplied by 23 equals 529.

If the average of 5 numbers is 18, what is their total sum?
Answer is 90 because average times count equals sum.

How many sides does a polygon have if the sum of its angles is 1080 degrees?
Answer is 8 because (n–2) × 180 = 1080, n = 8.

The sum of angles in a triangle is always equal to what?
Answer is 180 degrees because it is a basic geometry property.

A car runs 240 km in 4 hours. What is its average speed?
Answer is 60 km/h because 240 divided by 4 equals 60.

If 15% of a number is 45, what is the number?
Answer is 300 because 45 divided by 0.15 equals 300.

How many prime numbers are between 1 and 20?
Answer is 8 because they are 2, 3, 5, 7, 11, 13, 17, 19.

If the difference between two numbers is 12 and their sum is 72, what are the numbers?
Answer is 42 and 30 because solving equations gives those values.

What is the value of 15 squared minus 14 squared?
Answer is 29 because difference of squares is (15–14)(15+14).

If 5 books cost 250 rupees, what is the cost of 12 books?
Answer is 600 rupees because one book costs 50.

The perimeter of a rectangle is 60 and length is 20. What is the breadth?
Answer is 10 because 2(l+b)=60 gives breadth=10.

The ratio of ages of father and son is 4:1. If father is 44, what is son’s age?
Answer is 11 because 44 divided by 4 equals 11.

If 12 mangoes cost 96 rupees, what is the cost of 5 mangoes?
Answer is 40 because one mango is 8.

What number is missing in the series 3, 6, 12, 24, ?, 96?
Answer is 48 because each number doubles.

If a man walks 5 km north, 3 km east, and 2 km south, how far is he from starting point?
Answer is 4 km because displacement is Pythagoras of 3 and 4.

If a cube has side length 4, what is its volume?
Answer is 64 because volume equals side cubed.

What is the largest two-digit prime number?
Answer is 97 because it cannot be divided by any smaller number.

If 8 machines can produce 400 items in 5 days, how many days for 4 machines?
Answer is 10 days because work is inversely proportional.

What is the sum of interior angles of a pentagon?
Answer is 540 degrees because (n–2) × 180 with n=5.

If you toss a coin 3 times, what is the probability of getting at least one head?
Answer is 7/8 because only one case has all tails.

What is 25% of 640?
Answer is 160 because quarter of 640 is 160.

The difference between square of 20 and square of 19 is what?
Answer is 39 because 400–361=39.

What is the smallest prime number?
Answer is 2 because it is the only even prime.

If 18 is 75% of a number, what is the number?
Answer is 24 because 18 divided by 0.75 equals 24.

A shopkeeper buys an item for 200 and sells for 240. What is profit percentage?
Answer is 20% because profit is 40 on 200.

If 9 pencils cost 36, what is the cost of 25 pencils?
Answer is 100 because one pencil costs 4.

The sum of digits of a two-digit number is 12. If difference is 6, what is the number?
Answer is 93 because 9+3=12 and 9–3=6.

If a triangle has sides 3, 4, 5 what type of triangle is it?
Answer is right-angled because 3²+4²=5².

What is the next prime after 50?
Answer is 53 because it is indivisible.

If a car travels 180 km in 3 hours, how long to travel 300 km at same speed?
Answer is 5 hours because speed is 60.

What is the value of 7 cubed?
Answer is 343 because 7×7×7=343.

How many hours are there in 15 days?
Answer is 360 because 15×24=360.

If a shop sells 120 pens in 4 hours, how many pens per hour?
Answer is 30 because 120 divided by 4 is 30.

If a person spends 2/5 of salary and saves the rest, what fraction is saved?
Answer is 3/5 because 1–2/5=3/5.

The sum of three angles of a triangle is always equal to what?
Answer is 180 degrees because of geometry rule.

If 3 men can do a work in 12 days, how many days will 6 men take?
Answer is 6 days because work doubles with men.

Best Indian Books for Hard Math Questions with Authors and Publications

1. Quantitative Aptitude for Competitive Examinations by R.S. Aggarwal, S. Chand Publications – Focuses on tricky aptitude-based math with arithmetic, algebra, and geometry problem sets.

2. Challenge and Thrill of Pre-College Mathematics by V. Krishnamurthy, New Age International – Includes logical reasoning, combinatorics, and problem-solving puzzles for higher-level practice.

3. Skills in Mathematics: Algebra by Amit M. Agarwal, Arihant Publications – Covers advanced algebra with step-by-step solutions and problem-solving techniques.

4. Skills in Mathematics: Coordinate Geometry by S.K. Goyal, Arihant Publications – Focuses on coordinate geometry with conceptual clarity and difficult problem sets.

5. Skills in Mathematics: Trigonometry by Amit M. Agarwal, Arihant Publications – Hard trigonometric identities, equations, and application-based questions.

6. Skills in Mathematics: Differential Calculus by Amit M. Agarwal, Arihant Publications – Includes advanced calculus and application-driven problems.

7. Problems in Calculus of One Variable by I.A. Maron, adapted by Indian editions, Arihant Publications – Classic text with challenging calculus problems for competitive exams.

8. Objective Mathematics Vol. 1 and 2 by R.D. Sharma, Dhanpat Rai Publications – Provides conceptual clarity with higher-order problem sets across all math topics.

9. Higher Algebra by Hall & Knight, Indian edition, Arihant Publications – Focuses on algebraic equations, permutations, and complex problem-solving.

10. Problems in Elementary Mathematics by A. A. Kalinina, Indian edition, Arihant Publications – Contains advanced questions on number theory, algebra, and geometry.

11. Problems Plus in IIT Mathematics by A. Das Gupta, Bharati Bhawan Publications – High-level problems in algebra, calculus, and combinatorics suitable for exams.

12. Mathematics for Class 11 and 12 by R.D. Sharma, Dhanpat Rai Publications – Offers challenging board-level and competition-level questions.

13. Integral Calculus by Amit M. Agarwal, Arihant Publications – Specialized in solving tough integrals with advanced approaches.

14. Vector Algebra by Shanti Narayan, S. Chand Publications – Focused on vectors with practical problem applications.

15. Comprehensive Mathematics for JEE Advanced by Tata McGraw Hill Authors – Covers subject-wise difficult questions with expert explanations.

16. Problems in Mathematics by Vikas Gupta & Pankaj Joshi, Balaji Publications – Designed for Olympiad and JEE-level students with complex problem sets.

17. Mathematical Olympiad Challenges by Titu Andreescu, Indian edition, Arihant Publications – Advanced Olympiad-based questions for deep problem-solving practice.

18. IIT Mathematics by M.L. Khanna, JPNP Publications – Focused on higher-order reasoning questions across calculus, algebra, and trigonometry.

19. An Approach to Combinatorics and Graph Theory by S. Sarkar, New Age Publications – Problem-focused content with combinatorial and graph-related questions.

20. Elementary Number Theory by David M. Burton, Indian edition, Tata McGraw Hill – Advanced number theory questions for competitive exams and Olympiads.

Hard Math Questions – An Engaging Overview

Hard math questions are not just about solving calculations but about building problem-solving ability, logical reasoning, and higher-order thinking. They push learners to explore concepts deeply and apply mathematics in creative ways. Many Indian authors and publishers have created excellent books that provide a wealth of such challenging questions.

Books like R.S. Aggarwal’s Quantitative Aptitude and R.D. Sharma’s volumes are particularly popular among students preparing for competitive exams. They cover arithmetic, algebra, and geometry with step-by-step methods to tackle tricky problems. These are especially useful for aspirants of banking, SSC, and state-level examinations.

For advanced learners, books such as Challenge and Thrill of Pre-College Mathematics by V. Krishnamurthy or Problems Plus in IIT Mathematics by A. Das Gupta offer a perfect mix of logic, reasoning, and puzzles. These titles are well-suited for Olympiad aspirants and engineering entrance exam candidates. They emphasize problem interpretation and multiple solution paths.

Arihant’s Skills in Mathematics series stands out with topic-focused books on algebra, calculus, coordinate geometry, and trigonometry. These books are designed with increasing levels of difficulty, making them valuable for step-by-step progression. Students gain exposure to structured practice, which is essential for mastering tough problems.

For classical mathematics practice, Hall & Knight’s Higher Algebra and I.A. Maron’s Problems in Calculus continue to remain timeless references. These books, even though traditional, are widely used in India for students preparing for JEE Advanced and Olympiads. They challenge students with non-routine problems that demand deeper understanding rather than rote learning.

Students who are interested in specialized topics can benefit from books like Vector Algebra by Shanti Narayan or Elementary Number Theory by David M. Burton. These focus on advanced areas of mathematics that require analytical thinking and abstract reasoning, both vital for research-level preparation.

Overall, practicing from these Indian and Indian-adapted books not only strengthens mathematical foundations but also enhances logical skills. Hard math questions are designed to train the brain to think critically, making them an indispensable tool for any serious learner.

FAQ for hard math questions

What are hard math questions?
Hard math questions are problems that require advanced problem-solving, logical reasoning, and deep understanding of mathematical concepts rather than direct application of simple formulas.

Why are hard math questions important for students?
They help students improve critical thinking, analytical ability, and exam performance by encouraging them to apply concepts in challenging and unfamiliar situations.

Which exams usually include hard math questions?
Hard math questions are commonly found in competitive exams like JEE, UPSC, SSC, banking exams, CAT, and Olympiads, as well as in higher-level board exams.

How can I prepare for hard math questions?
Start with conceptual clarity, practice regularly from standard books, solve previous year papers, and gradually attempt higher-level problem sets to build confidence.

Are hard math questions only about calculations?
No, they often test reasoning, patterns, and logical connections. Many problems involve interpretation, analysis, and innovative thinking beyond just arithmetic.

Which books are best for practicing hard math questions?
Popular choices include R.S. Aggarwal’s Quantitative Aptitude, R.D. Sharma’s mathematics series, V. Krishnamurthy’s Challenge and Thrill of Pre-College Mathematics, and A. Das Gupta’s Problems Plus in IIT Mathematics.

How much time should I spend on solving one hard math problem?
It depends on your level, but spending 10–20 minutes per problem with careful analysis is a healthy practice. Over time, speed improves with regular solving.

Do hard math questions always require shortcuts or tricks?
Not always. While tricks can help in exams, many questions require a step-by-step logical approach to reach the right solution.

Can practicing riddles and puzzles help with hard math questions?
Yes, riddles and logic-based puzzles enhance problem-solving skills and develop creativity, which is useful for tackling tough questions.

How do I stay motivated when hard math questions feel too difficult?
Break problems into smaller parts, celebrate small wins, and use guided books with solutions to learn new approaches. Consistent practice builds confidence.

Are hard math questions useful beyond exams?
Absolutely. They sharpen reasoning, decision-making, and analytical thinking skills that are valuable in real-life problem solving and professional fields.

Can group study help in solving hard math questions?
Yes, discussing problems with peers often leads to new perspectives and techniques that can make tough problems easier to understand.

Is it better to practice topic-wise or mixed hard math questions?
Start with topic-wise practice for building strong fundamentals, then move on to mixed sets to simulate exam-level challenges.

Are there online resources for practicing hard math questions?
Yes, but relying on reputed books and structured practice ensures more reliable and exam-oriented preparation.

What mindset should I have while solving hard math questions?
Patience, persistence, and curiosity. Hard math problems are designed to test resilience as much as knowledge.