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Mathematical Reasoning aptitude questions

Data sufficiency, number puzzles and statement-based deduction.

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Formula sheetMathematical Reasoning: the essentialsFormula sheetMathematical Reasoning: the essentialsShow

A mixed section: clocks, calendars and puzzles, plus statement-based questions.

Clocks, calendars and counting

  • Clock angle at H hours M minutes = |30H − 5.5M|; above 180°, take 360° minus it. In 12 hours the hands coincide 11 times, point opposite 11 times and are at right angles 22 times.
  • Calendars: odd days are the remainder on division by 7. An ordinary year has 1 odd day and a leap year 2; a 31-day month 3, a 30-day month 2. A century year is a leap year only if it divides by 400.
  • Painted cube cut into n × n × n: 8 small cubes have three painted faces, 12(n − 2) two, 6(n − 2)² one and (n − 2)³ none.
  • Sets: n(A or B) = n(A) + n(B) − n(both); exactly one = n(A or B) − n(both).
  • Inequalities: a conclusion is definite only if every link points the same way; one strict link makes it strict.

Data sufficiency and statements

  • Test statement I alone, then II alone, then both, carrying nothing across. Sufficient means exactly one answer.
  • An assumption must be necessary for the statement, not merely consistent with it; 'only', 'always' and 'never' usually make an option too strong.
  • A conclusion must follow from the statement itself; a course of action must be relevant and practical.
  • In truth-teller puzzles, assume each person honest in turn and keep the case with no contradiction.

Questions35

1

Data sufficiency: how old is Ravi?

Question: How old is Ravi? Statement I: Ravi is 5 years older than his brother. Statement II: Ravi's brother will be 20 next year. Which statements are needed?

data sufficiency

75s

Medium2

Angle between the hands

What is the angle between the hour and minute hands of a clock at 3:30?

clocks

60s

Medium3

What day was it?

If 15 August 2026 is a Saturday, what day is 15 September 2026?

calendar

60s

Medium4

Statement and assumption

Statement: 'Use our coaching, and you will clear the exam in your first attempt.' Assumptions: I. People want to clear the exam in the first attempt. II.

assumptions

75s

Medium5

Angle between the hands at 4:20

What is the angle between the hour and minute hands at 4:20?

clocks

60s

Medium6

Angle at 7:20

What is the angle between the hands of a clock at 7:20?

clocks

60s

Medium7

When the hands coincide

How many times do the hands of a clock coincide in 24 hours?

clocks

90s

Hard8

Right angles in a day

How many times in 24 hours are the hands of a clock at right angles?

clocks

90s

Hard9

A clock that gains time

A clock gains 5 minutes every hour. If it is set correctly at 8 a.m., what will it show when the correct time is 2 p.m.?

clocks

60s

Medium10

The day after 61 days

If today is Monday, what day will it be after 61 days?

calendar

45s

Easy11

The same date a year later

If 15 March 2027 is a Monday, what day will 15 March 2028 be? (2028 is a leap year.)

calendar

105s

Hard12

A month later

If 10 June is a Wednesday, what day is 10 July of the same year?

calendar

60s

Medium13

Counting backwards

If today is Thursday, what day was it 45 days ago?

calendar

60s

Medium14

Data sufficiency: Ravi's age from his older brother's

Question: What is Ravi's age? Statement I: Ravi is 8 years younger than his brother. Statement II: Ravi's brother will be 30 in five years.

data sufficiency

90s

Medium15

Data sufficiency: the area

Question: What is the area of a square? Statement I: Its perimeter is 40 cm. Statement II: Its diagonal is 10√2 cm. Which statements are sufficient?

data sufficiency

90s

Medium16

Data sufficiency: not enough

Question: What is the value of x? Statement I: x is an even number. Statement II: x is greater than 4 and less than 10. Which statements are sufficient?

data sufficiency

90s

Medium17

Statement and assumption: the institute's advertisement

Statement: 'Enrol in our institute and secure a government job.' Assumptions: I. People are interested in government jobs. II.

assumptions

90s

Medium18

Statement and conclusion

Statement: 'The company has decided to increase the salaries of all its employees by 15% from next month.' Conclusions: I.

conclusions

90s

Medium19

Cause and effect

Statements: I. The government has raised the price of petrol sharply. II. Sales of electric scooters have risen this quarter. Which best describes the relation?

cause and effect

90s

Medium20

Course of action

Statement: 'Many students fail mathematics every year in this school.' Courses of action: I. The school should arrange extra coaching in mathematics. II.

course of action

75s

Easy21

Sum and product

Two numbers have a sum of 15 and a product of 56. What is the difference between them?

number puzzle

75s

Medium22

Reversing a two-digit number

A two-digit number has digits summing to 9. Reversing the digits gives a number 27 less than the original. What is the number?

number puzzle

120s

Hard23

Three consecutive integers

The sum of three consecutive integers is 72. What is the largest of them?

number puzzle

45s

Easy24

Two subjects

In a class of 60 students, 35 study physics, 30 study chemistry and 15 study both. How many study neither?

set theory

75s

Medium25

Only one of the two

Of 100 people, 60 read newspaper A, 45 read newspaper B and 20 read both. How many read exactly one newspaper?

set theory

75s

Medium26

Two sports, one overlap

In a group, 30 play cricket, 25 play football and 20 play both. How many play at least one of the two?

set theory

45s

Easy27

Which box holds the prize

Three boxes are labelled A, B and C, and exactly one holds a prize.

deduction

135s

Hard28

Who is telling the truth

A says 'B is lying', and B says 'C is lying', while C says 'both A and B are lying'. Who is telling the truth?

deduction

150s

Hard29

Matchsticks in a pattern

Squares are built in a row from matchsticks: 1 square needs 4 sticks, 2 squares need 7, and 3 squares need 10. How many sticks are needed for 10 squares?

pattern

75s

Medium30

A painted cube cut up

A cube is painted on all faces and then cut into 27 smaller cubes of equal size. How many small cubes have exactly two faces painted?

cubes

120s

Hard31

Cubes with one painted face

A painted cube is cut into 64 equal smaller cubes. How many have exactly one face painted?

cubes

105s

Hard32

Matches in a tournament

In a tournament where every team plays every other team once, 45 matches are played. How many teams took part?

counting

60s

Medium33

Reasoning about an average

A student's average across 4 tests is 80. What is the lowest possible score on any one test, if each test is marked out of 100?

extremes

105s

Hard34

Reading a chain of inequalities

If A > B, B > C, C = D and D > E, which of the following is definitely true?

inequalities

75s

Medium35

Which inequality must hold

If P ≥ Q, Q > R and S < R, which of the following is definitely true?

inequalities

75s

Medium

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Mathematical Reasoning Aptitude Questions with Solutions

35 Mathematical Reasoning questions with hints, worked solutions and time targets — 4 easy · 22 medium · 9 hard.

  • Section: Logical Reasoning
  • Questions: 35
  • By difficulty: 4 easy · 22 medium · 9 hard
  • Cost: Free, unlimited on every plan

Data sufficiency, number puzzles and statement-based deduction.

Mathematical Reasoning: the essentials

A mixed section: clocks, calendars and puzzles, plus statement-based questions.

Clocks, calendars and counting

  • Clock angle at H hours M minutes = |30H − 5.5M|; above 180°, take 360° minus it. In 12 hours the hands coincide 11 times, point opposite 11 times and are at right angles 22 times.
  • Calendars: odd days are the remainder on division by 7. An ordinary year has 1 odd day and a leap year 2; a 31-day month 3, a 30-day month 2. A century year is a leap year only if it divides by 400.
  • Painted cube cut into n × n × n: 8 small cubes have three painted faces, 12(n − 2) two, 6(n − 2)² one and (n − 2)³ none.
  • Sets: n(A or B) = n(A) + n(B) − n(both); exactly one = n(A or B) − n(both).
  • Inequalities: a conclusion is definite only if every link points the same way; one strict link makes it strict.

Data sufficiency and statements

  • Test statement I alone, then II alone, then both, carrying nothing across. Sufficient means exactly one answer.
  • An assumption must be necessary for the statement, not merely consistent with it; 'only', 'always' and 'never' usually make an option too strong.
  • A conclusion must follow from the statement itself; a course of action must be relevant and practical.
  • In truth-teller puzzles, assume each person honest in turn and keep the case with no contradiction.

Questions

  1. Data sufficiency: how old is Ravi? Medium
    Question: How old is Ravi? Statement I: Ravi is 5 years older than his brother. Statement II: Ravi's brother will be 20 next year. Which statements are needed?
  2. Angle between the hands Medium
    What is the angle between the hour and minute hands of a clock at 3:30?
  3. What day was it? Medium
    If 15 August 2026 is a Saturday, what day is 15 September 2026?
  4. Statement and assumption Medium
    Statement: 'Use our coaching, and you will clear the exam in your first attempt.' Assumptions: I. People want to clear the exam in the first attempt. II.
  5. Angle between the hands at 4:20 Medium
    What is the angle between the hour and minute hands at 4:20?
  6. Angle at 7:20 Medium
    What is the angle between the hands of a clock at 7:20?
  7. When the hands coincide Hard
    How many times do the hands of a clock coincide in 24 hours?
  8. Right angles in a day Hard
    How many times in 24 hours are the hands of a clock at right angles?
  9. A clock that gains time Medium
    A clock gains 5 minutes every hour. If it is set correctly at 8 a.m., what will it show when the correct time is 2 p.m.?
  10. The day after 61 days Easy
    If today is Monday, what day will it be after 61 days?
  11. The same date a year later Hard
    If 15 March 2027 is a Monday, what day will 15 March 2028 be? (2028 is a leap year.)
  12. A month later Medium
    If 10 June is a Wednesday, what day is 10 July of the same year?
  13. Counting backwards Medium
    If today is Thursday, what day was it 45 days ago?
  14. Data sufficiency: Ravi's age from his older brother's Medium
    Question: What is Ravi's age? Statement I: Ravi is 8 years younger than his brother. Statement II: Ravi's brother will be 30 in five years.
  15. Data sufficiency: the area Medium
    Question: What is the area of a square? Statement I: Its perimeter is 40 cm. Statement II: Its diagonal is 10√2 cm. Which statements are sufficient?
  16. Data sufficiency: not enough Medium
    Question: What is the value of x? Statement I: x is an even number. Statement II: x is greater than 4 and less than 10. Which statements are sufficient?
  17. Statement and assumption: the institute's advertisement Medium
    Statement: 'Enrol in our institute and secure a government job.' Assumptions: I. People are interested in government jobs. II.
  18. Statement and conclusion Medium
    Statement: 'The company has decided to increase the salaries of all its employees by 15% from next month.' Conclusions: I.
  19. Cause and effect Medium
    Statements: I. The government has raised the price of petrol sharply. II. Sales of electric scooters have risen this quarter. Which best describes the relation?
  20. Course of action Easy
    Statement: 'Many students fail mathematics every year in this school.' Courses of action: I. The school should arrange extra coaching in mathematics. II.
  21. Sum and product Medium
    Two numbers have a sum of 15 and a product of 56. What is the difference between them?
  22. Reversing a two-digit number Hard
    A two-digit number has digits summing to 9. Reversing the digits gives a number 27 less than the original. What is the number?
  23. Three consecutive integers Easy
    The sum of three consecutive integers is 72. What is the largest of them?
  24. Two subjects Medium
    In a class of 60 students, 35 study physics, 30 study chemistry and 15 study both. How many study neither?
  25. Only one of the two Medium
    Of 100 people, 60 read newspaper A, 45 read newspaper B and 20 read both. How many read exactly one newspaper?
  26. Two sports, one overlap Easy
    In a group, 30 play cricket, 25 play football and 20 play both. How many play at least one of the two?
  27. Which box holds the prize Hard
    Three boxes are labelled A, B and C, and exactly one holds a prize.
  28. Who is telling the truth Hard
    A says 'B is lying', and B says 'C is lying', while C says 'both A and B are lying'. Who is telling the truth?
  29. Matchsticks in a pattern Medium
    Squares are built in a row from matchsticks: 1 square needs 4 sticks, 2 squares need 7, and 3 squares need 10. How many sticks are needed for 10 squares?
  30. A painted cube cut up Hard
    A cube is painted on all faces and then cut into 27 smaller cubes of equal size. How many small cubes have exactly two faces painted?
  31. Cubes with one painted face Hard
    A painted cube is cut into 64 equal smaller cubes. How many have exactly one face painted?
  32. Matches in a tournament Medium
    In a tournament where every team plays every other team once, 45 matches are played. How many teams took part?
  33. Reasoning about an average Hard
    A student's average across 4 tests is 80. What is the lowest possible score on any one test, if each test is marked out of 100?
  34. Reading a chain of inequalities Medium
    If A > B, B > C, C = D and D > E, which of the following is definitely true?
  35. Which inequality must hold Medium
    If P ≥ Q, Q > R and S < R, which of the following is definitely true?

Placement tests that ask mathematical reasoning