MATHCOUNTS Chapter Lab

Lesson 13: Logic, Sets & Venn Diagrams

Solve • Strategize • Check • Explain

Chapter-level practice with 50 multiple-choice problems and complete solutions.

MATHCOUNTS Chapter Lab — Lesson 13

Lesson Method

Use a contest-style checklist before choosing an answer.

1. Decode

Read for hidden constraints: integer, positive, inclusive, least, greatest.

2. Choose a Tool

Factorization, ratios, equations, diagrams, counting, or modular arithmetic.

3. Estimate

Use mental checks to eliminate unreasonable answers.

4. Solve Cleanly

Write only enough work to avoid mistakes under time pressure.

5. Verify

Check units, signs, parity, and whether the answer satisfies the question.

MATHCOUNTS Chapter Lab — Lesson 13

Topic Focus

The goal is fast recognition of structure, not long computation.

• Identify the mathematical structure before calculating.

• Use multiple approaches: arithmetic, algebra, diagrams and estimation.

• Keep choices hidden from your reasoning until you have a target answer.

• When stuck, test answer choices or use a smaller case.

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 1-2

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1. In a group, 22 students play chess, 22 play tennis, and 9 play both. How many play at least one of the two?��A. 53�B. 26�C. 44�D. 35

2. A lunch has 6 sandwich choices, 7 drink choices and 4 dessert choices. How many lunches are possible?��A. 168�B. 42�C. 17�D. 172

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 3-4

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3. A bag has 7 red marbles and 7 blue marbles. What is the probability of drawing a red marble?��A. 7/14�B. 7/14�C. 7/15�D. 7/7

4. How many committees of 3 students can be chosen from 6 students?��A. 40�B. 20�C. 120�D. 23

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 5-6

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5. How many ways can first, second and third place be awarded among 7 runners?��A. 210�B. 35�C. 21�D. 105

6. A ratio is 4:6. If the total is 70, what is the smaller/larger first part?��A. 28�B. 32�C. 42�D. 22

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 7-8

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7. What is 30% of 150?��A. 40�B. 50�C. 500�D. 45

8. The average of 15, 20, 25 and x is 22. What is x?��A. 24�B. 28�C. 32�D. 22

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 9-10

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9. If 3x + 1 = 31, what is x?��A. 9�B. 11�C. 10�D. 12

10. What is the greatest common divisor of 84, 168 and 150?��A. 3�B. 6�C. 18�D. 12

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 11-12

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11. In a group, 22 students play chess, 26 play tennis, and 7 play both. How many play at least one of the two?��A. 34�B. 41�C. 55�D. 48

12. A lunch has 4 sandwich choices, 7 drink choices and 2 dessert choices. How many lunches are possible?��A. 28�B. 13�C. 58�D. 56

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 13-14

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13. A bag has 7 red marbles and 7 blue marbles. What is the probability of drawing a red marble?��A. 7/14�B. 7/14�C. 7/7�D. 7/15

14. How many committees of 3 students can be chosen from 6 students?��A. 40�B. 120�C. 20�D. 23

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 15-16

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15. How many ways can first, second and third place be awarded among 5 runners?��A. 30�B. 15�C. 60�D. 10

16. A ratio is 4:4. If the total is 56, what is the smaller/larger first part?��A. 29�B. 24�C. 32�D. 28

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 17-18

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17. What is 30% of 160?��A. 43�B. 533�C. 53�D. 48

18. The average of 11, 20, 23 and x is 22. What is x?��A. 30�B. 22�C. 38�D. 34

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 19-20

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19. If 3x + 9 = 27, what is x?��A. 7�B. 5�C. 8�D. 6

20. What is the greatest common divisor of 84, 120 and 180?��A. 6�B. 18�C. 12�D. 24

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 21-22

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21. In a group, 22 students play chess, 30 play tennis, and 5 play both. How many play at least one of the two?��A. 42�B. 47�C. 57�D. 52

22. A lunch has 6 sandwich choices, 7 drink choices and 3 dessert choices. How many lunches are possible?��A. 126�B. 42�C. 16�D. 129

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 23-24

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23. A bag has 7 red marbles and 7 blue marbles. What is the probability of drawing a red marble?��A. 7/14�B. 7/15�C. 7/14�D. 7/7

24. How many committees of 3 students can be chosen from 6 students?��A. 20�B. 40�C. 120�D. 23

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 25-26

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25. How many ways can first, second and third place be awarded among 7 runners?��A. 35�B. 105�C. 21�D. 210

26. A ratio is 4:6. If the total is 70, what is the smaller/larger first part?��A. 32�B. 28�C. 42�D. 22

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 27-28

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27. What is 30% of 80?��A. 24�B. 29�C. 266�D. 19

28. The average of 14, 20, 21 and x is 22. What is x?��A. 33�B. 22�C. 29�D. 37

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 29-30

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29. If 3x + 5 = 32, what is x?��A. 9�B. 10�C. 11�D. 8

30. What is the greatest common divisor of 84, 168 and 120?��A. 6�B. 24�C. 12�D. 18

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 31-32

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31. In a group, 22 students play chess, 18 play tennis, and 9 play both. How many play at least one of the two?��A. 49�B. 31�C. 40�D. 22

32. A lunch has 4 sandwich choices, 7 drink choices and 4 dessert choices. How many lunches are possible?��A. 15�B. 112�C. 116�D. 28

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 33-34

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33. A bag has 7 red marbles and 7 blue marbles. What is the probability of drawing a red marble?��A. 7/7�B. 7/15�C. 7/14�D. 7/14

34. How many committees of 3 students can be chosen from 6 students?��A. 120�B. 20�C. 23�D. 40

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 35-36

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35. How many ways can first, second and third place be awarded among 5 runners?��A. 15�B. 10�C. 30�D. 60

36. A ratio is 4:4. If the total is 56, what is the smaller/larger first part?��A. 32�B. 28�C. 24�D. 29

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 37-38

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37. What is 30% of 90?��A. 22�B. 27�C. 300�D. 32

38. The average of 10, 20, 25 and x is 22. What is x?��A. 33�B. 22�C. 37�D. 29

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 39-40

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39. If 3x + 1 = 16, what is x?��A. 7�B. 6�C. 5�D. 4

40. What is the greatest common divisor of 84, 120 and 150?��A. 12�B. 3�C. 6�D. 18

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 41-42

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41. In a group, 22 students play chess, 22 play tennis, and 7 play both. How many play at least one of the two?��A. 51�B. 37�C. 30�D. 44

42. A lunch has 6 sandwich choices, 7 drink choices and 2 dessert choices. How many lunches are possible?��A. 15�B. 42�C. 84�D. 86

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 43-44

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43. A bag has 7 red marbles and 7 blue marbles. What is the probability of drawing a red marble?��A. 7/14�B. 7/14�C. 7/15�D. 7/7

44. How many committees of 3 students can be chosen from 6 students?��A. 20�B. 40�C. 120�D. 23

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 45-46

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45. How many ways can first, second and third place be awarded among 7 runners?��A. 21�B. 35�C. 210�D. 105

46. A ratio is 4:6. If the total is 70, what is the smaller/larger first part?��A. 42�B. 22�C. 32�D. 28

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 47-48

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47. What is 30% of 100?��A. 25�B. 30�C. 35�D. 333

48. The average of 13, 20, 23 and x is 22. What is x?��A. 32�B. 36�C. 28�D. 22

MATHCOUNTS Chapter Lab — Lesson 13

Sprint Practice 49-50

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49. If 3x + 9 = 33, what is x?��A. 7�B. 8�C. 10�D. 9

50. What is the greatest common divisor of 84, 168 and 180?��A. 24�B. 12�C. 18�D. 6

MATHCOUNTS Chapter Lab — Lesson 13

Answer Key

Multiple-choice questions 1-50

1. D�2. A�3. A�4. B�5. A�6. A�7. D�8. B�9. C�10. B

11. B�12. D�13. A�14. C�15. C�16. D�17. D�18. D�19. D�20. C

21. B�22. A�23. A�24. A�25. D�26. B�27. A�28. A�29. A�30. C

31. B�32. B�33. C�34. B�35. D�36. B�37. B�38. A�39. C�40. C

41. B�42. C�43. A�44. A�45. C�46. D�47. B�48. A�49. B�50. B

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 1-2

Read after completing the practice section.

1. Answer: D. 35�Use inclusion-exclusion: A ∪ B = 22+22-9 = 35.

2. Answer: A. 168�Use the multiplication principle: 6 × 7 × 4 = 168 possible lunches.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 3-4

Read after completing the practice section.

3. Answer: A. 7/14�There are 14 total marbles and 7 favorable red marbles. Probability = 7/14.

4. Answer: B. 20�Order does not matter, so use combinations: C(6,3) = 20.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 5-6

Read after completing the practice section.

5. Answer: A. 210�Order matters. The number is P(7,3) = 7×6×5 = 210.

6. Answer: A. 28�The total ratio parts are 10. One part is 70/10 = 7. The first part is 4 parts, or 28.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 7-8

Read after completing the practice section.

7. Answer: D. 45�30% = 30/100. So 30% of 150 is 30/100 × 150 = 45.

8. Answer: B. 28�Total for four numbers is 4×22 = 88. The known sum is 60, so x = 88-60 = 28.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 9-10

Read after completing the practice section.

9. Answer: C. 10�Subtract 1: 3x = 30. Divide by 3: x = 10.

10. Answer: B. 6�Find the greatest number dividing all three. Repeated gcd gives gcd(84,168,150) = 6.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 11-12

Read after completing the practice section.

11. Answer: B. 41�Use inclusion-exclusion: A ∪ B = 22+26-7 = 41.

12. Answer: D. 56�Use the multiplication principle: 4 × 7 × 2 = 56 possible lunches.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 13-14

Read after completing the practice section.

13. Answer: A. 7/14�There are 14 total marbles and 7 favorable red marbles. Probability = 7/14.

14. Answer: C. 20�Order does not matter, so use combinations: C(6,3) = 20.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 15-16

Read after completing the practice section.

15. Answer: C. 60�Order matters. The number is P(5,3) = 5×4×3 = 60.

16. Answer: D. 28�The total ratio parts are 8. One part is 56/8 = 7. The first part is 4 parts, or 28.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 17-18

Read after completing the practice section.

17. Answer: D. 48�30% = 30/100. So 30% of 160 is 30/100 × 160 = 48.

18. Answer: D. 34�Total for four numbers is 4×22 = 88. The known sum is 54, so x = 88-54 = 34.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 19-20

Read after completing the practice section.

19. Answer: D. 6�Subtract 9: 3x = 18. Divide by 3: x = 6.

20. Answer: C. 12�Find the greatest number dividing all three. Repeated gcd gives gcd(84,120,180) = 12.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 21-22

Read after completing the practice section.

21. Answer: B. 47�Use inclusion-exclusion: A ∪ B = 22+30-5 = 47.

22. Answer: A. 126�Use the multiplication principle: 6 × 7 × 3 = 126 possible lunches.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 23-24

Read after completing the practice section.

23. Answer: A. 7/14�There are 14 total marbles and 7 favorable red marbles. Probability = 7/14.

24. Answer: A. 20�Order does not matter, so use combinations: C(6,3) = 20.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 25-26

Read after completing the practice section.

25. Answer: D. 210�Order matters. The number is P(7,3) = 7×6×5 = 210.

26. Answer: B. 28�The total ratio parts are 10. One part is 70/10 = 7. The first part is 4 parts, or 28.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 27-28

Read after completing the practice section.

27. Answer: A. 24�30% = 30/100. So 30% of 80 is 30/100 × 80 = 24.

28. Answer: A. 33�Total for four numbers is 4×22 = 88. The known sum is 55, so x = 88-55 = 33.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 29-30

Read after completing the practice section.

29. Answer: A. 9�Subtract 5: 3x = 27. Divide by 3: x = 9.

30. Answer: C. 12�Find the greatest number dividing all three. Repeated gcd gives gcd(84,168,120) = 12.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 31-32

Read after completing the practice section.

31. Answer: B. 31�Use inclusion-exclusion: A ∪ B = 22+18-9 = 31.

32. Answer: B. 112�Use the multiplication principle: 4 × 7 × 4 = 112 possible lunches.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 33-34

Read after completing the practice section.

33. Answer: C. 7/14�There are 14 total marbles and 7 favorable red marbles. Probability = 7/14.

34. Answer: B. 20�Order does not matter, so use combinations: C(6,3) = 20.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 35-36

Read after completing the practice section.

35. Answer: D. 60�Order matters. The number is P(5,3) = 5×4×3 = 60.

36. Answer: B. 28�The total ratio parts are 8. One part is 56/8 = 7. The first part is 4 parts, or 28.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 37-38

Read after completing the practice section.

37. Answer: B. 27�30% = 30/100. So 30% of 90 is 30/100 × 90 = 27.

38. Answer: A. 33�Total for four numbers is 4×22 = 88. The known sum is 55, so x = 88-55 = 33.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 39-40

Read after completing the practice section.

39. Answer: C. 5�Subtract 1: 3x = 15. Divide by 3: x = 5.

40. Answer: C. 6�Find the greatest number dividing all three. Repeated gcd gives gcd(84,120,150) = 6.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 41-42

Read after completing the practice section.

41. Answer: B. 37�Use inclusion-exclusion: A ∪ B = 22+22-7 = 37.

42. Answer: C. 84�Use the multiplication principle: 6 × 7 × 2 = 84 possible lunches.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 43-44

Read after completing the practice section.

43. Answer: A. 7/14�There are 14 total marbles and 7 favorable red marbles. Probability = 7/14.

44. Answer: A. 20�Order does not matter, so use combinations: C(6,3) = 20.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 45-46

Read after completing the practice section.

45. Answer: C. 210�Order matters. The number is P(7,3) = 7×6×5 = 210.

46. Answer: D. 28�The total ratio parts are 10. One part is 70/10 = 7. The first part is 4 parts, or 28.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 47-48

Read after completing the practice section.

47. Answer: B. 30�30% = 30/100. So 30% of 100 is 30/100 × 100 = 30.

48. Answer: A. 32�Total for four numbers is 4×22 = 88. The known sum is 56, so x = 88-56 = 32.

MATHCOUNTS Chapter Lab — Lesson 13

Solutions 49-50

Read after completing the practice section.

49. Answer: B. 8�Subtract 9: 3x = 24. Divide by 3: x = 8.

50. Answer: B. 12�Find the greatest number dividing all three. Repeated gcd gives gcd(84,168,180) = 12.

MATHCOUNTS Chapter Lab — Lesson 13