The average of fifteen numbers is 49. The average of the first four numbers is…

2020

The average of fifteen numbers is 49. The average of the first four numbers is 51, and that of the next eight numbers is 47. The 13th number is two times the 14th number, and the 14th number is 3 less than the 15th number. What is the average of the 13th and 15th numbers?

  1. A.

    53.5

  2. B.

    49.5

  3. C.

    55.5

  4. D.

    58.5

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Show answer & explanation

Correct answer: D

CONCEPT: Average means total divided by the number of terms, so total = average × number of terms.

When a total is split into known and unknown groups, subtract the known group totals first, then translate the relation among the unknown terms into an equation.

APPLICATION:

  1. Total of all fifteen numbers = 49 × 15 = 735.

  2. Total of the first four numbers = 51 × 4 = 204, and total of the next eight numbers = 47 × 8 = 376.

  3. So the total of the 13th, 14th, and 15th numbers = 735 - 204 - 376 = 155.

  4. Let the 14th number be x. Then the 13th number is 2x, and the 15th number is x + 3.

  5. Their sum is 2x + x + (x + 3) = 155, so 4x + 3 = 155.

  6. Therefore 4x = 152 and x = 38. The 13th number is 76, and the 15th number is 41.

  7. Average of the 13th and 15th numbers = (76 + 41) / 2 = 117 / 2 = 58.5.

CROSS-CHECK: The final three numbers are 76, 38, and 41; their sum is 155, which matches the total left after the first twelve numbers.

Result: 58.5

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