MATHCOUNTS Chapter Lab
Lesson 11: Rates: Speed, Work & Units
Solve • Strategize • Check • Explain
Chapter-level practice with 50 multiple-choice problems and complete solutions.
MATHCOUNTS Chapter Lab — Lesson 11
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 11
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 11
Sprint Practice 1-2
Choose the best answer. Answers are not shown in this section.
1. A car travels at 40 miles per hour for 2 hours. How many miles does it travel?��A. 40�B. 80�C. 42�D. 120
2. One pipe fills a tank in 4 hours and another in 8 hours. How many hours do they take together?��A. 12�B. 4�C. 8�D. 8/3
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 3-4
Choose the best answer. Answers are not shown in this section.
3. A ratio is 2:5. If the total is 35, what is the smaller/larger first part?��A. 10�B. 12�C. 25�D. 5
4. What is 20% of 100?��A. 20�B. 25�C. 500�D. 15
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 5-6
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5. If 5x + 1 = 46, what is x?��A. 9�B. 11�C. 10�D. 8
6. If x + y = 9 and x - y = -3, what is xy?��A. 20�B. -3�C. 9�D. 18
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 7-8
Choose the best answer. Answers are not shown in this section.
7. The average of 10, 20, 22 and x is 22. What is x?��A. 22�B. 32�C. 40�D. 36
8. What is the least positive integer divisible by 8, 21 and 18?��A. 512�B. 504�C. 1008�D. 252
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 9-10
Choose the best answer. Answers are not shown in this section.
9. What is the greatest common divisor of 84, 96 and 150?��A. 6�B. 3�C. 18�D. 12
10. The sequence begins 5, 12, 19, ... . What is the 17th term?��A. 24�B. 110�C. 117�D. 124
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 11-12
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11. A car travels at 50 miles per hour for 4 hours. How many miles does it travel?��A. 150�B. 54�C. 250�D. 200
12. One pipe fills a tank in 6 hours and another in 12 hours. How many hours do they take together?��A. 18�B. 4/1�C. 12�D. 6
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 13-14
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13. A ratio is 2:3. If the total is 25, what is the smaller/larger first part?��A. 12�B. 7�C. 15�D. 10
14. What is 20% of 110?��A. 550�B. 17�C. 22�D. 27
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 15-16
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15. If 5x + 9 = 34, what is x?��A. 7�B. 4�C. 6�D. 5
16. If x + y = 12 and x - y = 2, what is xy?��A. 12�B. 35�C. 2�D. 37
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 17-18
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17. The average of 13, 20, 20 and x is 22. What is x?��A. 22�B. 39�C. 35�D. 31
18. What is the least positive integer divisible by 8, 15 and 20?��A. 240�B. 120�C. 128�D. 60
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 19-20
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19. What is the greatest common divisor of 84, 144 and 180?��A. 6�B. 18�C. 12�D. 24
20. The sequence begins 5, 10, 15, ... . What is the 17th term?��A. 80�B. 22�C. 85�D. 90
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 21-22
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21. A car travels at 60 miles per hour for 2 hours. How many miles does it travel?��A. 62�B. 60�C. 120�D. 180
22. One pipe fills a tank in 4 hours and another in 8 hours. How many hours do they take together?��A. 12�B. 8/3�C. 4�D. 8
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 23-24
Choose the best answer. Answers are not shown in this section.
23. A ratio is 2:5. If the total is 35, what is the smaller/larger first part?��A. 5�B. 10�C. 25�D. 12
24. What is 20% of 120?��A. 24�B. 19�C. 600�D. 29
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 25-26
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25. If 5x + 5 = 45, what is x?��A. 10�B. 8�C. 9�D. 7
26. If x + y = 9 and x - y = 1, what is xy?��A. 9�B. 1�C. 22�D. 20
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 27-28
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27. The average of 16, 20, 24 and x is 22. What is x?��A. 22�B. 24�C. 32�D. 28
28. What is the least positive integer divisible by 8, 21 and 22?��A. 1848�B. 3696�C. 1856�D. 924
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 29-30
Choose the best answer. Answers are not shown in this section.
29. What is the greatest common divisor of 84, 96 and 120?��A. 18�B. 12�C. 24�D. 6
30. The sequence begins 5, 8, 11, ... . What is the 17th term?��A. 50�B. 56�C. 53�D. 20
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 31-32
Choose the best answer. Answers are not shown in this section.
31. A car travels at 70 miles per hour for 4 hours. How many miles does it travel?��A. 350�B. 74�C. 280�D. 210
32. One pipe fills a tank in 6 hours and another in 12 hours. How many hours do they take together?��A. 18�B. 4/1�C. 12�D. 6
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 33-34
Choose the best answer. Answers are not shown in this section.
33. A ratio is 2:3. If the total is 25, what is the smaller/larger first part?��A. 10�B. 7�C. 15�D. 12
34. What is 20% of 130?��A. 26�B. 650�C. 21�D. 31
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 35-36
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35. If 5x + 1 = 21, what is x?��A. 6�B. 3�C. 5�D. 4
36. If x + y = 6 and x - y = 0, what is xy?��A. 11�B. 6�C. 9�D. 0
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 37-38
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37. The average of 12, 20, 22 and x is 22. What is x?��A. 30�B. 34�C. 38�D. 22
38. What is the least positive integer divisible by 8, 15 and 18?��A. 360�B. 368�C. 720�D. 180
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 39-40
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39. What is the greatest common divisor of 84, 144 and 150?��A. 12�B. 3�C. 18�D. 6
40. The sequence begins 5, 12, 19, ... . What is the 17th term?��A. 24�B. 110�C. 124�D. 117
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 41-42
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41. A car travels at 40 miles per hour for 2 hours. How many miles does it travel?��A. 40�B. 120�C. 80�D. 42
42. One pipe fills a tank in 4 hours and another in 8 hours. How many hours do they take together?��A. 8/3�B. 8�C. 4�D. 12
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 43-44
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43. A ratio is 2:5. If the total is 35, what is the smaller/larger first part?��A. 5�B. 10�C. 12�D. 25
44. What is 20% of 140?��A. 33�B. 28�C. 700�D. 23
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 45-46
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45. If 5x + 9 = 44, what is x?��A. 7�B. 6�C. 9�D. 8
46. If x + y = 9 and x - y = 5, what is xy?��A. 5�B. 16�C. 14�D. 9
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 47-48
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47. The average of 15, 20, 20 and x is 22. What is x?��A. 29�B. 37�C. 22�D. 33
48. What is the least positive integer divisible by 8, 21 and 20?��A. 1680�B. 840�C. 848�D. 420
MATHCOUNTS Chapter Lab — Lesson 11
Sprint Practice 49-50
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49. What is the greatest common divisor of 84, 96 and 180?��A. 12�B. 18�C. 6�D. 24
50. The sequence begins 5, 10, 15, ... . What is the 17th term?��A. 85�B. 90�C. 80�D. 22
MATHCOUNTS Chapter Lab — Lesson 11
Answer Key
Multiple-choice questions 1-50
1. B�2. D�3. A�4. A�5. A�6. D�7. D�8. B�9. A�10. C
11. D�12. B�13. D�14. C�15. D�16. B�17. C�18. B�19. C�20. C
21. C�22. B�23. B�24. A�25. B�26. D�27. D�28. A�29. B�30. C
31. C�32. B�33. A�34. A�35. D�36. C�37. B�38. A�39. D�40. D
41. C�42. A�43. B�44. B�45. A�46. C�47. D�48. B�49. A�50. A
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 1-2
Read after completing the practice section.
1. Answer: B. 80�Distance = rate × time = 40 × 2 = 80 miles.
2. Answer: D. 8/3�Together rate = 1/4 + 1/8 = 12/32. Time = reciprocal = 32/12 hours, simplified as 8/3.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 3-4
Read after completing the practice section.
3. Answer: A. 10�The total ratio parts are 7. One part is 35/7 = 5. The first part is 2 parts, or 10.
4. Answer: A. 20�20% = 20/100. So 20% of 100 is 20/100 × 100 = 20.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 5-6
Read after completing the practice section.
5. Answer: A. 9�Subtract 1: 5x = 45. Divide by 5: x = 9.
6. Answer: D. 18�Add the equations: 2x = 6, so x = 3. Then y = 9-3 = 6. Thus xy = 18.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 7-8
Read after completing the practice section.
7. Answer: D. 36�Total for four numbers is 4×22 = 88. The known sum is 52, so x = 88-52 = 36.
8. Answer: B. 504�The least positive integer divisible by all three numbers is their LCM. Using prime factors gives LCM = 504.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 9-10
Read after completing the practice section.
9. Answer: A. 6�Find the greatest number dividing all three. Repeated gcd gives gcd(84,96,150) = 6.
10. Answer: C. 117�This is arithmetic with first term 5 and common difference 7. Term 17 is 5+(17-1)7 = 117.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 11-12
Read after completing the practice section.
11. Answer: D. 200�Distance = rate × time = 50 × 4 = 200 miles.
12. Answer: B. 4/1�Together rate = 1/6 + 1/12 = 18/72. Time = reciprocal = 72/18 hours, simplified as 4/1.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 13-14
Read after completing the practice section.
13. Answer: D. 10�The total ratio parts are 5. One part is 25/5 = 5. The first part is 2 parts, or 10.
14. Answer: C. 22�20% = 20/100. So 20% of 110 is 20/100 × 110 = 22.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 15-16
Read after completing the practice section.
15. Answer: D. 5�Subtract 9: 5x = 25. Divide by 5: x = 5.
16. Answer: B. 35�Add the equations: 2x = 14, so x = 7. Then y = 12-7 = 5. Thus xy = 35.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 17-18
Read after completing the practice section.
17. Answer: C. 35�Total for four numbers is 4×22 = 88. The known sum is 53, so x = 88-53 = 35.
18. Answer: B. 120�The least positive integer divisible by all three numbers is their LCM. Using prime factors gives LCM = 120.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 19-20
Read after completing the practice section.
19. Answer: C. 12�Find the greatest number dividing all three. Repeated gcd gives gcd(84,144,180) = 12.
20. Answer: C. 85�This is arithmetic with first term 5 and common difference 5. Term 17 is 5+(17-1)5 = 85.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 21-22
Read after completing the practice section.
21. Answer: C. 120�Distance = rate × time = 60 × 2 = 120 miles.
22. Answer: B. 8/3�Together rate = 1/4 + 1/8 = 12/32. Time = reciprocal = 32/12 hours, simplified as 8/3.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 23-24
Read after completing the practice section.
23. Answer: B. 10�The total ratio parts are 7. One part is 35/7 = 5. The first part is 2 parts, or 10.
24. Answer: A. 24�20% = 20/100. So 20% of 120 is 20/100 × 120 = 24.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 25-26
Read after completing the practice section.
25. Answer: B. 8�Subtract 5: 5x = 40. Divide by 5: x = 8.
26. Answer: D. 20�Add the equations: 2x = 10, so x = 5. Then y = 9-5 = 4. Thus xy = 20.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 27-28
Read after completing the practice section.
27. Answer: D. 28�Total for four numbers is 4×22 = 88. The known sum is 60, so x = 88-60 = 28.
28. Answer: A. 1848�The least positive integer divisible by all three numbers is their LCM. Using prime factors gives LCM = 1848.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 29-30
Read after completing the practice section.
29. Answer: B. 12�Find the greatest number dividing all three. Repeated gcd gives gcd(84,96,120) = 12.
30. Answer: C. 53�This is arithmetic with first term 5 and common difference 3. Term 17 is 5+(17-1)3 = 53.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 31-32
Read after completing the practice section.
31. Answer: C. 280�Distance = rate × time = 70 × 4 = 280 miles.
32. Answer: B. 4/1�Together rate = 1/6 + 1/12 = 18/72. Time = reciprocal = 72/18 hours, simplified as 4/1.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 33-34
Read after completing the practice section.
33. Answer: A. 10�The total ratio parts are 5. One part is 25/5 = 5. The first part is 2 parts, or 10.
34. Answer: A. 26�20% = 20/100. So 20% of 130 is 20/100 × 130 = 26.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 35-36
Read after completing the practice section.
35. Answer: D. 4�Subtract 1: 5x = 20. Divide by 5: x = 4.
36. Answer: C. 9�Add the equations: 2x = 6, so x = 3. Then y = 6-3 = 3. Thus xy = 9.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 37-38
Read after completing the practice section.
37. Answer: B. 34�Total for four numbers is 4×22 = 88. The known sum is 54, so x = 88-54 = 34.
38. Answer: A. 360�The least positive integer divisible by all three numbers is their LCM. Using prime factors gives LCM = 360.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 39-40
Read after completing the practice section.
39. Answer: D. 6�Find the greatest number dividing all three. Repeated gcd gives gcd(84,144,150) = 6.
40. Answer: D. 117�This is arithmetic with first term 5 and common difference 7. Term 17 is 5+(17-1)7 = 117.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 41-42
Read after completing the practice section.
41. Answer: C. 80�Distance = rate × time = 40 × 2 = 80 miles.
42. Answer: A. 8/3�Together rate = 1/4 + 1/8 = 12/32. Time = reciprocal = 32/12 hours, simplified as 8/3.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 43-44
Read after completing the practice section.
43. Answer: B. 10�The total ratio parts are 7. One part is 35/7 = 5. The first part is 2 parts, or 10.
44. Answer: B. 28�20% = 20/100. So 20% of 140 is 20/100 × 140 = 28.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 45-46
Read after completing the practice section.
45. Answer: A. 7�Subtract 9: 5x = 35. Divide by 5: x = 7.
46. Answer: C. 14�Add the equations: 2x = 14, so x = 7. Then y = 9-7 = 2. Thus xy = 14.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 47-48
Read after completing the practice section.
47. Answer: D. 33�Total for four numbers is 4×22 = 88. The known sum is 55, so x = 88-55 = 33.
48. Answer: B. 840�The least positive integer divisible by all three numbers is their LCM. Using prime factors gives LCM = 840.
MATHCOUNTS Chapter Lab — Lesson 11
Solutions 49-50
Read after completing the practice section.
49. Answer: A. 12�Find the greatest number dividing all three. Repeated gcd gives gcd(84,96,180) = 12.
50. Answer: A. 85�This is arithmetic with first term 5 and common difference 5. Term 17 is 5+(17-1)5 = 85.
MATHCOUNTS Chapter Lab — Lesson 11