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Jack, Tom, Beth, and Ruth each have a number of marbles. Jack has 1010 marbles, the least, while Tom has 2727 marbles, the highest. Which of the following could be the average of the number of marbles present with all of them, given that no two of them has the same number of marbles?\newlineA) $13A)\ \$13\newlineB) $17B)\ \$17\newlineC) $23C)\ \$23\newlineD) $26D)\ \$26

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Q. Jack, Tom, Beth, and Ruth each have a number of marbles. Jack has 1010 marbles, the least, while Tom has 2727 marbles, the highest. Which of the following could be the average of the number of marbles present with all of them, given that no two of them has the same number of marbles?\newlineA) $13A)\ \$13\newlineB) $17B)\ \$17\newlineC) $23C)\ \$23\newlineD) $26D)\ \$26
  1. Denote Marbles: Let's denote the number of marbles Jack, Tom, Beth, and Ruth have as JJ, TT, BB, and RR respectively. We know that J=10J = 10 and T=27T = 27. Since no two of them have the same number of marbles, BB and RR must have a number of marbles that is between 1010 and 2727, but not equal to 1010 or 2727.
  2. Calculate Average: The average number of marbles AA can be calculated by adding the number of marbles each person has and dividing by 44, since there are 44 people. So, A=(J+T+B+R)/4A = (J + T + B + R) / 4.
  3. Check Option (A): We know that J+T=10+27=37J + T = 10 + 27 = 37. Now, we need to find possible values for BB and RR such that when added to 3737 and divided by 44, the result is one of the given options (A, B, C, D).
  4. Check Option (B): Let's check each option to see if it could be the average:\newlineFor option (A) 1313: A=(J+T+B+R)/4=(37+B+R)/4=13A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 13. Multiplying both sides by 44 gives us 37+B+R=5237 + B + R = 52. This means B+R=52−37=15B + R = 52 - 37 = 15. Since BB and RR must be different and between 1010 and 2727, they cannot add up to 1515. This option is not possible.
  5. Check Option (C): For option (B) 1717: A=(J+T+B+R)/4=(37+B+R)/4=17A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 17. Multiplying both sides by 44 gives us 37+B+R=6837 + B + R = 68. This means B+R=68−37=31B + R = 68 - 37 = 31. Since BB and RR must be different and between 1010 and 2727, they could potentially add up to 3131 with A=(J+T+B+R)/4=(37+B+R)/4=17A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 1700 and A=(J+T+B+R)/4=(37+B+R)/4=17A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 1711, for example. This option is possible.
  6. Check Option (D): For option (C) 2323: A=(J+T+B+R)/4=(37+B+R)/4=23A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 23. Multiplying both sides by 44 gives us 37+B+R=9237 + B + R = 92. This means B+R=92−37=55B + R = 92 - 37 = 55. Since the maximum number of marbles RR can have is 2626 (one less than Tom), and the minimum for BB is 1111 (one more than Jack), the maximum sum of BB and RR is A=(J+T+B+R)/4=(37+B+R)/4=23A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 2311, which is less than A=(J+T+B+R)/4=(37+B+R)/4=23A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 2322. This option is not possible.
  7. Check Option (D): For option (C) 2323: A=(J+T+B+R)/4=(37+B+R)/4=23A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 23. Multiplying both sides by 44 gives us 37+B+R=9237 + B + R = 92. This means B+R=92−37=55B + R = 92 - 37 = 55. Since the maximum number of marbles RR can have is 2626 (one less than Tom), and the minimum for BB is 1111 (one more than Jack), the maximum sum of BB and RR is A=(J+T+B+R)/4=(37+B+R)/4=23A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 2311, which is less than A=(J+T+B+R)/4=(37+B+R)/4=23A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 2322. This option is not possible.For option (D) 2626: A=(J+T+B+R)/4=(37+B+R)/4=23A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 2344. Multiplying both sides by 44 gives us A=(J+T+B+R)/4=(37+B+R)/4=23A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 2366. This means A=(J+T+B+R)/4=(37+B+R)/4=23A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 2377. Since the maximum number of marbles RR can have is 2626 and the minimum for BB is 1111, the maximum sum of BB and RR is A=(J+T+B+R)/4=(37+B+R)/4=23A = (J + T + B + R) / 4 = (37 + B + R) / 4 = 2311, which is less than 4455. This option is not possible.

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