MATHCOUNTS Chapter Lab

Lesson 4: Algebra: Equations, Expressions & Systems

Solve • Strategize • Check • Explain

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

MATHCOUNTS Chapter Lab — Lesson 4

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 4

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 4

Sprint Practice 1-2

Choose the best answer. Answers are not shown in this section.

1. If 3x + 9 = 36, what is x?��A. 10�B. 8�C. 11�D. 9

2. If x + y = 13 and x - y = 1, what is xy?��A. 1�B. 42�C. 13�D. 44

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 3-4

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3. If x^2 - 11x + 28 = 0 has positive integer roots, what is the sum of the roots?��A. 28�B. 11�C. 12�D. 3

4. The average of 11, 18, 21 and x is 21. What is x?��A. 21�B. 30�C. 34�D. 38

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 5-6

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5. The sequence begins 7, 11, 15, ... . What is the 14th term?��A. 63�B. 55�C. 18�D. 59

6. A ratio is 2:4. If the total is 30, what is the smaller/larger first part?��A. 10�B. 12�C. 20�D. 6

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 7-8

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7. What is 20% of 90?��A. 13�B. 18�C. 23�D. 450

8. What is 5^5 × 5^5?��A. 298023223876953125�B. 9765625�C. 6250�D. 1953125

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 9-10

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9. What is the sum of the digits of 7576?��A. 25�B. 26�C. 24�D. 50

10. What is the remainder when 11 × 11 is divided by 6?��A. 2�B. 3�C. 1�D. 5

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 11-12

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11. If 3x + 5 = 20, what is x?��A. 4�B. 7�C. 5�D. 6

12. If x + y = 10 and x - y = 0, what is xy?��A. 27�B. 25�C. 0�D. 10

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 13-14

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13. If x^2 - 11x + 28 = 0 has positive integer roots, what is the sum of the roots?��A. 12�B. 28�C. 11�D. 3

14. The average of 14, 18, 25 and x is 21. What is x?��A. 21�B. 23�C. 27�D. 31

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 15-16

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15. The sequence begins 7, 9, 11, ... . What is the 14th term?��A. 33�B. 35�C. 16�D. 31

16. A ratio is 2:6. If the total is 40, what is the smaller/larger first part?��A. 30�B. 4�C. 12�D. 10

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 17-18

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17. What is 20% of 100?��A. 15�B. 25�C. 500�D. 20

18. What is 3^3 × 3^3?��A. 243�B. 729�C. 54�D. 19683

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 19-20

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19. What is the sum of the digits of 8946?��A. 26�B. 27�C. 54�D. 28

20. What is the remainder when 12 × 17 is divided by 8?��A. 6�B. 4�C. 5�D. 7

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 21-22

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21. If 3x + 1 = 25, what is x?��A. 9�B. 10�C. 7�D. 8

22. If x + y = 7 and x - y = -1, what is xy?��A. 12�B. 14�C. -1�D. 7

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 23-24

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23. If x^2 - 11x + 28 = 0 has positive integer roots, what is the sum of the roots?��A. 3�B. 28�C. 11�D. 12

24. The average of 10, 18, 23 and x is 21. What is x?��A. 37�B. 33�C. 29�D. 21

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 25-26

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25. The sequence begins 7, 13, 19, ... . What is the 14th term?��A. 85�B. 79�C. 20�D. 91

26. A ratio is 2:4. If the total is 30, what is the smaller/larger first part?��A. 12�B. 20�C. 10�D. 6

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 27-28

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27. What is 20% of 110?��A. 17�B. 550�C. 27�D. 22

28. What is 5^4 × 5^5?��A. 3750�B. 95367431640625�C. 390625�D. 1953125

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 29-30

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29. What is the sum of the digits of 10316?��A. 22�B. 11�C. 12�D. 10

30. What is the remainder when 13 × 23 is divided by 10?��A. 9�B. 0�C. 1�D. 3

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 31-32

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31. If 3x + 9 = 21, what is x?��A. 6�B. 5�C. 3�D. 4

32. If x + y = 10 and x - y = 4, what is xy?��A. 10�B. 4�C. 21�D. 23

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 33-34

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33. If x^2 - 11x + 28 = 0 has positive integer roots, what is the sum of the roots?��A. 12�B. 3�C. 28�D. 11

34. The average of 13, 18, 21 and x is 21. What is x?��A. 32�B. 21�C. 36�D. 28

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 35-36

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35. The sequence begins 7, 11, 15, ... . What is the 14th term?��A. 63�B. 55�C. 59�D. 18

36. A ratio is 2:6. If the total is 40, what is the smaller/larger first part?��A. 4�B. 12�C. 10�D. 30

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 37-38

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37. What is 20% of 120?��A. 19�B. 29�C. 24�D. 600

38. What is 3^5 × 3^3?��A. 270�B. 14348907�C. 6561�D. 2187

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 39-40

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39. What is the sum of the digits of 11686?��A. 21�B. 23�C. 22�D. 44

40. What is the remainder when 14 × 15 is divided by 12?��A. 6�B. 7�C. 2�D. 8

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 41-42

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41. If 3x + 5 = 26, what is x?��A. 7�B. 6�C. 8�D. 9

42. If x + y = 7 and x - y = 3, what is xy?��A. 3�B. 10�C. 12�D. 7

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 43-44

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43. If x^2 - 11x + 28 = 0 has positive integer roots, what is the sum of the roots?��A. 12�B. 28�C. 3�D. 11

44. The average of 16, 18, 25 and x is 21. What is x?��A. 21�B. 29�C. 19�D. 25

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 45-46

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45. The sequence begins 7, 9, 11, ... . What is the 14th term?��A. 35�B. 33�C. 16�D. 31

46. A ratio is 2:4. If the total is 30, what is the smaller/larger first part?��A. 12�B. 6�C. 20�D. 10

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 47-48

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47. What is 20% of 130?��A. 26�B. 21�C. 650�D. 31

48. What is 5^3 × 5^5?��A. 3250�B. 390625�C. 78125�D. 30517578125

MATHCOUNTS Chapter Lab — Lesson 4

Sprint Practice 49-50

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49. What is the sum of the digits of 13056?��A. 14�B. 15�C. 16�D. 30

50. What is the remainder when 15 × 21 is divided by 6?��A. 5�B. 4�C. 6�D. 3

MATHCOUNTS Chapter Lab — Lesson 4

Answer Key

Multiple-choice questions 1-50

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

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

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

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

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

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 1-2

Read after completing the practice section.

1. Answer: D. 9�Subtract 9: 3x = 27. Divide by 3: x = 9.

2. Answer: B. 42�Add the equations: 2x = 14, so x = 7. Then y = 13-7 = 6. Thus xy = 42.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 3-4

Read after completing the practice section.

3. Answer: B. 11�For x^2 - Sx + P = 0, the sum of the roots is S. Here S = 11.

4. Answer: C. 34�Total for four numbers is 4×21 = 84. The known sum is 50, so x = 84-50 = 34.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 5-6

Read after completing the practice section.

5. Answer: D. 59�This is arithmetic with first term 7 and common difference 4. Term 14 is 7+(14-1)4 = 59.

6. Answer: A. 10�The total ratio parts are 6. One part is 30/6 = 5. The first part is 2 parts, or 10.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 7-8

Read after completing the practice section.

7. Answer: B. 18�20% = 20/100. So 20% of 90 is 20/100 × 90 = 18.

8. Answer: B. 9765625�When multiplying powers with the same base, add exponents: 5^5×5^5=5^10=9765625.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 9-10

Read after completing the practice section.

9. Answer: A. 25�Add the digits of 7576: 7 + 5 + 7 + 6 = 25.

10. Answer: C. 1�Reduce modulo 6: 11 ≡ 5 and 11 ≡ 5. The product remainder is 25 mod 6 = 1.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 11-12

Read after completing the practice section.

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

12. Answer: B. 25�Add the equations: 2x = 10, so x = 5. Then y = 10-5 = 5. Thus xy = 25.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 13-14

Read after completing the practice section.

13. Answer: C. 11�For x^2 - Sx + P = 0, the sum of the roots is S. Here S = 11.

14. Answer: C. 27�Total for four numbers is 4×21 = 84. The known sum is 57, so x = 84-57 = 27.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 15-16

Read after completing the practice section.

15. Answer: A. 33�This is arithmetic with first term 7 and common difference 2. Term 14 is 7+(14-1)2 = 33.

16. Answer: D. 10�The total ratio parts are 8. One part is 40/8 = 5. The first part is 2 parts, or 10.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 17-18

Read after completing the practice section.

17. Answer: D. 20�20% = 20/100. So 20% of 100 is 20/100 × 100 = 20.

18. Answer: B. 729�When multiplying powers with the same base, add exponents: 3^3×3^3=3^6=729.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 19-20

Read after completing the practice section.

19. Answer: B. 27�Add the digits of 8946: 8 + 9 + 4 + 6 = 27.

20. Answer: B. 4�Reduce modulo 8: 12 ≡ 4 and 17 ≡ 1. The product remainder is 4 mod 8 = 4.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 21-22

Read after completing the practice section.

21. Answer: D. 8�Subtract 1: 3x = 24. Divide by 3: x = 8.

22. Answer: A. 12�Add the equations: 2x = 6, so x = 3. Then y = 7-3 = 4. Thus xy = 12.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 23-24

Read after completing the practice section.

23. Answer: C. 11�For x^2 - Sx + P = 0, the sum of the roots is S. Here S = 11.

24. Answer: B. 33�Total for four numbers is 4×21 = 84. The known sum is 51, so x = 84-51 = 33.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 25-26

Read after completing the practice section.

25. Answer: A. 85�This is arithmetic with first term 7 and common difference 6. Term 14 is 7+(14-1)6 = 85.

26. Answer: C. 10�The total ratio parts are 6. One part is 30/6 = 5. The first part is 2 parts, or 10.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 27-28

Read after completing the practice section.

27. Answer: D. 22�20% = 20/100. So 20% of 110 is 20/100 × 110 = 22.

28. Answer: D. 1953125�When multiplying powers with the same base, add exponents: 5^4×5^5=5^9=1953125.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 29-30

Read after completing the practice section.

29. Answer: B. 11�Add the digits of 10316: 1 + 0 + 3 + 1 + 6 = 11.

30. Answer: A. 9�Reduce modulo 10: 13 ≡ 3 and 23 ≡ 3. The product remainder is 9 mod 10 = 9.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 31-32

Read after completing the practice section.

31. Answer: D. 4�Subtract 9: 3x = 12. Divide by 3: x = 4.

32. Answer: C. 21�Add the equations: 2x = 14, so x = 7. Then y = 10-7 = 3. Thus xy = 21.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 33-34

Read after completing the practice section.

33. Answer: D. 11�For x^2 - Sx + P = 0, the sum of the roots is S. Here S = 11.

34. Answer: A. 32�Total for four numbers is 4×21 = 84. The known sum is 52, so x = 84-52 = 32.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 35-36

Read after completing the practice section.

35. Answer: C. 59�This is arithmetic with first term 7 and common difference 4. Term 14 is 7+(14-1)4 = 59.

36. Answer: C. 10�The total ratio parts are 8. One part is 40/8 = 5. The first part is 2 parts, or 10.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 37-38

Read after completing the practice section.

37. Answer: C. 24�20% = 20/100. So 20% of 120 is 20/100 × 120 = 24.

38. Answer: C. 6561�When multiplying powers with the same base, add exponents: 3^5×3^3=3^8=6561.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 39-40

Read after completing the practice section.

39. Answer: C. 22�Add the digits of 11686: 1 + 1 + 6 + 8 + 6 = 22.

40. Answer: A. 6�Reduce modulo 12: 14 ≡ 2 and 15 ≡ 3. The product remainder is 6 mod 12 = 6.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 41-42

Read after completing the practice section.

41. Answer: A. 7�Subtract 5: 3x = 21. Divide by 3: x = 7.

42. Answer: B. 10�Add the equations: 2x = 10, so x = 5. Then y = 7-5 = 2. Thus xy = 10.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 43-44

Read after completing the practice section.

43. Answer: D. 11�For x^2 - Sx + P = 0, the sum of the roots is S. Here S = 11.

44. Answer: D. 25�Total for four numbers is 4×21 = 84. The known sum is 59, so x = 84-59 = 25.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 45-46

Read after completing the practice section.

45. Answer: B. 33�This is arithmetic with first term 7 and common difference 2. Term 14 is 7+(14-1)2 = 33.

46. Answer: D. 10�The total ratio parts are 6. One part is 30/6 = 5. The first part is 2 parts, or 10.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 47-48

Read after completing the practice section.

47. Answer: A. 26�20% = 20/100. So 20% of 130 is 20/100 × 130 = 26.

48. Answer: B. 390625�When multiplying powers with the same base, add exponents: 5^3×5^5=5^8=390625.

MATHCOUNTS Chapter Lab — Lesson 4

Solutions 49-50

Read after completing the practice section.

49. Answer: B. 15�Add the digits of 13056: 1 + 3 + 0 + 5 + 6 = 15.

50. Answer: D. 3�Reduce modulo 6: 15 ≡ 3 and 21 ≡ 3. The product remainder is 9 mod 6 = 3.

MATHCOUNTS Chapter Lab — Lesson 4